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第 8 章 &nbsp; 堆
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第 10 章 &nbsp; 搜索
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10.2 &nbsp; 二分查找插入点
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10.4 &nbsp; 哈希优化策略
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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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第 12 章 &nbsp; 分治
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第 14 章 &nbsp; 动态规划
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14.6 &nbsp; 编辑距离问题
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<h1 id="1112">11.12 &nbsp; 练习<a class="headerlink" href="#1112" title="Permanent link">&para;</a></h1>
<h2 id="11121">11.12.1 &nbsp; 知识巩固<a class="headerlink" href="#11121" title="Permanent link">&para;</a></h2>
<h3 id="1">1. &nbsp; 选择排序和冒泡排序的前几轮<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>给定数组 <code>[4, 2, 5, 1, 3]</code>,以下都按从小到大排序。</p>
<!-- numbered-subquestions -->
<ol>
<li>按选择排序模拟前两轮,写出每轮结束后的数组,并指出此时哪些位置已经确定。</li>
<li>按冒泡排序模拟第一轮,写出数组状态和交换次数,并指出哪个位置已经确定。</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>前两轮为:</p>
<table>
<thead>
<tr>
<th>轮次</th>
<th>数组状态</th>
<th>说明</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td><code>[1, 2, 5, 4, 3]</code></td>
<td>最小元素 1 与首位交换</td>
</tr>
<tr>
<td>2</td>
<td><code>[1, 2, 5, 4, 3]</code></td>
<td>数值 2 已位于索引 1,无须交换</td>
</tr>
</tbody>
</table>
<p>此时前两个位置已经确定,后续只需在 <code>[5, 4, 3]</code> 中继续选择最小元素。</p>
</li>
<li>
<p>依次比较相邻元素:4 与 2 交换、4 与 5 不换、5 与 1 交换、5 与 3 交换,
得 <code>[2, 4, 1, 3, 5]</code> ,共 3 次交换;最大元素 5 已移动到数组末尾,因此最后一个位置已经确定。</p>
</li>
</ol>
</details>
<h3 id="2">2. &nbsp; 相同元素的先后次序会变吗<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p>数组 <span class="arithmatex">\([2_a, 2_b, 1]\)</span> 中,<span class="arithmatex">\(2_a\)</span> 和 <span class="arithmatex">\(2_b\)</span> 的数值相同,但用下标标出了原有次序。</p>
<!-- numbered-subquestions -->
<ol>
<li>写出选择排序第一轮结束后的数组。<span class="arithmatex">\(2_a\)</span> 和 <span class="arithmatex">\(2_b\)</span> 的相对次序是否改变?</li>
<li>写出冒泡排序第一轮结束后的数组。<span class="arithmatex">\(2_a\)</span> 和 <span class="arithmatex">\(2_b\)</span> 的相对次序是否改变?</li>
<li>根据前两问,说明两种排序在保持相等元素原有次序方面有什么不同。</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>选择排序第一轮选出最小元素 1,并与首位 <span class="arithmatex">\(2_a\)</span> 交换,得
<span class="arithmatex">\([1, 2_b, 2_a]\)</span> 。<span class="arithmatex">\(2_a\)</span> 被换到了 <span class="arithmatex">\(2_b\)</span> 之后,两者的相对次序已经改变。</p>
</li>
<li>
<p>冒泡排序先比较 <span class="arithmatex">\(2_a\)</span> 和 <span class="arithmatex">\(2_b\)</span>,因为两者相等,不交换;再比较 <span class="arithmatex">\(2_b\)</span> 和 1 并交换,
第一轮结束后为 <span class="arithmatex">\([2_a, 1, 2_b]\)</span> 。<span class="arithmatex">\(2_a\)</span> 仍在 <span class="arithmatex">\(2_b\)</span> 之前,相对次序没有改变。</p>
</li>
<li>
<p>这个例子中,选择排序会改变相等元素的原有次序;冒泡排序只在左边元素大于右边元素时
交换相邻元素。相等元素之间不交换,因此能保持它们原有的先后次序。</p>
</li>
</ol>
</details>
<h3 id="3">3. &nbsp; 比较计数排序和基数排序<a class="headerlink" href="#3" title="Permanent link">&para;</a></h3>
<p>学校要排序许多固定为 8 位的学号。请回答:</p>
<!-- numbered-subquestions -->
<ol>
<li>从最低位开始的基数排序需要处理几轮?</li>
<li>如果把学号直接当作整数使用计数排序,为什么会准备大量没有用到的计数位置?</li>
<li>根据前两问,如果在计数排序和基数排序之间选择一种处理大量固定为 8 位的学号,你会选择哪一种?为什么?</li>
</ol>
<details class="success">
<summary>参考答案</summary>
<ol>
<li>
<p>学号有 8 个数位,因此从最低位到最高位共处理 8 轮;每轮只按 0~9 分组。</p>
</li>
<li>
<p>直接计数需要为所有可能的 8 位取值预留位置,但实际学生只使用其中很少一部分,
大多数计数位置会一直为 0。</p>
</li>
<li>
<p>应选择基数排序。它利用了“位数固定、每位只有 10 种取值”的特点,只需重复 8 轮稳定分组。
计数排序若把整个 8 位学号当作整数下标,则需要为大量实际不会出现的数值预留计数位置。</p>
</li>
</ol>
</details>
<h2 id="11122">11.12.2 &nbsp; 编程练习<a class="headerlink" href="#11122" title="Permanent link">&para;</a></h2>
<h3 id="1_1">1. &nbsp; 用归并排序排列数组<a class="headerlink" href="#1_1" title="Permanent link">&para;</a></h3>
<p>给定一个整数数组 <code>nums</code>,请自行实现归并排序,把其中的元素按非递减顺序排列并返回。不要调用语言自带的排序函数。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>区间长度不超过 1 时,它已经有序</li>
<li>从中点把区间分成两半,并分别递归排序</li>
<li>用两个指针合并两个有序半区,再把结果写回原数组</li>
</ol>
</details>
<p><a class="rounded-button exercise-button" href="https://leetcode.cn/problems/sort-an-array/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
<h3 id="2_1">2. &nbsp; 用计数排序排列整数数组<a class="headerlink" href="#2_1" title="Permanent link">&para;</a></h3>
<p>给定整数数组 <code>nums</code> 和非负整数 <span class="arithmatex">\(K\)</span>,数组中的每个元素都在 <span class="arithmatex">\(0\)</span> 到 <span class="arithmatex">\(K\)</span> 之间。</p>
<p>请实现计数排序,把结果按非递减顺序写回 <code>nums</code> 并返回 <code>nums</code>。
不要使用元素之间的大小比较来决定顺序,也不要调用语言自带的排序函数。</p>
<details class="tip">
<summary>解题提示</summary>
<ol>
<li>因为每个元素都在 0 到 K 之间,可以直接用元素值作为计数数组的索引</li>
<li>第一遍扫描 nums,令对应位置的计数加 1</li>
<li>再从 0 到 K 扫描计数数组;数值 x 出现多少次,就向 nums 中连续写入多少个 x</li>
</ol>
</details>
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