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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 11 章 &nbsp; 排序
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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<a href="../../chapter_paperbook/" class="md-nav__link ">
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纸质书
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<!-- Page content -->
<h1 id="113">11.3 &nbsp; 冒泡排序<a class="headerlink" href="#113" title="Permanent link">&para;</a></h1>
<p>「冒泡排序 bubble sort」通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>
<p><u>冒泡排序（bubble sort）</u>通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>
<p>如图 11-4 所示，冒泡过程可以利用元素交换操作来模拟：从数组最左端开始向右遍历，依次比较相邻元素大小，如果“左元素 &gt; 右元素”就交换二者。遍历完成后，最大的元素会被移动到数组的最右端。</p>
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<h1 id="118">11.8 &nbsp; 桶排序<a class="headerlink" href="#118" title="Permanent link">&para;</a></h1>
<p>前述几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>
<p>「桶排序 bucket sort」是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
<p><u>桶排序（bucket sort）</u>是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
<h2 id="1181">11.8.1 &nbsp; 算法流程<a class="headerlink" href="#1181" title="Permanent link">&para;</a></h2>
<p>考虑一个长度为 <span class="arithmatex">\(n\)</span> 的数组，其元素是范围 <span class="arithmatex">\([0, 1)\)</span> 内的浮点数。桶排序的流程如图 11-13 所示。</p>
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第 0 章 &nbsp; 前言
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<!-- Page content -->
<h1 id="119">11.9 &nbsp; 计数排序<a class="headerlink" href="#119" title="Permanent link">&para;</a></h1>
<p>「计数排序 counting sort」通过统计元素数量来实现排序，通常应用于整数数组。</p>
<p><u>计数排序（counting sort）</u>通过统计元素数量来实现排序，通常应用于整数数组。</p>
<h2 id="1191">11.9.1 &nbsp; 简单实现<a class="headerlink" href="#1191" title="Permanent link">&para;</a></h2>
<p>先来看一个简单的例子。给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，其中的元素都是“非负整数”，计数排序的整体流程如图 11-16 所示。</p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 8 章 &nbsp; 堆
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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<p class="admonition-title">Tip</p>
<p>阅读本节前，请确保已学完“堆“章节。</p>
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<p>「堆排序 heap sort」是一种基于堆数据结构实现的高效排序算法。我们可以利用已经学过的“建堆操作”和“元素出堆操作”实现堆排序。</p>
<p><u>堆排序（heap sort）</u>是一种基于堆数据结构实现的高效排序算法。我们可以利用已经学过的“建堆操作”和“元素出堆操作”实现堆排序。</p>
<ol>
<li>输入数组并建立小顶堆，此时最小元素位于堆顶。</li>
<li>不断执行出堆操作，依次记录出堆元素，即可得到从小到大排序的序列。</li>
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 11 章 &nbsp; 排序
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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<span class="md-ellipsis">
纸质书
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纸质书
</label>
@@ -3741,7 +3798,7 @@ aria-label="页脚"
<div class="md-copyright">
<div class="md-copyright__highlight">
Copyright &copy; 2022-2024 krahets</br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
Copyright &copy; 2022-2024 krahets<br>The website content is licensed under <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a>
</div>
chapter_sorting/insertion_sort/index.html
+134 -77
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序
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序
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<ul class="md-nav__list" data-md-scrollfix>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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</a>
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第 5 章 &nbsp; 栈与队列
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</a>
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第 6 章 &nbsp; 哈希表
</label>
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</a>
