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<title>8.1. 堆（Heap） - Hello 算法</title>
<title>8.1.   堆（Heap） - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
8.1. 堆（Heap）
8.1. &nbsp; 堆（Heap）
</span>
</div>
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
0. 写在前面
0. &nbsp; &nbsp; 写在前面
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0. 写在前面
0. &nbsp; &nbsp; 写在前面
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<li class="md-nav__item">
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
0.1. 关于本书
0.1. &nbsp; 关于本书
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<li class="md-nav__item">
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
0.2. 如何使用本书
0.2. &nbsp; 如何使用本书
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<li class="md-nav__item">
<a href="../../chapter_preface/installation/" class="md-nav__link">
0.3. 编程环境安装
0.3. &nbsp; 编程环境安装
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<a href="../../chapter_preface/contribution/" class="md-nav__link">
0.4. 一起参与创作
0.4. &nbsp; 一起参与创作
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
1. 引言
1. &nbsp; &nbsp; 引言
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1. 引言
1. &nbsp; &nbsp; 引言
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1.1. 算法无处不在
1.1. &nbsp; 算法无处不在
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1.2. 算法是什么
1.2. &nbsp; 算法是什么
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2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
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2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
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2.1. 算法效率评估
2.1. &nbsp; 算法效率评估
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2.2. 时间复杂度
2.2. &nbsp; 时间复杂度
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<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
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<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
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<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. 小结
2.5. &nbsp; 小结
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3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
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3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
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<a href="../../chapter_data_structure/data_and_memory/" class="md-nav__link">
3.1. 数据与内存
3.1. &nbsp; 数据与内存
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<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
3.2. 数据结构分类
3.2. &nbsp; 数据结构分类
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3.3. 小结
3.3. &nbsp; 小结
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4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
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4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
4.1. 数组（Array）
4.1. &nbsp; 数组（Array）
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<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
4.2. 链表（LinkedList）
4.2. &nbsp; 链表（LinkedList）
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
4.3. 列表（List）
4.3. &nbsp; 列表（List）
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
4.4. 小结
4.4. &nbsp; 小结
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
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5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
5.1. 栈（Stack）
5.1. &nbsp; 栈（Stack）
</a>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
5.2. 队列（Queue）
5.2. &nbsp; 队列（Queue）
</a>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
5.3. 双向队列（Deque）
5.3. &nbsp; 双向队列（Deque）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
5.4. 小结
5.4. &nbsp; 小结
</a>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. 散列表
6. &nbsp; &nbsp; 散列表
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6. 散列表
6. &nbsp; &nbsp; 散列表
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. 哈希表（HashMap）
6.1. &nbsp; 哈希表（HashMap）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. 小结
6.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. 二叉树
7. &nbsp; &nbsp; 二叉树
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7. 二叉树
7. &nbsp; &nbsp; 二叉树
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. 二叉树（Binary Tree）
7.1. &nbsp; 二叉树（Binary Tree）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. 二叉树遍历
7.2. &nbsp; 二叉树遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. 二叉搜索树
7.3. &nbsp; 二叉搜索树
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. AVL 树 *
7.4. &nbsp; AVL 树 *
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. 小结
7.5. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. 堆
8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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8. 堆
8. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<label class="md-nav__link md-nav__link--active" for="__toc">
8.1. 堆（Heap）
8.1. &nbsp; 堆（Heap）
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
8.1. 堆（Heap）
8.1. &nbsp; 堆（Heap）
</a>
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<li class="md-nav__item">
<a href="#811" class="md-nav__link">
8.1.1. 堆术语与性质
8.1.1. &nbsp; 堆术语与性质
</a>
</li>
<li class="md-nav__item">
<a href="#812" class="md-nav__link">
8.1.2. 堆常用操作
8.1.2. &nbsp; 堆常用操作
</a>
</li>
<li class="md-nav__item">
