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<title>9.2.   建堆操作 * - Hello 算法</title>
<title>8.2.   建堆操作 * - Hello 算法</title>
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9.2. &nbsp; 建堆操作 *
8.2. &nbsp; 建堆操作 *
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6.1. &nbsp; 二分查找
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6.1. &nbsp; 哈希表
</a>
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6.2. &nbsp; 二分查找边界
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6.2. &nbsp; 哈希冲突处理
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6.3. &nbsp; 小结
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7. &nbsp; &nbsp; 散列表
7. &nbsp; &nbsp; 树
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7.1. &nbsp; 哈希表
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7.2. &nbsp; 哈希冲突处理
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7.3. &nbsp; 小结
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<li class="md-nav__item">
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8.1. &nbsp; 二叉树
7.1. &nbsp; 二叉树
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
8.2. &nbsp; 二叉树遍历
7.2. &nbsp; 二叉树遍历
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<li class="md-nav__item">
<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
8.3. &nbsp; 二叉树数组表示
7.3. &nbsp; 二叉树数组表示
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<li class="md-nav__item">
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8.4. &nbsp; 二叉搜索树
7.4. &nbsp; 二叉搜索树
</a>
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
8.5. &nbsp; AVL 树 *
7.5. &nbsp; AVL 树 *
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<li class="md-nav__item">
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8.6. &nbsp; 小结
7.6. &nbsp; 小结
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9. &nbsp; &nbsp; 堆
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8. &nbsp; &nbsp; 堆
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9. &nbsp; &nbsp; 堆
8. &nbsp; &nbsp; 堆
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9.1. &nbsp; 堆
8.1. &nbsp; 堆
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<label class="md-nav__link md-nav__link--active" for="__toc">
9.2. &nbsp; 建堆操作 *
8.2. &nbsp; 建堆操作 *
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9.2. &nbsp; 建堆操作 *
8.2. &nbsp; 建堆操作 *
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
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9.2.1. &nbsp; 借助入堆方法实现
<a href="#821" class="md-nav__link">
8.2.1. &nbsp; 借助入堆方法实现
</a>
</li>
<li class="md-nav__item">
<a href="#922" class="md-nav__link">
9.2.2. &nbsp; 基于堆化操作实现
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8.2.2. &nbsp; 基于堆化操作实现
</a>
</li>
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9.2.3. &nbsp; 复杂度分析
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8.2.3. &nbsp; 复杂度分析
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</li>
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9.3. &nbsp; 小结
8.3. &nbsp; 小结
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9.1. &nbsp; 图
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9.2. &nbsp; 图基础操作
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</li>
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9.3. &nbsp; 图的遍历
</a>
</li>
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9.4. &nbsp; 小结
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10. &nbsp; &nbsp; 图
10. &nbsp; &nbsp; 搜索算法
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10.1. &nbsp; 图
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10.1. &nbsp; 二分查找（New）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
10.2. &nbsp; 图基础操作
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10.2. &nbsp; 二分查找边界（New）
</a>
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10.3. &nbsp; 哈希优化策略
</a>
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<li class="md-nav__item">
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10.4. &nbsp; 小结
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10.4. &nbsp; 重识搜索算法
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10.5. &nbsp; 小结
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12. &nbsp; &nbsp; 搜索算法
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13.1. &nbsp; 回溯算法（New）
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13.1. &nbsp; 编程环境安装
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13.3. &nbsp; N 皇后问题（New）
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13.2. &nbsp; 一起参与创作
