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<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，主要可分为下图所示的两种类型：</p>
<p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为图 8-1 所示的两种类型：</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>
<li>「小顶堆 min heap」：任意节点的值 <span class="arithmatex">\(\leq\)</span> 其子节点的值。</li>
</ul>
<p><img alt="小顶堆与大顶堆" src="../heap.assets/min_heap_and_max_heap.png" /></p>
<p align="center"> 图：小顶堆与大顶堆 </p>
<p align="center"> 图 8-1 &nbsp; 小顶堆与大顶堆 </p>
<p>堆作为完全二叉树的一个特例，具有以下特性：</p>
<ul>
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<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>
<p align="center"> 表：堆的操作效率 </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>
<div class="center-table">
<table>
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<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="cm">/* 堆顶元素出堆 */</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1">// 出堆元素会形成一个从大到小的序列</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 4</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 3</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 2</span>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 1</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">peek</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">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="n">peek</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">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 4</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">peek</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">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 3</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">peek</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">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 2</span>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">peek</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">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 1</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="cm">/* 获取堆大小 */</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</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">size</span><span class="p">();</span>
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<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>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>如下图所示，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>（向下取整）。当索引越界时，表示空节点或节点不存在。</p>
<p><img alt="堆的表示与存储" src="../heap.assets/representation_of_heap.png" /></p>
<p align="center"> 图：堆的表示与存储 </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>
<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">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
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</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>考虑从入堆节点开始，<strong>从底至顶执行堆化</strong>。如下图所示，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>
<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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<p align="center"> 图：元素入堆步骤 </p>
<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>
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<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>从根节点开始，<strong>从顶至底执行堆化</strong>。</li>
</ol>
<p>如下图所示，<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>
<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>
<div class="tabbed-set tabbed-alternate" data-tabs="6:10"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">&lt;1&gt;</label><label for="__tabbed_6_2">&lt;2&gt;</label><label for="__tabbed_6_3">&lt;3&gt;</label><label for="__tabbed_6_4">&lt;4&gt;</label><label for="__tabbed_6_5">&lt;5&gt;</label><label for="__tabbed_6_6">&lt;6&gt;</label><label for="__tabbed_6_7">&lt;7&gt;</label><label for="__tabbed_6_8">&lt;8&gt;</label><label for="__tabbed_6_9">&lt;9&gt;</label><label for="__tabbed_6_10">&lt;10&gt;</label></div>
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<p align="center"> 图：堆顶元素出堆步骤 </p>
<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>
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