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chapter_sorting/quick_sort/index.html

@@ -25,7 +25,7 @@
     
     
       
-        <title>11.4.   快速排序 - Hello 算法</title>
+        <title>11.5.   快速排序 - Hello 算法</title>
       
     
     
@@ -79,7 +79,7 @@
     <div data-md-component="skip">
       
         
-        <a href="#114" class="md-skip">
+        <a href="#115" class="md-skip">
           跳转至
         </a>
       
@@ -113,7 +113,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
             
-              11.4. &nbsp; 快速排序
+              11.5. &nbsp; 快速排序
             
           </span>
         </div>
@@ -1409,6 +1409,8 @@
           
         
           
+        
+          
         
           
         
@@ -1445,9 +1447,23 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../selection_sort/" class="md-nav__link">
+        11.2. &nbsp; 选择排序
+      </a>
+    </li>
+  
+
+            
+          
+            
+              
+  
+  
+  
     <li class="md-nav__item">
       <a href="../bubble_sort/" class="md-nav__link">
-        11.2. &nbsp; 冒泡排序
+        11.3. &nbsp; 冒泡排序
       </a>
     </li>
   
@@ -1461,7 +1477,7 @@
   
     <li class="md-nav__item">
       <a href="../insertion_sort/" class="md-nav__link">
-        11.3. &nbsp; 插入排序
+        11.4. &nbsp; 插入排序
       </a>
     </li>
   
@@ -1484,12 +1500,12 @@
       
       
         <label class="md-nav__link md-nav__link--active" for="__toc">
-          11.4. &nbsp; 快速排序
+          11.5. &nbsp; 快速排序
           <span class="md-nav__icon md-icon"></span>
         </label>
       
       <a href="./" class="md-nav__link md-nav__link--active">
-        11.4. &nbsp; 快速排序
+        11.5. &nbsp; 快速排序
       </a>
       
         
@@ -1508,36 +1524,36 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
       
         <li class="md-nav__item">
-  <a href="#1141" class="md-nav__link">
-    11.4.1. &nbsp; 算法流程
+  <a href="#1151" class="md-nav__link">
+    11.5.1. &nbsp; 算法流程
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1142" class="md-nav__link">
-    11.4.2. &nbsp; 算法特性
+  <a href="#1152" class="md-nav__link">
+    11.5.2. &nbsp; 算法特性
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1143" class="md-nav__link">
-    11.4.3. &nbsp; 快排为什么快？
+  <a href="#1153" class="md-nav__link">
+    11.5.3. &nbsp; 快排为什么快？
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1144" class="md-nav__link">
-    11.4.4. &nbsp; 基准数优化
+  <a href="#1154" class="md-nav__link">
+    11.5.4. &nbsp; 基准数优化
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1145" class="md-nav__link">
-    11.4.5. &nbsp; 尾递归优化
+  <a href="#1155" class="md-nav__link">
+    11.5.5. &nbsp; 尾递归优化
   </a>
   
 </li>
@@ -1558,7 +1574,7 @@
   
     <li class="md-nav__item">
       <a href="../merge_sort/" class="md-nav__link">
-        11.5. &nbsp; 归并排序
+        11.6. &nbsp; 归并排序
       </a>
     </li>
   
@@ -1572,7 +1588,7 @@
   
     <li class="md-nav__item">
       <a href="../bucket_sort/" class="md-nav__link">
-        11.6. &nbsp; 桶排序
+        11.7. &nbsp; 桶排序
       </a>
     </li>
   
@@ -1586,7 +1602,7 @@
   
     <li class="md-nav__item">
       <a href="../counting_sort/" class="md-nav__link">
-        11.7. &nbsp; 计数排序
+        11.8. &nbsp; 计数排序
       </a>
     </li>
   
@@ -1600,7 +1616,7 @@
   
     <li class="md-nav__item">
       <a href="../radix_sort/" class="md-nav__link">
-        11.8. &nbsp; 基数排序
+        11.9. &nbsp; 基数排序
       </a>
     </li>
   
@@ -1614,7 +1630,7 @@
   
     <li class="md-nav__item">
       <a href="../summary/" class="md-nav__link">
-        11.9. &nbsp; 小结
+        11.10. &nbsp; 小结
       </a>
     </li>
   
