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8.2 &nbsp; 建堆操作
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8.3 &nbsp; Top-k 问题
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1. &nbsp; 数字 10 入堆后怎样调整
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2. &nbsp; 检查小顶堆的父子关系
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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9.2 &nbsp; 图基础操作
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第 10 章 &nbsp; 搜索
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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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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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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10.6 &nbsp; 小结
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10.7 &nbsp; 练习
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序算法
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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11.10 &nbsp; 基数排序
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11.11 &nbsp; 小结
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11.12 &nbsp; 练习
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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 13 章 &nbsp; 回溯
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13.2 &nbsp; 全排列问题
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13.3 &nbsp; 子集和问题
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13.4 &nbsp; N 皇后问题
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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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<h1 id="85">8.5 &nbsp; 练习<a class="headerlink" href="#85" title="Permanent link">&para;</a></h1>
<h2 id="851">8.5.1 &nbsp; 知识巩固<a class="headerlink" href="#851" title="Permanent link">&para;</a></h2>
<h3 id="1-10">1. &nbsp; 数字 10 入堆后怎样调整<a class="headerlink" href="#1-10" title="Permanent link">&para;</a></h3>
<p>数组 <code>[9, 7, 8, 3, 5]</code> 表示一个大顶堆。现在将数字 10 入堆。</p>
<!-- numbered-subquestions -->
<ol>
<li>先把 10 添加到数组末尾，它的父节点值是多少？</li>
<li>从新节点开始向上堆化，写出每次交换后的数组。</li>
<li>最终的堆顶元素是什么？一共交换了几次？</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>10 添加后的索引为 5，其父节点索引为
<span class="arithmatex">\(\lfloor(5-1)/2\rfloor=2\)</span>，父节点值为 8。</p>
</li>
<li>
<p>10 大于 8，第一次交换后为 <code>[9, 7, 10, 3, 5, 8]</code>；
10 又大于父节点 9，第二次交换后为 <code>[10, 7, 9, 3, 5, 8]</code>。
此时 10 已到达根节点，堆化结束。</p>
</li>
<li>
<p>最终堆顶为 10，共交换 2 次。</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 检查小顶堆的父子关系<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>数组 <code>[1, 4, 3, 7, 6, 2]</code> 表示一棵完全二叉树。小顶堆要求每个父节点都不大于它的孩子。
对索引 <span class="arithmatex">\(i\)</span>，左、右孩子索引分别为 <span class="arithmatex">\(2i+1\)</span> 和 <span class="arithmatex">\(2i+2\)</span>。</p>
<!-- numbered-subquestions -->
<ol>
<li>索引 2 的孩子索引和值分别是什么？</li>
<li>索引 2 处的节点值为 3，它与孩子节点之间是否违反小顶堆规则？如果违反，应交换哪两个元素？</li>
<li>根据第 2 问的判断：若违反规则，写出交换后的数组；若不违反，说明无须交换。最后检查其余父子关系。</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>索引 2 的左孩子索引是 5、值为 2；右孩子索引是 6，但数组长度为 6，因此右孩子不存在。</p>
</li>
<li>
<p>父节点值 3 大于孩子值 2，违反小顶堆规则，应交换索引 2 和索引 5 的元素。</p>
</li>
<li>
<p>交换后得到 <code>[1, 4, 2, 7, 6, 3]</code>。逐一检查：
<code>1 ≤ 4</code>、<code>1 ≤ 2</code>；<code>4 ≤ 7</code>、<code>4 ≤ 6</code>；<code>2 ≤ 3</code>。
所有父节点都不大于自己的孩子，因此现在满足小顶堆规则。</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 用小顶堆保留最大的三个数<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>要从数据流 <code>[4, 1, 7, 3, 8]</code> 中保留最大的 3 个数，可以维护一个大小不超过 3 的小顶堆。</p>
<p>先把前 3 个数依次放入小顶堆。堆满后，每读入一个新数：
如果它大于堆顶，就删除堆顶并放入新数；否则堆保持不变。</p>
<p>请写出每次读入数字后堆中保留的数字和堆顶。
只需把保留的数字写成集合，不要求写出它们在堆数组中的排列。</p>
<details class="success">
<summary>参考答案</summary>
<p>每次读入数字后的结果如下：</p>
<table>
<thead>
<tr>
<th>读入数字</th>
<th>保留的数字</th>
<th>堆顶</th>
</tr>
</thead>
<tbody>
<tr>
<td>4</td>
<td><code>{4}</code></td>
<td>4</td>
</tr>
<tr>
<td>1</td>
<td><code>{1, 4}</code></td>
<td>1</td>
</tr>
<tr>
<td>7</td>
<td><code>{1, 4, 7}</code></td>
<td>1</td>
</tr>
<tr>
<td>3</td>
<td><code>{3, 4, 7}</code></td>
<td>3</td>
</tr>
<tr>
<td>8</td>
<td><code>{4, 7, 8}</code></td>
<td>4</td>
</tr>
</tbody>
</table>
<p>堆满后，堆顶是当前保留数字中最小的一个。只有新数字大于堆顶时，
才用新数字替换堆顶。最终保留的 <code>{4, 7, 8}</code> 正是最大的 3 个数。</p>
</details>
<h2 id="852">8.5.2 &nbsp; 编程练习<a class="headerlink" href="#852" title="Permanent link">&para;</a></h2>
<h3 id="1-k">1. &nbsp; 寻找数组中的第 k 大元素<a class="headerlink" href="#1-k" title="Permanent link">&para;</a></h3>
<p>给定整数数组 <code>nums</code> 和整数 <span class="arithmatex">\(k\)</span>，其中 <span class="arithmatex">\(1 \le k \le n\)</span>，<span class="arithmatex">\(n\)</span> 是数组长度。把数组按从大到小排列后，请返回第 <span class="arithmatex">\(k\)</span> 个位置上的元素。</p>
<p>重复元素需要分别计数。例如，<code>[5, 5, 2]</code> 的第 2 大元素仍是 5。请使用一个大小不超过 <span class="arithmatex">\(k\)</span> 的小顶堆完成。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>第 k 大元素是最大的 k 个数中最小的那个</li>
<li>每个数先入小顶堆，大小超过 k 时弹出最小值</li>
<li>遍历结束时堆中保留最大的 k 个数，堆顶就是答案</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.cn/problems/kth-largest-element-in-an-array/" rel="noopener noreferrer" target="_blank">LeetCode</a> <a class="rounded-button exercise-button" href="https://leetcode.cn/problems/kth-largest-element-in-an-array/solutions/2361969/215-shu-zu-zhong-de-di-k-ge-zui-da-yuan-d786p/" rel="noopener noreferrer" target="_blank">题目解析</a></p>
<blockquote>
<p><strong>说明：</strong> 链接中的题解讲解了快速排序和快速选择，没有使用堆；本练习请按照正文 8.3 节使用大小不超过 k 的小顶堆完成</p>
</blockquote>
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