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<span class="md-nav__icon md-icon"></span>
第 7 章 &nbsp; 树
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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</a>
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<span class="md-nav__icon md-icon"></span>
第 8 章 &nbsp; 堆
</label>
@@ -1814,7 +1871,7 @@
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
@@ -1832,14 +1889,14 @@
</a>
<label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
第 9 章 &nbsp; 图
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@@ -1972,7 +2029,7 @@
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
@@ -1990,14 +2047,14 @@
</a>
<label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
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</label>
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<span class="md-nav__icon md-icon"></span>
第 10 章 &nbsp; 搜索
</label>
@@ -2184,7 +2241,7 @@
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" checked>
@@ -2202,14 +2259,14 @@
</a>
<label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
<span class="md-nav__icon md-icon"></span>
</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_12">
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_13">
<span class="md-nav__icon md-icon"></span>
第 11 章 &nbsp; 排序
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@@ -2554,7 +2611,7 @@
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
@@ -2572,14 +2629,14 @@
</a>
<label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
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<span class="md-nav__icon md-icon"></span>
</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_13">
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<label class="md-nav__title" for="__nav_14">
<span class="md-nav__icon md-icon"></span>
第 12 章 &nbsp; 分治
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</a>
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</label>
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第 13 章 &nbsp; 回溯
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@@ -2912,7 +2969,7 @@
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@@ -2930,14 +2987,14 @@
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_16_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
第 14 章 &nbsp; 动态规划
</label>
@@ -3131,7 +3188,7 @@
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_17" >
@@ -3149,14 +3206,14 @@
</a>
<label class="md-nav__link " for="__nav_16" id="__nav_16_label" tabindex="0">
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第 15 章 &nbsp; 贪心
</label>
@@ -3304,7 +3361,7 @@
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</a>
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第 16 章 &nbsp; 附录
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_18_label" aria-expanded="false">
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参考文献
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<!-- Page content -->
<h1 id="114">11.4 &nbsp; 插入排序<a class="headerlink" href="#114" title="Permanent link">&para;</a></h1>
<p>「插入排序 insertion sort」是一种简单的排序算法，它的工作原理与手动整理一副牌的过程非常相似。</p>
<p><u>插入排序（insertion sort）</u>是一种简单的排序算法，它的工作原理与手动整理一副牌的过程非常相似。</p>
<p>具体来说，我们在未排序区间选择一个基准元素，将该元素与其左侧已排序区间的元素逐一比较大小，并将该元素插入到正确的位置。</p>
<p>图 11-6 展示了数组插入元素的操作流程。设基准元素为 <code>base</code> ，我们需要将从目标索引到 <code>base</code> 之间的所有元素向右移动一位，然后将 <code>base</code> 赋值给目标索引。</p>
<p><a class="glightbox" href="../insertion_sort.assets/insertion_operation.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="单次插入操作" class="animation-figure" src="../insertion_sort.assets/insertion_operation.png" /></a></p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 4 章 &nbsp; 数组与链表