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14.1. &nbsp; 编程环境安装
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14.2. &nbsp; 一起参与创作
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参考文献
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
<li class="md-nav__item">
<a href="#921" class="md-nav__link">
9.2.1. &nbsp; 借助入堆方法实现
<a href="#821" class="md-nav__link">
8.2.1. &nbsp; 借助入堆方法实现
</a>
</li>
<li class="md-nav__item">
<a href="#922" class="md-nav__link">
9.2.2. &nbsp; 基于堆化操作实现
<a href="#822" class="md-nav__link">
8.2.2. &nbsp; 基于堆化操作实现
</a>
</li>
<li class="md-nav__item">
<a href="#923" class="md-nav__link">
9.2.3. &nbsp; 复杂度分析
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8.2.3. &nbsp; 复杂度分析
</a>
</li>
@@ -1927,12 +1888,12 @@
<h1 id="92">9.2. &nbsp; 建堆操作 *<a class="headerlink" href="#92" title="Permanent link">&para;</a></h1>
<h1 id="82">8.2. &nbsp; 建堆操作 *<a class="headerlink" href="#82" title="Permanent link">&para;</a></h1>
<p>如果我们想要根据输入列表生成一个堆，这个过程被称为「建堆」。</p>
<h2 id="921">9.2.1. &nbsp; 借助入堆方法实现<a class="headerlink" href="#921" title="Permanent link">&para;</a></h2>
<h2 id="821">8.2.1. &nbsp; 借助入堆方法实现<a class="headerlink" href="#821" title="Permanent link">&para;</a></h2>
<p>最直接的方法是借助“元素入堆操作”实现，首先创建一个空堆，然后将列表元素依次添加到堆中。</p>
<p>设元素数量为 <span class="arithmatex">\(n\)</span> ，则最后一个元素入堆的时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> 。在依次添加元素时，堆的平均长度为 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此该方法的总体时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
<h2 id="922">9.2.2. &nbsp; 基于堆化操作实现<a class="headerlink" href="#922" title="Permanent link">&para;</a></h2>
<h2 id="822">8.2.2. &nbsp; 基于堆化操作实现<a class="headerlink" href="#822" title="Permanent link">&para;</a></h2>
<p>有趣的是，存在一种更高效的建堆方法，其时间复杂度仅为 <span class="arithmatex">\(O(n)\)</span> 。我们先将列表所有元素原封不动添加到堆中，<strong>然后迭代地对各个节点执行“从顶至底堆化”</strong>。当然，<strong>我们不需要对叶节点执行堆化操作</strong>，因为它们没有子节点。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label></div>
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</div>
</div>
</div>
<h2 id="923">9.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#923" title="Permanent link">&para;</a></h2>
<h2 id="823">8.2.3. &nbsp; 复杂度分析<a class="headerlink" href="#823" title="Permanent link">&para;</a></h2>
<p>为什么第二种建堆方法的时间复杂度是 <span class="arithmatex">\(O(n)\)</span> ？我们来展开推算一下。</p>
<ul>
<li>完全二叉树中，设节点总数为 <span class="arithmatex">\(n\)</span> ，则叶节点数量为 <span class="arithmatex">\((n + 1) / 2\)</span> ，其中 <span class="arithmatex">\(/\)</span> 为向下整除。因此，在排除叶节点后，需要堆化的节点数量为 <span class="arithmatex">\((n - 1)/2\)</span> ，复杂度为 <span class="arithmatex">\(O(n)\)</span> ；</li>
@@ -2176,7 +2137,7 @@ T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
<nav class="md-footer__inner md-grid" aria-label="页脚" >
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9.2. &nbsp; 建堆操作 *
8.2. &nbsp; 建堆操作 *
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9.1.1. &nbsp; 堆常用操作
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8.1.1. &nbsp; 堆常用操作
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9.1.2. &nbsp; 堆的实现
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8.1.3. &nbsp; 堆常见应用
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<h1 id="91">9.1. &nbsp; 堆<a class="headerlink" href="#91" title="Permanent link">&para;</a></h1>
<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，可分为两种类型：</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>
@@ -2010,7 +1971,7 @@
<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>
</ul>
<h2 id="911">9.1.1. &nbsp; 堆常用操作<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
<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>堆的常用操作见下表，方法名需要根据编程语言来确定。</p>
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</div>
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<h2 id="912">9.1.2. &nbsp; 堆的实现<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<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">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>
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<h2 id="913">9.1.3. &nbsp; 堆常见应用<a class="headerlink" href="#913" title="Permanent link">&para;</a></h2>
<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，然后依次将所有元素弹出，从而得到一个有序序列。当然，堆排序的实现方法并不需要弹出元素，而是每轮将堆顶元素交换至数组尾部并缩小堆的长度。</li>
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<li>堆是一棵完全二叉树，根据成立条件可分为大顶堆和小顶堆。大（小）顶堆的堆顶元素是最大（小）的。</li>
<li>优先队列的定义是具有出队优先级的队列，通常使用堆来实现。</li>
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