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</head>
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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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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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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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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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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14.7 &nbsp; 小结
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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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15.2 &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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<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 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>
<div class="tabbed-content">
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<p><img alt="元素入堆步骤" src="../heap.assets/heap_push_step1.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="元素入堆步骤" src="../heap.assets/heap_push_step1.png" /></a></p>
</div>
<div class="tabbed-block">
<p><img alt="heap_push_step2" src="../heap.assets/heap_push_step2.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_push_step2" src="../heap.assets/heap_push_step2.png" /></a></p>
</div>
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<p><img alt="heap_push_step3" src="../heap.assets/heap_push_step3.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step3.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_push_step3" src="../heap.assets/heap_push_step3.png" /></a></p>
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<p><img alt="heap_push_step4" src="../heap.assets/heap_push_step4.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step4.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_push_step4" src="../heap.assets/heap_push_step4.png" /></a></p>
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<p><img alt="heap_push_step5" src="../heap.assets/heap_push_step5.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_push_step5" src="../heap.assets/heap_push_step5.png" /></a></p>
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<p><img alt="heap_push_step6" src="../heap.assets/heap_push_step6.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_push_step6.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_push_step6" src="../heap.assets/heap_push_step6.png" /></a></p>
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<p><img alt="heap_push_step7" src="../heap.assets/heap_push_step7.png" /></p>
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<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><img alt="堆顶元素出堆步骤" src="../heap.assets/heap_pop_step1.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_pop_step1.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="堆顶元素出堆步骤" src="../heap.assets/heap_pop_step1.png" /></a></p>
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<p><img alt="heap_pop_step2" src="../heap.assets/heap_pop_step2.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_pop_step2.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step2" src="../heap.assets/heap_pop_step2.png" /></a></p>
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<p><img alt="heap_pop_step5" src="../heap.assets/heap_pop_step5.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_pop_step5.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step5" src="../heap.assets/heap_pop_step5.png" /></a></p>
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<p><img alt="heap_pop_step6" src="../heap.assets/heap_pop_step6.png" /></p>
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<p><a class="glightbox" href="../heap.assets/heap_pop_step7.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step7" src="../heap.assets/heap_pop_step7.png" /></a></p>
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<p><img alt="heap_pop_step8" src="../heap.assets/heap_pop_step8.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_pop_step8.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step8" src="../heap.assets/heap_pop_step8.png" /></a></p>
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<p><a class="glightbox" href="../heap.assets/heap_pop_step9.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step9" src="../heap.assets/heap_pop_step9.png" /></a></p>
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<p><img alt="heap_pop_step10" src="../heap.assets/heap_pop_step10.png" /></p>
<p><a class="glightbox" href="../heap.assets/heap_pop_step10.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="heap_pop_step10" src="../heap.assets/heap_pop_step10.png" /></a></p>
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