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<p>将上述两者相乘，可得到建堆过程的时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。<strong>但这个估算结果并不准确，因为我们没有考虑到二叉树底层节点数量远多于顶层节点的性质</strong>。</p>
<p>接下来我们来进行更为准确的计算。为了减小计算难度，假设给定一个节点数量为 <span class="arithmatex">\(n\)</span> ，高度为 <span class="arithmatex">\(h\)</span> 的“完美二叉树”，该假设不会影响计算结果的正确性。</p>
<p><img alt="完美二叉树的各层节点数量" src="../build_heap.assets/heapify_operations_count.png" /></p>
<p><a class="glightbox" href="../build_heap.assets/heapify_operations_count.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="完美二叉树的各层节点数量" src="../build_heap.assets/heapify_operations_count.png" /></a></p>
<p align="center"> 图 8-5 &nbsp; 完美二叉树的各层节点数量 </p>
<p>如图 8-5 所示，节点“从顶至底堆化”的最大迭代次数等于该节点到叶节点的距离，而该距离正是“节点高度”。因此，我们可以将各层的“节点数量 <span class="arithmatex">\(\times\)</span> 节点高度”求和，<strong>从而得到所有节点的堆化迭代次数的总和</strong>。</p>
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<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2023 Krahets
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Made with
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<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><a class="glightbox" href="../heap.assets/min_heap_and_max_heap.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="小顶堆与大顶堆" src="../heap.assets/min_heap_and_max_heap.png" /></a></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>
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<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><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" 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>
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</div>
</div>
</div>
@@ -5197,10 +5032,15 @@ aria-label="页脚"
<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2023 Krahets
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Made with
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<!-- Social links -->
@@ -5269,5 +5109,5 @@ aria-label="页脚"
</body>
<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
</html>
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</head>
<link href="../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
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10.2 &nbsp; 二分查找插入点
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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第 14 章 &nbsp; 动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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15.2 &nbsp; 分数背包问题
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15.4 &nbsp; 最大切分乘积问题
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<!-- Page content -->
<h1 id="8">第 8 章 &nbsp; 堆<a class="headerlink" href="#8" title="Permanent link">&para;</a></h1>
<div class="center-table">
<p><img alt="堆" src="../assets/covers/chapter_heap.jpg" width="600" /></p>
<p><a class="glightbox" href="../assets/covers/chapter_heap.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="堆" src="../assets/covers/chapter_heap.jpg" width="600" /></a></p>
</div>
<div class="admonition abstract">
<p class="admonition-title">Abstract</p>
@@ -3591,10 +3426,15 @@ aria-label="页脚"
<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2023 Krahets
Copyright &copy; 2022 - 2023 Krahets
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Made with
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<!-- Social links -->
@@ -3663,5 +3503,5 @@ aria-label="页脚"
</body>
<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
</html>
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</head>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
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</style> <script src="../../assets/javascripts/glightbox.min.js"></script></head>
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10.2 &nbsp; 二分查找插入点
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12.4 &nbsp; 汉诺塔问题
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第 14 章 &nbsp; 动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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</li>
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第 15 章 &nbsp; 贪心
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15.2 &nbsp; 分数背包问题
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@@ -3652,10 +3487,15 @@ aria-label="页脚"
<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2023 Krahets
Copyright &copy; 2022 - 2023 Krahets
</div>
Made with
<a href="https://squidfunk.github.io/mkdocs-material/" target="_blank" rel="noopener">
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</a>
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<!-- Social links -->
@@ -3724,5 +3564,5 @@ aria-label="页脚"
</body>
