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2026-08-15 05:00:59 +00:00 · 2023-12-02 06:24:11 +08:00
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 <!-- Page content -->
 <h1 id="81">8.1 &nbsp; 堆<a class="headerlink" href="#81" title="Permanent link">&para;</a></h1>
-<p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为图 8-1 所示的两种类型。</p>
+<p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为两种类型，如图 8-1 所示。</p>
 <ul>
 <li>「大顶堆 max heap」：任意节点的值 <span class="arithmatex">\(\geq\)</span> 其子节点的值。</li>
 <li>「小顶堆 min heap」：任意节点的值 <span class="arithmatex">\(\leq\)</span> 其子节点的值。</li>
@@ -3476,11 +3476,11 @@
 <ul>
 <li>最底层节点靠左填充，其他层的节点都被填满。</li>
 <li>我们将二叉树的根节点称为“堆顶”，将底层最靠右的节点称为“堆底”。</li>
-<li>对于大顶堆（小顶堆），堆顶元素（即根节点）的值分别是最大（最小）的。</li>
+<li>对于大顶堆（小顶堆），堆顶元素（根节点）的值分别是最大（最小）的。</li>
 </ul>
 <h2 id="811">8.1.1 &nbsp; 堆常用操作<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
 <p>需要指出的是，许多编程语言提供的是「优先队列 priority queue」，这是一种抽象数据结构，定义为具有优先级排序的队列。</p>
-<p>实际上，<strong>堆通常用作实现优先队列，大顶堆相当于元素按从大到小顺序出队的优先队列</strong>。从使用角度来看，我们可以将“优先队列”和“堆”看作等价的数据结构。因此，本书对两者不做特别区分，统一使用“堆“来命名。</p>
+<p>实际上，<strong>堆通常用于实现优先队列，大顶堆相当于元素按从大到小的顺序出队的优先队列</strong>。从使用角度来看，我们可以将“优先队列”和“堆”看作等价的数据结构。因此，本书对两者不做特别区分，统一称作“堆”。</p>
 <p>堆的常用操作见表 8-1 ，方法名需要根据编程语言来确定。</p>
 <p align="center"> 表 8-1 &nbsp; 堆的操作效率 </p>

@@ -3523,10 +3523,7 @@
 </table>
 </div>
 <p>在实际应用中，我们可以直接使用编程语言提供的堆类（或优先队列类）。</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>类似于排序算法中的“从小到大排列”和“从大到小排列”，我们可以通过修改 Comparator 来实现“小顶堆”与“大顶堆”之间的转换。</p>
-</div>
+<p>类似于排序算法中的“从小到大排列”和“从大到小排列”，我们可以通过设置一个 <code>flag</code> 或修改 <code>Comparator</code> 实现“小顶堆”与“大顶堆”之间的转换。代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
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@@ -3818,13 +3815,13 @@
 <h2 id="812">8.1.2 &nbsp; 堆的实现<a class="headerlink" href="#812" title="Permanent link">&para;</a></h2>
 <p>下文实现的是大顶堆。若要将其转换为小顶堆，只需将所有大小逻辑判断取逆（例如，将 <span class="arithmatex">\(\geq\)</span> 替换为 <span class="arithmatex">\(\leq\)</span> ）。感兴趣的读者可以自行实现。</p>
 <h3 id="1">1. &nbsp; 堆的存储与表示<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
-<p>我们在二叉树章节中学习到，完全二叉树非常适合用数组来表示。由于堆正是一种完全二叉树，<strong>我们将采用数组来存储堆</strong>。</p>
+<p>“二叉树”章节讲过，完全二叉树非常适合用数组来表示。由于堆正是一种完全二叉树，<strong>因此我们将采用数组来存储堆</strong>。</p>
 <p>当使用数组表示二叉树时，元素代表节点值，索引代表节点在二叉树中的位置。<strong>节点指针通过索引映射公式来实现</strong>。</p>
-<p>如图 8-2 所示，给定索引 <span class="arithmatex">\(i\)</span> ，其左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span> ，父节点索引为 <span class="arithmatex">\((i - 1) / 2\)</span>（向下取整）。当索引越界时，表示空节点或节点不存在。</p>
+<p>如图 8-2 所示，给定索引 <span class="arithmatex">\(i\)</span> ，其左子节点索引为 <span class="arithmatex">\(2i + 1\)</span> ，右子节点索引为 <span class="arithmatex">\(2i + 2\)</span> ，父节点索引为 <span class="arithmatex">\((i - 1) / 2\)</span>（向下整除）。当索引越界时，表示空节点或节点不存在。</p>
 <p><a class="glightbox" href="../heap.assets/representation_of_heap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="堆的表示与存储" class="animation-figure" src="../heap.assets/representation_of_heap.png" /></a></p>
 <p align="center"> 图 8-2 &nbsp; 堆的表示与存储 </p>

-<p>我们可以将索引映射公式封装成函数，方便后续使用。</p>
+<p>我们可以将索引映射公式封装成函数，方便后续使用：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Zig</label></div>
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 <div class="tabbed-block">
@@ -3854,7 +3851,7 @@
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="cm">/* 获取父节点索引 */</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</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="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 向下取整</span>
+<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</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="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 向下整除</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -4033,7 +4030,7 @@
 </div>
 </div>
 <h3 id="2">2. &nbsp; 访问堆顶元素<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
-<p>堆顶元素即为二叉树的根节点，也就是列表的首个元素。</p>
+<p>堆顶元素即为二叉树的根节点，也就是列表的首个元素：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Zig</label></div>
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@@ -4122,7 +4119,7 @@
 </div>
 </div>
 <h3 id="3">3. &nbsp; 元素入堆<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
