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<title>8.1.   堆 - Hello 算法</title>
<title>9.1.   堆 - Hello 算法</title>
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8.1. &nbsp; 堆
9.1. &nbsp; 堆
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2.5. &nbsp; 小结
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7.2. &nbsp; 二叉树遍历
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7.4. &nbsp; AVL 树 *
8.4. &nbsp; AVL 树 *
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8.1. &nbsp; 堆
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8.1. &nbsp; 堆
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8.1.3. &nbsp; 堆常见应用
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9.1.3. &nbsp; 堆常见应用
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8.2. &nbsp; 建堆操作 *
9.2. &nbsp; 建堆操作 *
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9.1. &nbsp; 图
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9.2. &nbsp; 图基础操作
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9.3. &nbsp; 图的遍历
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9.4. &nbsp; 小结
9.3. &nbsp; 小结
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10.1. &nbsp; 线性查找
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10.1. &nbsp; 图
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10.2. &nbsp; 二分查找
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10.2. &nbsp; 图基础操作
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10.3. &nbsp; 哈希查找
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10.3. &nbsp; 图的遍历
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11.6. &nbsp; 桶排序（New）
11.6. &nbsp; 桶排序
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11.7. &nbsp; 计数排序（New）
11.7. &nbsp; 计数排序
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11.8. &nbsp; 基数排序（New）
11.8. &nbsp; 基数排序
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8.1.1. &nbsp; 堆常用操作
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9.1.1. &nbsp; 堆常用操作
</a>
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<a href="#812" class="md-nav__link">
8.1.2. &nbsp; 堆的实现
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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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9.1.3. &nbsp; 堆常见应用
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<h1 id="81">8.1. &nbsp; 堆<a class="headerlink" href="#81" title="Permanent link">&para;</a></h1>
<h1 id="91">9.1. &nbsp; 堆<a class="headerlink" href="#91" 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>
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<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="811">8.1.1. &nbsp; 堆常用操作<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
<h2 id="911">9.1.1. &nbsp; 堆常用操作<a class="headerlink" href="#911" 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>
</div>
</div>
<h2 id="812">8.1.2. &nbsp; 堆的实现<a class="headerlink" href="#812" title="Permanent link">&para;</a></h2>
<h2 id="912">9.1.2. &nbsp; 堆的实现<a class="headerlink" href="#912" 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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</div>
</div>
</div>
<h2 id="813">8.1.3. &nbsp; 堆常见应用<a class="headerlink" href="#813" title="Permanent link">&para;</a></h2>
<h2 id="913">9.1.3. &nbsp; 堆常见应用<a class="headerlink" href="#913" 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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