<script>document$.subscribe(() => {const lightbox = GLightbox({"touchNavigation": true, "loop": false, "zoomable": true, "draggable": false, "openEffect": "zoom", "closeEffect": "zoom", "slideEffect": "none"});})</script></body>
</html>
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@@ -60,7 +60,18 @@
</head>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
.gslide-desc { color: #666; user-select: text; }
.gslide-image img { background: white; }
.gscrollbar-fixer { padding-right: 15px; }
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body[data-md-color-scheme="slate"] .gslide-desc { color: var(--md-default-fg-color);}
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@@ -1938,14 +1949,6 @@
10.2 &nbsp; 二分查找插入点
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.4 &nbsp; 最大切分乘积问题
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@@ -3496,7 +3331,7 @@
<h2 id="831">8.3.1 &nbsp; 方法一：遍历选择<a class="headerlink" href="#831" title="Permanent link">&para;</a></h2>
<p>我们可以进行图 8-6 所示的 <span class="arithmatex">\(k\)</span> 轮遍历，分别在每轮中提取第 <span class="arithmatex">\(1\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(k\)</span> 大的元素，时间复杂度为 <span class="arithmatex">\(O(nk)\)</span> 。</p>
<p>此方法只适用于 <span class="arithmatex">\(k \ll n\)</span> 的情况，因为当 <span class="arithmatex">\(k\)</span> 与 <span class="arithmatex">\(n\)</span> 比较接近时，其时间复杂度趋向于 <span class="arithmatex">\(O(n^2)\)</span> ，非常耗时。</p>
<p><img alt="遍历寻找最大的 k 个元素" src="../top_k.assets/top_k_traversal.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_traversal.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="遍历寻找最大的 k 个元素" src="../top_k.assets/top_k_traversal.png" /></a></p>
<p align="center"> 图 8-6 &nbsp; 遍历寻找最大的 k 个元素 </p>
<div class="admonition tip">
@@ -3506,7 +3341,7 @@
<h2 id="832">8.3.2 &nbsp; 方法二：排序<a class="headerlink" href="#832" title="Permanent link">&para;</a></h2>
<p>如图 8-7 所示，我们可以先对数组 <code>nums</code> 进行排序，再返回最右边的 <span class="arithmatex">\(k\)</span> 个元素，时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
<p>显然，该方法“超额”完成任务了，因为我们只需要找出最大的 <span class="arithmatex">\(k\)</span> 个元素即可，而不需要排序其他元素。</p>
<p><img alt="排序寻找最大的 k 个元素" src="../top_k.assets/top_k_sorting.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_sorting.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="排序寻找最大的 k 个元素" src="../top_k.assets/top_k_sorting.png" /></a></p>
<p align="center"> 图 8-7 &nbsp; 排序寻找最大的 k 个元素 </p>
<h2 id="833">8.3.3 &nbsp; 方法三：堆<a class="headerlink" href="#833" title="Permanent link">&para;</a></h2>
@@ -3520,31 +3355,31 @@
<div class="tabbed-set tabbed-alternate" data-tabs="1:9"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
<div class="tabbed-content">
<div class="tabbed-block">
<p><img alt="基于堆寻找最大的 k 个元素" src="../top_k.assets/top_k_heap_step1.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="基于堆寻找最大的 k 个元素" src="../top_k.assets/top_k_heap_step1.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step2" src="../top_k.assets/top_k_heap_step2.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step2" src="../top_k.assets/top_k_heap_step2.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step3" src="../top_k.assets/top_k_heap_step3.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step3" src="../top_k.assets/top_k_heap_step3.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step4" src="../top_k.assets/top_k_heap_step4.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step4" src="../top_k.assets/top_k_heap_step4.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step5" src="../top_k.assets/top_k_heap_step5.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step5" src="../top_k.assets/top_k_heap_step5.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step6" src="../top_k.assets/top_k_heap_step6.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step6" src="../top_k.assets/top_k_heap_step6.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step7" src="../top_k.assets/top_k_heap_step7.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step7" src="../top_k.assets/top_k_heap_step7.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step8" src="../top_k.assets/top_k_heap_step8.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step8" src="../top_k.assets/top_k_heap_step8.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="top_k_heap_step9" src="../top_k.assets/top_k_heap_step9.png" /></p>
<p><a class="glightbox" href="../top_k.assets/top_k_heap_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="top_k_heap_step9" src="../top_k.assets/top_k_heap_step9.png" /></a></p>
</div>
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@@ -3925,10 +3760,15 @@ aria-label="页脚"
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