-<p>给定元素 <code>val</code> ，我们首先将其添加到堆底。添加之后，由于 val 可能大于堆中其他元素，堆的成立条件可能已被破坏。因此，<strong>需要修复从插入节点到根节点的路径上的各个节点</strong>，这个操作被称为「堆化 heapify」。</p>
+<p>给定元素 <code>val</code> ，我们首先将其添加到堆底。添加之后，由于 val 可能大于堆中其他元素，堆的成立条件可能已被破坏，<strong>因此需要修复从插入节点到根节点的路径上的各个节点</strong>，这个操作被称为「堆化 heapify」。</p>
 <p>考虑从入堆节点开始，<strong>从底至顶执行堆化</strong>。如图 8-3 所示，我们比较插入节点与其父节点的值，如果插入节点更大，则将它们交换。然后继续执行此操作，从底至顶修复堆中的各个节点，直至越过根节点或遇到无须交换的节点时结束。</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:9"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label></div>
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@@ -4157,7 +4154,7 @@
 </div>
 <p align="center"> 图 8-3 &nbsp; 元素入堆步骤 </p>

-<p>设节点总数为 <span class="arithmatex">\(n\)</span> ，则树的高度为 <span class="arithmatex">\(O(\log n)\)</span> 。由此可知，堆化操作的循环轮数最多为 <span class="arithmatex">\(O(\log n)\)</span> ，<strong>元素入堆操作的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span></strong> 。</p>
+<p>设节点总数为 <span class="arithmatex">\(n\)</span> ，则树的高度为 <span class="arithmatex">\(O(\log n)\)</span> 。由此可知，堆化操作的循环轮数最多为 <span class="arithmatex">\(O(\log n)\)</span> ，<strong>元素入堆操作的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span></strong> 。代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:12"><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" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Zig</label></div>
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@@ -4475,10 +4472,10 @@
 </div>
 </div>
 <h3 id="4">4. &nbsp; 堆顶元素出堆<a class="headerlink" href="#4" title="Permanent link">&para;</a></h3>
-<p>堆顶元素是二叉树的根节点，即列表首元素。如果我们直接从列表中删除首元素，那么二叉树中所有节点的索引都会发生变化，这将使得后续使用堆化修复变得困难。为了尽量减少元素索引的变动，我们采用以下操作步骤。</p>
+<p>堆顶元素是二叉树的根节点，即列表首元素。如果我们直接从列表中删除首元素，那么二叉树中所有节点的索引都会发生变化，这将使得后续使用堆化进行修复变得困难。为了尽量减少元素索引的变动，我们采用以下操作步骤。</p>
 <ol>
-<li>交换堆顶元素与堆底元素（即交换根节点与最右叶节点）。</li>
-<li>交换完成后，将堆底从列表中删除（注意，由于已经交换，实际上删除的是原来的堆顶元素）。</li>
+<li>交换堆顶元素与堆底元素（交换根节点与最右叶节点）。</li>
+<li>交换完成后，将堆底从列表中删除（注意，由于已经交换，因此实际上删除的是原来的堆顶元素）。</li>
 <li>从根节点开始，<strong>从顶至底执行堆化</strong>。</li>
 </ol>
 <p>如图 8-4 所示，<strong>“从顶至底堆化”的操作方向与“从底至顶堆化”相反</strong>，我们将根节点的值与其两个子节点的值进行比较，将最大的子节点与根节点交换。然后循环执行此操作，直到越过叶节点或遇到无须交换的节点时结束。</p>
@@ -4518,7 +4515,7 @@
 </div>
 <p align="center"> 图 8-4 &nbsp; 堆顶元素出堆步骤 </p>

-<p>与元素入堆操作相似，堆顶元素出堆操作的时间复杂度也为 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
+<p>与元素入堆操作相似，堆顶元素出堆操作的时间复杂度也为 <span class="arithmatex">\(O(\log n)\)</span> 。代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Zig</label></div>
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@@ -4527,7 +4524,7 @@
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a>    <span class="c1"># 判空处理</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a>    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a>        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">&quot;堆为空&quot;</span><span class="p">)</span>
-<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a>    <span class="c1"># 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a>    <span class="c1"># 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a>    <span class="c1"># 删除节点</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a>    <span class="n">val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
@@ -4561,7 +4558,7 @@
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">&quot;堆为空&quot;</span><span class="p">);</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
@@ -4594,7 +4591,7 @@
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// 判空处理</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
-<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -4630,7 +4627,7 @@
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="c1">// 判空处理</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">();</span>
-<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Last</span><span class="p">();</span>
@@ -4668,7 +4665,7 @@
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">&quot;error&quot;</span><span class="p">)</span>
 <a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a><span class="w">    </span><span class="nx">val</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
@@ -4710,7 +4707,7 @@