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 16 章 &nbsp; 附录
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<!-- Page content -->
<h1 id="116">11.6 &nbsp; 归并排序<a class="headerlink" href="#116" title="Permanent link">&para;</a></h1>
<p>「归并排序 merge sort」是一种基于分治策略的排序算法，包含图 11-10 所示的“划分”和“合并”阶段。</p>
<p><u>归并排序（merge sort）</u>是一种基于分治策略的排序算法，包含图 11-10 所示的“划分”和“合并”阶段。</p>
<ol>
<li><strong>划分阶段</strong>：通过递归不断地将数组从中点处分开，将长数组的排序问题转换为短数组的排序问题。</li>
<li><strong>合并阶段</strong>：当子数组长度为 1 时终止划分，开始合并，持续地将左右两个较短的有序数组合并为一个较长的有序数组，直至结束。</li>
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序
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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<!-- Page content -->
<h1 id="115">11.5 &nbsp; 快速排序<a class="headerlink" href="#115" title="Permanent link">&para;</a></h1>
<p>「快速排序 quick sort」是一种基于分治策略的排序算法，运行高效，应用广泛。</p>
<p><u>快速排序（quick sort）</u>是一种基于分治策略的排序算法，运行高效，应用广泛。</p>
<p>快速排序的核心操作是“哨兵划分”，其目标是：选择数组中的某个元素作为“基准数”，将所有小于基准数的元素移到其左侧，而大于基准数的元素移到其右侧。具体来说，哨兵划分的流程如图 11-8 所示。</p>
<ol>
<li>选取数组最左端元素作为基准数，初始化两个指针 <code>i</code> 和 <code>j</code> 分别指向数组的两端。</li>
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序
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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<!-- Page content -->
<h1 id="1110">11.10 &nbsp; 基数排序<a class="headerlink" href="#1110" title="Permanent link">&para;</a></h1>
<p>上一节介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>
<p>「基数排序 radix sort」的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
<p><u>基数排序（radix sort）</u>的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
<h2 id="11101">11.10.1 &nbsp; 算法流程<a class="headerlink" href="#11101" title="Permanent link">&para;</a></h2>
<p>以学号数据为例，假设数字的最低位是第 <span class="arithmatex">\(1\)</span> 位，最高位是第 <span class="arithmatex">\(8\)</span> 位，基数排序的流程如图 11-18 所示。</p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 2 章 &nbsp; 复杂度分析
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第 3 章 &nbsp; 数据结构
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第 4 章 &nbsp; 数组与链表
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第 5 章 &nbsp; 栈与队列
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第 6 章 &nbsp; 哈希表
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第 7 章 &nbsp; 树
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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<!-- Page content -->
<h1 id="112">11.2 &nbsp; 选择排序<a class="headerlink" href="#112" title="Permanent link">&para;</a></h1>
<p>「选择排序 selection sort」的工作原理非常简单：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>
<p><u>选择排序（selection sort）</u>的工作原理非常简单：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>
<p>设数组的长度为 <span class="arithmatex">\(n\)</span> ，选择排序的算法流程如图 11-2 所示。</p>
<ol>
<li>初始状态下，所有元素未排序，即未排序（索引）区间为 <span class="arithmatex">\([0, n-1]\)</span> 。</li>
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第 0 章 &nbsp; 前言
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第 3 章 &nbsp; 数据结构
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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<!-- Page content -->
<h1 id="111">11.1 &nbsp; 排序算法<a class="headerlink" href="#111" title="Permanent link">&para;</a></h1>
<p>「排序算法 sorting algorithm」用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更高效地查找、分析和处理。</p>
<p><u>排序算法（sorting algorithm）</u>用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更高效地查找、分析和处理。</p>
<p>如图 11-1 所示，排序算法中的数据类型可以是整数、浮点数、字符或字符串等。排序的判断规则可根据需求设定，如数字大小、字符 ASCII 码顺序或自定义规则。</p>
<p><a class="glightbox" href="../sorting_algorithm.assets/sorting_examples.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="数据类型和判断规则示例" class="animation-figure" src="../sorting_algorithm.assets/sorting_examples.png" /></a></p>