@@ -1859,36 +1875,36 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
       
         <li class="md-nav__item">
-  <a href="#1141" class="md-nav__link">
-    11.4.1. &nbsp; 算法流程
+  <a href="#1151" class="md-nav__link">
+    11.5.1. &nbsp; 算法流程
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1142" class="md-nav__link">
-    11.4.2. &nbsp; 算法特性
+  <a href="#1152" class="md-nav__link">
+    11.5.2. &nbsp; 算法特性
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1143" class="md-nav__link">
-    11.4.3. &nbsp; 快排为什么快？
+  <a href="#1153" class="md-nav__link">
+    11.5.3. &nbsp; 快排为什么快？
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1144" class="md-nav__link">
-    11.4.4. &nbsp; 基准数优化
+  <a href="#1154" class="md-nav__link">
+    11.5.4. &nbsp; 基准数优化
   </a>
   
 </li>
       
         <li class="md-nav__item">
-  <a href="#1145" class="md-nav__link">
-    11.4.5. &nbsp; 尾递归优化
+  <a href="#1155" class="md-nav__link">
+    11.5.5. &nbsp; 尾递归优化
   </a>
   
 </li>
@@ -1916,7 +1932,7 @@
   
 
 
-<h1 id="114">11.4. &nbsp; 快速排序<a class="headerlink" href="#114" title="Permanent link">&para;</a></h1>
+<h1 id="115">11.5. &nbsp; 快速排序<a class="headerlink" href="#115" title="Permanent link">&para;</a></h1>
 <p>「快速排序 Quick Sort」是一种基于分治思想的排序算法，运行高效，应用广泛。</p>
 <p>快速排序的核心操作是「哨兵划分」，其目标是：选择数组中的某个元素作为“基准数”，将所有小于基准数的元素移到其左侧，而大于基准数的元素移到其右侧。具体来说，哨兵划分的流程为：</p>
 <ol>
@@ -2212,9 +2228,9 @@
 </div>
 </div>
 </div>
-<h2 id="1141">11.4.1. &nbsp; 算法流程<a class="headerlink" href="#1141" title="Permanent link">&para;</a></h2>
+<h2 id="1151">11.5.1. &nbsp; 算法流程<a class="headerlink" href="#1151" title="Permanent link">&para;</a></h2>
 <ol>
-<li>首先，对原数组执行一次「哨兵划分」，得到待排序的左子数组和右子数组；</li>
+<li>首先，对原数组执行一次「哨兵划分」，得到未排序的左子数组和右子数组；</li>
 <li>然后，对左子数组和右子数组分别递归执行「哨兵划分」；</li>
 <li>持续递归，直至子数组长度为 1 时终止，从而完成整个数组的排序；</li>
 </ol>
@@ -2389,19 +2405,20 @@
 </div>
 </div>
 </div>
-<h2 id="1142">11.4.2. &nbsp; 算法特性<a class="headerlink" href="#1142" title="Permanent link">&para;</a></h2>
-<p><strong>时间复杂度 <span class="arithmatex">\(O(n \log n)\)</span></strong> ：在平均情况下，哨兵划分的递归层数为 <span class="arithmatex">\(\log n\)</span> ，每层中的总循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n \log n)\)</span> 时间。</p>
-<p>在最差情况下，每轮哨兵划分操作都将长度为 <span class="arithmatex">\(n\)</span> 的数组划分为长度为 <span class="arithmatex">\(0\)</span> 和 <span class="arithmatex">\(n - 1\)</span> 的两个子数组，此时递归层数达到 <span class="arithmatex">\(n\)</span> 层，每层中的循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n^2)\)</span> 时间；因此快速排序是“自适应排序”。</p>
-<p><strong>空间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ：在输入数组完全倒序的情况下，达到最差递归深度 <span class="arithmatex">\(n\)</span> 。由于未使用辅助数组，因此算法是“原地排序”。</p>
-<p>在哨兵划分的最后一步，基准数可能会被交换至相等元素的右侧，因此是“非稳定排序”。</p>
-<h2 id="1143">11.4.3. &nbsp; 快排为什么快？<a class="headerlink" href="#1143" title="Permanent link">&para;</a></h2>
+<h2 id="1152">11.5.2. &nbsp; 算法特性<a class="headerlink" href="#1152" title="Permanent link">&para;</a></h2>
+<ul>
+<li><strong>时间复杂度 <span class="arithmatex">\(O(n \log n)\)</span> 、自适应排序</strong> ：在平均情况下，哨兵划分的递归层数为 <span class="arithmatex">\(\log n\)</span> ，每层中的总循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n \log n)\)</span> 时间。在最差情况下，每轮哨兵划分操作都将长度为 <span class="arithmatex">\(n\)</span> 的数组划分为长度为 <span class="arithmatex">\(0\)</span> 和 <span class="arithmatex">\(n - 1\)</span> 的两个子数组，此时递归层数达到 <span class="arithmatex">\(n\)</span> 层，每层中的循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n^2)\)</span> 时间。</li>
+<li><strong>空间复杂度 <span class="arithmatex">\(O(n)\)</span> 、原地排序</strong> ：在输入数组完全倒序的情况下，达到最差递归深度 <span class="arithmatex">\(n\)</span> ，使用 <span class="arithmatex">\(O(n)\)</span> 栈帧空间。排序操作是在原数组上进行的，未借助额外数组。</li>
+<li><strong>非稳定排序</strong>：在哨兵划分的最后一步，基准数可能会被交换至相等元素的右侧。</li>
+</ul>
+<h2 id="1153">11.5.3. &nbsp; 快排为什么快？<a class="headerlink" href="#1153" title="Permanent link">&para;</a></h2>
 <p>从名称上就能看出，快速排序在效率方面应该具有一定的优势。尽管快速排序的平均时间复杂度与「归并排序」和「堆排序」相同，但通常快速排序的效率更高，原因如下：</p>
 <ul>
 <li><strong>出现最差情况的概率很低</strong>：虽然快速排序的最差时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，没有归并排序稳定，但在绝大多数情况下，快速排序能在 <span class="arithmatex">\(O(n \log n)\)</span> 的时间复杂度下运行。</li>