<a href="#813" class="md-nav__link">
8.1.3. 堆的实现
8.1.3. &nbsp; 堆的实现
</a>
<nav class="md-nav" aria-label="8.1.3. 堆的实现">
<nav class="md-nav" aria-label="8.1.3. &nbsp; 堆的实现">
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<li class="md-nav__item">
<a href="#814" class="md-nav__link">
8.1.4. 堆常见应用
8.1.4. &nbsp; 堆常见应用
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. 图
9. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
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9. 图
9. &nbsp; &nbsp; 图
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. 图（Graph）
9.1. &nbsp; 图（Graph）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. 图基础操作
9.2. &nbsp; 图基础操作
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. 图的遍历
9.3. &nbsp; 图的遍历
</a>
</li>
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. 查找算法
10. &nbsp; &nbsp; 查找算法
<span class="md-nav__icon md-icon"></span>
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10. 查找算法
10. &nbsp; &nbsp; 查找算法
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<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. 线性查找
10.1. &nbsp; 线性查找
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. 二分查找
10.2. &nbsp; 二分查找
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. 哈希查找
10.3. &nbsp; 哈希查找
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. 小结
10.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
11. 排序算法
11. &nbsp; &nbsp; 排序算法
<span class="md-nav__icon md-icon"></span>
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11. 排序算法
11. &nbsp; &nbsp; 排序算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
11.1. 排序简介
11.1. &nbsp; 排序简介
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
11.3. 插入排序
11.3. &nbsp; 插入排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
11.4. 快速排序
11.4. &nbsp; 快速排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
11.5. 归并排序
11.5. &nbsp; 归并排序
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.6. 小结
11.6. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="#811" class="md-nav__link">
8.1.1. 堆术语与性质
8.1.1. &nbsp; 堆术语与性质
</a>
</li>
<li class="md-nav__item">
<a href="#812" class="md-nav__link">
8.1.2. 堆常用操作
8.1.2. &nbsp; 堆常用操作
</a>
</li>
<li class="md-nav__item">
<a href="#813" class="md-nav__link">
8.1.3. 堆的实现
8.1.3. &nbsp; 堆的实现
</a>
<nav class="md-nav" aria-label="8.1.3. 堆的实现">
<nav class="md-nav" aria-label="8.1.3. &nbsp; 堆的实现">
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<li class="md-nav__item">
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<li class="md-nav__item">
<a href="#814" class="md-nav__link">
8.1.4. 堆常见应用
8.1.4. &nbsp; 堆常见应用
</a>
</li>
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<h1 id="81">8.1. 堆<a class="headerlink" href="#81" 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」是一棵限定条件下的「完全二叉树」。根据成立条件，堆主要分为两种类型：</p>
<ul>
<li>「大顶堆 Max Heap」，任意结点的值 <span class="arithmatex">\(\geq\)</span> 其子结点的值；</li>
<li>「小顶堆 Min Heap」，任意结点的值 <span class="arithmatex">\(\leq\)</span> 其子结点的值；</li>
</ul>
<p><img alt="min_heap_and_max_heap" src="../heap.assets/min_heap_and_max_heap.png" /></p>
<h2 id="811">8.1.1. 堆术语与性质<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
<h2 id="811">8.1.1. &nbsp; 堆术语与性质<a class="headerlink" href="#811" title="Permanent link">&para;</a></h2>
<ul>
<li>由于堆是完全二叉树，因此最底层结点靠左填充，其它层结点皆被填满。</li>
<li>二叉树中的根结点对应「堆顶」，底层最靠右结点对应「堆底」。</li>
<li>对于大顶堆 / 小顶堆，其堆顶元素（即根结点）的值最大 / 最小。</li>
</ul>
<h2 id="812">8.1.2. 堆常用操作<a class="headerlink" href="#812" title="Permanent link">&para;</a></h2>
<h2 id="812">8.1.2. &nbsp; 堆常用操作<a class="headerlink" href="#812" title="Permanent link">&para;</a></h2>
<p>值得说明的是，多数编程语言提供的是「优先队列 Priority Queue」，其是一种抽象数据结构，<strong>定义为具有出队优先级的队列</strong>。</p>
<p>而恰好，<strong>堆的定义与优先队列的操作逻辑完全吻合</strong>，大顶堆就是一个元素从大到小出队的优先队列。从使用角度看，我们可以将「优先队列」和「堆」理解为等价的数据结构。因此，本文与代码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（方法命名以 Java 为例）。</p>
@ -1988,7 +1988,7 @@
</div>
</div>
</div>
<h2 id="813">8.1.3. 堆的实现<a class="headerlink" href="#813" title="Permanent link">&para;</a></h2>
<h2 id="813">8.1.3. &nbsp; 堆的实现<a class="headerlink" href="#813" title="Permanent link">&para;</a></h2>
<p>下文实现的是「大顶堆」，若想转换为「小顶堆」，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> ）即可，有兴趣的同学可自行实现。</p>
<h3 id="_1">堆的存储与表示<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<p>在二叉树章节我们学过，「完全二叉树」非常适合使用「数组」来表示，而堆恰好是一棵完全二叉树，<strong>因而我们采用「数组」来存储「堆」</strong>。</p>
@ -3035,7 +3035,7 @@ T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
\end{aligned}
\]</div>
<p>进一步地，高度为 <span class="arithmatex">\(h\)</span> 的完美二叉树的结点数量为 <span class="arithmatex">\(n = 2^{h+1} - 1\)</span> ，易得复杂度为 <span class="arithmatex">\(O(2^h) = O(n)\)</span>。以上推算表明，<strong>输入列表并建堆的时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，非常高效</strong>。</p>
<h2 id="814">8.1.4. 堆常见应用<a class="headerlink" href="#814" title="Permanent link">&para;</a></h2>
<h2 id="814">8.1.4. &nbsp; 堆常见应用<a class="headerlink" href="#814" title="Permanent link">&para;</a></h2>
<ul>
<li><strong>优先队列</strong>。堆常作为实现优先队列的首选数据结构，入队和出队操作时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> ，建队操作为 <span class="arithmatex">\(O(n)\)</span> ，皆非常高效。</li>
<li><strong>堆排序</strong>。给定一组数据，我们使用其建堆，并依次全部弹出，则可以得到有序的序列。当然，堆排序一般无需弹出元素，仅需每轮将堆顶元素交换至数组尾部并减小堆的长度即可。</li>
@ -3118,7 +3118,7 @@ T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
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