<p align="center"> 图 11-1 &nbsp; 数据类型和判断规则示例 </p>
<h2 id="1111">11.1.1 &nbsp; 评价维度<a class="headerlink" href="#1111" title="Permanent link">&para;</a></h2>
<p><strong>运行效率</strong>：我们期望排序算法的时间复杂度尽量低，且总体操作数量较少（时间复杂度中的常数项变小）。对于大数据量的情况，运行效率显得尤为重要。</p>
<p><strong>就地性</strong>：顾名思义，「原地排序」通过在原数组上直接操作实现排序，无须借助额外的辅助数组，从而节省内存。通常情况下，原地排序的数据搬运操作较少，运行速度也更快。</p>
<p><strong>稳定性</strong>：「稳定排序」在完成排序后，相等元素在数组中的相对顺序不发生改变。</p>
<p>稳定排序是多级排序场景的必要条件。假设我们有一个存储学生信息的表格，第 1 列和第 2 列分别是姓名和年龄。在这种情况下，「非稳定排序」可能导致输入数据的有序性丧失：</p>
<p><strong>就地性</strong>：顾名思义，<u>原地排序</u>通过在原数组上直接操作实现排序，无须借助额外的辅助数组，从而节省内存。通常情况下，原地排序的数据搬运操作较少，运行速度也更快。</p>
<p><strong>稳定性</strong>：<u>稳定排序</u>在完成排序后，相等元素在数组中的相对顺序不发生改变。</p>
<p>稳定排序是多级排序场景的必要条件。假设我们有一个存储学生信息的表格，第 1 列和第 2 列分别是姓名和年龄。在这种情况下，<u>非稳定排序</u>可能导致输入数据的有序性丧失：</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># 输入数据是按照姓名排序好的</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># (name, age)</span>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="o">(</span><span class="s1">&#39;A&#39;</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
@@ -3628,9 +3685,9 @@
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="o">(</span><span class="s1">&#39;C&#39;</span>,<span class="w"> </span><span class="m">21</span><span class="o">)</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="o">(</span><span class="s1">&#39;E&#39;</span>,<span class="w"> </span><span class="m">23</span><span class="o">)</span>
</code></pre></div>
<p><strong>自适应性</strong>：「自适应排序」的时间复杂度会受输入数据的影响，即最佳时间复杂度、最差时间复杂度、平均时间复杂度并不完全相等。</p>
<p><strong>自适应性</strong>：<u>自适应排序</u>的时间复杂度会受输入数据的影响，即最佳时间复杂度、最差时间复杂度、平均时间复杂度并不完全相等。</p>
<p>自适应性需要根据具体情况来评估。如果最差时间复杂度差于平均时间复杂度，说明排序算法在某些数据下性能可能劣化，因此被视为负面属性；而如果最佳时间复杂度优于平均时间复杂度，则被视为正面属性。</p>
<p><strong>是否基于比较</strong>：「基于比较的排序」依赖比较运算符（<span class="arithmatex">\(&lt;\)</span>、<span class="arithmatex">\(=\)</span>、<span class="arithmatex">\(&gt;\)</span>）来判断元素的相对顺序，从而排序整个数组，理论最优时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。而「非比较排序」不使用比较运算符，时间复杂度可达 <span class="arithmatex">\(O(n)\)</span> ，但其通用性相对较差。</p>
<p><strong>是否基于比较</strong>：<u>基于比较的排序</u>依赖比较运算符（<span class="arithmatex">\(&lt;\)</span>、<span class="arithmatex">\(=\)</span>、<span class="arithmatex">\(&gt;\)</span>）来判断元素的相对顺序，从而排序整个数组，理论最优时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。而<u>非比较排序</u>不使用比较运算符，时间复杂度可达 <span class="arithmatex">\(O(n)\)</span> ，但其通用性相对较差。</p>
<h2 id="1112">11.1.2 &nbsp; 理想排序算法<a class="headerlink" href="#1112" title="Permanent link">&para;</a></h2>
<p><strong>运行快、原地、稳定、正向自适应、通用性好</strong>。显然，迄今为止尚未发现兼具以上所有特性的排序算法。因此，在选择排序算法时，需要根据具体的数据特点和问题需求来决定。</p>
<p>接下来，我们将共同学习各种排序算法，并基于上述评价维度对各个排序算法的优缺点进行分析。</p>
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第 0 章 &nbsp; 前言
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第 1 章 &nbsp; 初识算法
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第 3 章 &nbsp; 数据结构
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第 8 章 &nbsp; 堆
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第 9 章 &nbsp; 图
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第 11 章 &nbsp; 排序
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第 12 章 &nbsp; 分治
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第 13 章 &nbsp; 回溯
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第 14 章 &nbsp; 动态规划
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第 15 章 &nbsp; 贪心
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第 16 章 &nbsp; 附录
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参考文献
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纸质书
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纸质书
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