 <li><strong>缓存使用效率高</strong>：在执行哨兵划分操作时，系统可将整个子数组加载到缓存，因此访问元素的效率较高。而像「堆排序」这类算法需要跳跃式访问元素，从而缺乏这一特性。</li>
 <li><strong>复杂度的常数系数低</strong>：在上述三种算法中，快速排序的比较、赋值、交换等操作的总数量最少。这与「插入排序」比「冒泡排序」更快的原因类似。</li>
 </ul>
-<h2 id="1144">11.4.4. &nbsp; 基准数优化<a class="headerlink" href="#1144" title="Permanent link">&para;</a></h2>
+<h2 id="1154">11.5.4. &nbsp; 基准数优化<a class="headerlink" href="#1154" title="Permanent link">&para;</a></h2>
 <p><strong>快速排序在某些输入下的时间效率可能降低</strong>。举一个极端例子，假设输入数组是完全倒序的，由于我们选择最左端元素作为基准数，那么在哨兵划分完成后，基准数被交换至数组最右端，导致左子数组长度为 <span class="arithmatex">\(n - 1\)</span> 、右子数组长度为 <span class="arithmatex">\(0\)</span> 。如此递归下去，每轮哨兵划分后的右子数组长度都为 <span class="arithmatex">\(0\)</span> ，分治策略失效，快速排序退化为「冒泡排序」。</p>
 <p>为了尽量避免这种情况发生，<strong>我们可以优化哨兵划分中的基准数的选取策略</strong>。例如，我们可以随机选取一个元素作为基准数。然而，如果运气不佳，每次都选到不理想的基准数，效率仍然不尽如人意。</p>
 <p>需要注意的是，编程语言通常生成的是“伪随机数”。如果我们针对伪随机数序列构建一个特定的测试样例，那么快速排序的效率仍然可能劣化。</p>
@@ -2763,7 +2780,7 @@
 </div>
 </div>
 </div>
-<h2 id="1145">11.4.5. &nbsp; 尾递归优化<a class="headerlink" href="#1145" title="Permanent link">&para;</a></h2>
+<h2 id="1155">11.5.5. &nbsp; 尾递归优化<a class="headerlink" href="#1155" title="Permanent link">&para;</a></h2>
 <p><strong>在某些输入下，快速排序可能占用空间较多</strong>。以完全倒序的输入数组为例，由于每轮哨兵划分后右子数组长度为 <span class="arithmatex">\(0\)</span> ，递归树的高度会达到 <span class="arithmatex">\(n - 1\)</span> ，此时需要占用 <span class="arithmatex">\(O(n)\)</span> 大小的栈帧空间。</p>
 <p>为了防止栈帧空间的累积，我们可以在每轮哨兵排序完成后，比较两个子数组的长度，<strong>仅对较短的子数组进行递归</strong>。由于较短子数组的长度不会超过 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此这种方法能确保递归深度不超过 <span class="arithmatex">\(\log n\)</span> ，从而将最差空间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:10"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label></div>
@@ -2778,10 +2795,10 @@
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="p">}</span>
@@ -2797,10 +2814,10 @@
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="p">}</span>
@@ -2816,10 +2833,10 @@
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a>        <span class="c1"># 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a>        <span class="k">if</span> <span class="n">pivot</span> <span class="o">-</span> <span class="n">left</span> <span class="o">&lt;</span> <span class="n">right</span> <span class="o">-</span> <span class="n">pivot</span><span class="p">:</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># 递归排序左子数组</span>
-<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a>            <span class="n">left</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a>            <span class="n">left</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>  <span class="c1"># 递归排序右子数组</span>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a>            <span class="n">right</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a>            <span class="n">right</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># 剩余未排序区间为 [left, pivot - 1]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -2832,10 +2849,10 @@
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">pivot</span><span class="o">-</span><span class="nx">left</span><span class="w"> </span><span class="p">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="o">-</span><span class="nx">pivot</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">            </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">            </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="p">}</span>
@@ -2851,10 +2868,10 @@
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="p">}</span>
@@ -2870,10 +2887,10 @@
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="p">}</span>
@@ -2890,10 +2907,10 @@
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                         </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                         </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                         </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                         </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="p">}</span>
@@ -2909,10 +2926,10 @@
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="p">}</span>
@@ -2930,10 +2947,10 @@