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a>    <span class="k">if</span> <span class="bp">isEmpty</span><span class="p">()</span> <span class="p">{</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a>        <span class="bp">fatalError</span><span class="p">(</span><span class="s">&quot;堆为空&quot;</span><span class="p">)</span>
 <a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a>    <span class="p">}</span>
-<a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a>    <span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a>    <span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a>    <span class="bp">swap</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a>    <span class="c1">// 删除节点</span>
 <a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a>    <span class="kd">let</span> <span class="nv">val</span> <span class="p">=</span> <span class="n">maxHeap</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span> <span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
@@ -4751,7 +4748,7 @@
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// 判空处理</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">&#39;堆为空&#39;</span><span class="p">);</span>
-<a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">swap</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="err">#</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
@@ -4785,7 +4782,7 @@
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="nx">pop</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// 判空处理</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">&#39;Heap is empty.&#39;</span><span class="p">);</span>
-<a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
@@ -4819,7 +4816,7 @@
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">  </span><span class="c1">// 判空处理</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s1">&#39;堆为空&#39;</span><span class="p">);</span>
-<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">  </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">  </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">  </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
@@ -4855,7 +4852,7 @@
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">        </span><span class="fm">panic!</span><span class="p">(</span><span class="s">&quot;index out of bounds&quot;</span><span class="p">);</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -4896,7 +4893,7 @@
 <a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">&quot;heap is empty!&quot;</span><span class="p">);</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
@@ -4938,7 +4935,7 @@
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="n">self</span><span class="o">:</span><span class="w"> </span><span class="o">*</span><span class="n">Self</span><span class="p">)</span><span class="w"> </span><span class="o">!</span><span class="n">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// 判断处理</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">self</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">unreachable</span><span class="p">;</span>
-<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（即交换首元素与尾元素）</span>
+<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// 交换根节点与最右叶节点（交换首元素与尾元素）</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">    </span><span class="k">try</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="c1">// 删除节点</span>
 <a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="o">?</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
@@ -4973,7 +4970,7 @@
 <h2 id="813">8.1.3 &nbsp; 堆常见应用<a class="headerlink" href="#813" title="Permanent link">&para;</a></h2>
 <ul>
 <li><strong>优先队列</strong>：堆通常作为实现优先队列的首选数据结构，其入队和出队操作的时间复杂度均为 <span class="arithmatex">\(O(\log n)\)</span> ，而建队操作为 <span class="arithmatex">\(O(n)\)</span> ，这些操作都非常高效。</li>
-<li><strong>堆排序</strong>：给定一组数据，我们可以用它们建立一个堆，然后不断地执行元素出堆操作，从而得到有序数据。然而，我们通常会使用一种更优雅的方式实现堆排序，详见后续的堆排序章节。</li>
+<li><strong>堆排序</strong>：给定一组数据，我们可以用它们建立一个堆，然后不断地执行元素出堆操作，从而得到有序数据。然而，我们通常会使用一种更优雅的方式实现堆排序，详见“堆排序”章节。</li>
 <li><strong>获取最大的 <span class="arithmatex">\(k\)</span> 个元素</strong>：这是一个经典的算法问题，同时也是一种典型应用，例如选择热度前 10 的新闻作为微博热搜，选取销量前 10 的商品等。</li>
 </ul>