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a>        <span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a>        <span class="k">if</span> <span class="p">(</span><span class="n">pivot</span> <span class="o">-</span> <span class="kr">left</span><span class="p">)</span> <span class="o">&lt;</span> <span class="p">(</span><span class="kr">right</span> <span class="o">-</span> <span class="n">pivot</span><span class="p">)</span> <span class="p">{</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a>            <span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span> <span class="kr">left</span><span class="p">:</span> <span class="kr">left</span><span class="p">,</span> <span class="kr">right</span><span class="p">:</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a>            <span class="kr">left</span> <span class="p">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a>            <span class="kr">left</span> <span class="p">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a>        <span class="p">}</span> <span class="k">else</span> <span class="p">{</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a>            <span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span> <span class="kr">left</span><span class="p">:</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="kr">right</span><span class="p">:</span> <span class="kr">right</span><span class="p">)</span> <span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a>            <span class="kr">right</span> <span class="p">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a>            <span class="kr">right</span> <span class="p">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a>        <span class="p">}</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a>    <span class="p">}</span>
 <a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="p">}</span>
@@ -2951,10 +2968,10 @@
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="w">        </span><span class="c1">// 对两个子数组中较短的那个执行快排</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w">   </span><span class="c1">// 递归排序左子数组</span>
-<a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                   </span><span class="c1">// 剩余待排序区间为 [pivot + 1, right]</span>
+<a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                   </span><span class="c1">// 剩余未排序区间为 [pivot + 1, right]</span>
 <a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w">  </span><span class="c1">// 递归排序右子数组</span>
-<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                  </span><span class="c1">// 剩余待排序区间为 [left, pivot - 1]</span>
+<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                  </span><span class="c1">// 剩余未排序区间为 [left, pivot - 1]</span>
 <a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a><span class="p">}</span>
@@ -3046,7 +3063,7 @@
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../insertion_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.3. &amp;nbsp; 插入排序" rel="prev">
+          <a href="../insertion_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.4. &amp;nbsp; 插入排序" rel="prev">
             <div class="md-footer__button md-icon">
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             </div>
@@ -3055,20 +3072,20 @@
                 上一页
               </span>
               <div class="md-ellipsis">
-                11.3. &nbsp; 插入排序
+                11.4. &nbsp; 插入排序
               </div>
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           </a>
         
         
           
-          <a href="../merge_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.5. &amp;nbsp; 归并排序" rel="next">
+          <a href="../merge_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.6. &amp;nbsp; 归并排序" rel="next">
             <div class="md-footer__title">
               <span class="md-footer__direction">
                 下一页
               </span>
               <div class="md-ellipsis">
-                11.5. &nbsp; 归并排序
+                11.6. &nbsp; 归并排序
               </div>
             </div>
             <div class="md-footer__button md-icon">