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<title>11.3. 插入排序 - Hello 算法</title>
<title>11.3.   插入排序 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
11.3. 插入排序
11.3. &nbsp; 插入排序
</span>
</div>
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<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
0. 写在前面
0. &nbsp; &nbsp; 写在前面
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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0. 写在前面
0. &nbsp; &nbsp; 写在前面
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<li class="md-nav__item">
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
0.1. 关于本书
0.1. &nbsp; 关于本书
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
0.2. 如何使用本书
0.2. &nbsp; 如何使用本书
</a>
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<li class="md-nav__item">
<a href="../../chapter_preface/installation/" class="md-nav__link">
0.3. 编程环境安装
0.3. &nbsp; 编程环境安装
</a>
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<li class="md-nav__item">
<a href="../../chapter_preface/contribution/" class="md-nav__link">
0.4. 一起参与创作
0.4. &nbsp; 一起参与创作
</a>
</li>
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
1. 引言
1. &nbsp; &nbsp; 引言
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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1. 引言
1. &nbsp; &nbsp; 引言
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<li class="md-nav__item">
<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
1.1. 算法无处不在
1.1. &nbsp; 算法无处不在
</a>
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<li class="md-nav__item">
<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
1.2. 算法是什么
1.2. &nbsp; 算法是什么
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<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
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2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
2.1. 算法效率评估
2.1. &nbsp; 算法效率评估
</a>
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/time_complexity/" class="md-nav__link">
2.2. 时间复杂度
2.2. &nbsp; 时间复杂度
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_complexity/" class="md-nav__link">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
</a>
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/space_time_tradeoff/" class="md-nav__link">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
</a>
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<li class="md-nav__item">
<a href="../../chapter_computational_complexity/summary/" class="md-nav__link">
2.5. 小结
2.5. &nbsp; 小结
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
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3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
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<li class="md-nav__item">
<a href="../../chapter_data_structure/data_and_memory/" class="md-nav__link">
3.1. 数据与内存
3.1. &nbsp; 数据与内存
</a>
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<li class="md-nav__item">
<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
3.2. 数据结构分类
3.2. &nbsp; 数据结构分类
</a>
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<li class="md-nav__item">
<a href="../../chapter_data_structure/summary/" class="md-nav__link">
3.3. 小结
3.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
<span class="md-nav__icon md-icon"></span>
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4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
4.1. 数组（Array）
4.1. &nbsp; 数组（Array）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
4.2. 链表（LinkedList）
4.2. &nbsp; 链表（LinkedList）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
4.3. 列表（List）
4.3. &nbsp; 列表（List）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
4.4. 小结
4.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
<span class="md-nav__icon md-icon"></span>
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<label class="md-nav__title" for="__nav_6">
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5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
5.1. 栈（Stack）
5.1. &nbsp; 栈（Stack）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
5.2. 队列（Queue）
5.2. &nbsp; 队列（Queue）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
5.3. 双向队列（Deque）
5.3. &nbsp; 双向队列（Deque）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
5.4. 小结
5.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. 散列表
6. &nbsp; &nbsp; 散列表
<span class="md-nav__icon md-icon"></span>
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6. 散列表
6. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. 哈希表（HashMap）
6.1. &nbsp; 哈希表（HashMap）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. 小结
6.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. 二叉树
7. &nbsp; &nbsp; 二叉树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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7. 二叉树
7. &nbsp; &nbsp; 二叉树
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. 二叉树（Binary Tree）
7.1. &nbsp; 二叉树（Binary Tree）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. 二叉树遍历
7.2. &nbsp; 二叉树遍历
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. 二叉搜索树
7.3. &nbsp; 二叉搜索树
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. AVL 树 *
7.4. &nbsp; AVL 树 *
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. 小结
7.5. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. 堆
8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
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8. 堆
8. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. 堆（Heap）
8.1. &nbsp; 堆（Heap）
</a>
</li>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. 图
9. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
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9. 图
9. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1131,7 +1131,7 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. 图（Graph）
9.1. &nbsp; 图（Graph）
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. 图基础操作
9.2. &nbsp; 图基础操作
</a>
</li>
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<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. 图的遍历
9.3. &nbsp; 图的遍历
</a>
</li>
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<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. 查找算法
10. &nbsp; &nbsp; 查找算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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10. 查找算法
10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1220,7 +1220,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. 线性查找
10.1. &nbsp; 线性查找
</a>
</li>
@@ -1234,7 +1234,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. 二分查找
10.2. &nbsp; 二分查找
</a>
</li>
@@ -1248,7 +1248,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. 哈希查找
10.3. &nbsp; 哈希查找
</a>
</li>
@@ -1262,7 +1262,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. 小结
10.4. &nbsp; 小结
</a>
</li>
@@ -1311,14 +1311,14 @@
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
11. 排序算法
11. &nbsp; &nbsp; 排序算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_12">
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11. 排序算法
11. &nbsp; &nbsp; 排序算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1329,7 +1329,7 @@
<li class="md-nav__item">
<a href="../intro_to_sort/" class="md-nav__link">
11.1. 排序简介
11.1. &nbsp; 排序简介
</a>
</li>
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<li class="md-nav__item">
<a href="../bubble_sort/" class="md-nav__link">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
</a>
</li>
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<label class="md-nav__link md-nav__link--active" for="__toc">
11.3. 插入排序
11.3. &nbsp; 插入排序
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
11.3. 插入排序
11.3. &nbsp; 插入排序
</a>
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<li class="md-nav__item">
<a href="#1131" class="md-nav__link">
11.3.1. 算法流程
11.3.1. &nbsp; 算法流程
</a>
</li>
<li class="md-nav__item">
<a href="#1132" class="md-nav__link">
11.3.2. 算法特性
11.3.2. &nbsp; 算法特性
</a>
</li>
<li class="md-nav__item">
<a href="#1133-vs" class="md-nav__link">
11.3.3. 插入排序 vs 冒泡排序
11.3.3. &nbsp; 插入排序 vs 冒泡排序
</a>
</li>
@@ -1426,7 +1426,7 @@
<li class="md-nav__item">
<a href="../quick_sort/" class="md-nav__link">
11.4. 快速排序
11.4. &nbsp; 快速排序
</a>
</li>
@@ -1440,7 +1440,7 @@
<li class="md-nav__item">
<a href="../merge_sort/" class="md-nav__link">
11.5. 归并排序
11.5. &nbsp; 归并排序
</a>
</li>
@@ -1454,7 +1454,7 @@
<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
11.6. 小结
11.6. &nbsp; 小结
</a>
</li>
@@ -1542,21 +1542,21 @@
<li class="md-nav__item">
<a href="#1131" class="md-nav__link">
11.3.1. 算法流程
11.3.1. &nbsp; 算法流程
</a>
</li>
<li class="md-nav__item">
<a href="#1132" class="md-nav__link">
11.3.2. 算法特性
11.3.2. &nbsp; 算法特性
</a>
</li>
<li class="md-nav__item">
<a href="#1133-vs" class="md-nav__link">
11.3.3. 插入排序 vs 冒泡排序
11.3.3. &nbsp; 插入排序 vs 冒泡排序
</a>
</li>
@@ -1585,14 +1585,14 @@
<h1 id="113">11.3. 插入排序<a class="headerlink" href="#113" title="Permanent link">&para;</a></h1>
<h1 id="113">11.3. &nbsp; 插入排序<a class="headerlink" href="#113" title="Permanent link">&para;</a></h1>
<p>「插入排序 Insertion Sort」是一种基于 <strong>数组插入操作</strong> 的排序算法。</p>
<p>「插入操作」原理：选定某个待排序元素为基准数 <code>base</code>，将 <code>base</code> 与其左侧已排序区间元素依次对比大小，并插入到正确位置。</p>
<p>回忆数组插入操作，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，然后再将 <code>base</code> 赋值给目标索引。</p>
<p><img alt="insertion_operation" src="../insertion_sort.assets/insertion_operation.png" /></p>
<p align="center"> Fig. 插入操作 </p>
<h2 id="1131">11.3.1. 算法流程<a class="headerlink" href="#1131" title="Permanent link">&para;</a></h2>
<h2 id="1131">11.3.1. &nbsp; 算法流程<a class="headerlink" href="#1131" title="Permanent link">&para;</a></h2>
<ol>
<li>第 1 轮先选取数组的 <strong>第 2 个元素</strong> 为 <code>base</code> ，执行「插入操作」后，<strong>数组前 2 个元素已完成排序</strong>。</li>
<li>第 2 轮选取 <strong>第 3 个元素</strong> 为 <code>base</code> ，执行「插入操作」后，<strong>数组前 3 个元素已完成排序</strong>。</li>
@@ -1758,13 +1758,13 @@
</div>
</div>
</div>
<h2 id="1132">11.3.2. 算法特性<a class="headerlink" href="#1132" title="Permanent link">&para;</a></h2>
<h2 id="1132">11.3.2. &nbsp; 算法特性<a class="headerlink" href="#1132" title="Permanent link">&para;</a></h2>
<p><strong>时间复杂度 <span class="arithmatex">\(O(n^2)\)</span></strong> ：最差情况下，各轮插入操作循环 <span class="arithmatex">\(n - 1\)</span> , <span class="arithmatex">\(n-2\)</span> , <span class="arithmatex">\(\cdots\)</span> , <span class="arithmatex">\(2\)</span> , <span class="arithmatex">\(1\)</span> 次，求和为 <span class="arithmatex">\(\frac{(n - 1) n}{2}\)</span> ，使用 <span class="arithmatex">\(O(n^2)\)</span> 时间。</p>
<p><strong>空间复杂度 <span class="arithmatex">\(O(1)\)</span></strong> ：指针 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> 使用常数大小的额外空间。</p>
<p><strong>原地排序</strong>：指针变量仅使用常数大小额外空间。</p>
<p><strong>稳定排序</strong>：不交换相等元素。</p>
<p><strong>自适应排序</strong>：最佳情况下，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</p>
<h2 id="1133-vs">11.3.3. 插入排序 vs 冒泡排序<a class="headerlink" href="#1133-vs" title="Permanent link">&para;</a></h2>
<h2 id="1133-vs">11.3.3. &nbsp; 插入排序 vs 冒泡排序<a class="headerlink" href="#1133-vs" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">Question</p>
<p>虽然「插入排序」和「冒泡排序」的时间复杂度皆为 <span class="arithmatex">\(O(n^2)\)</span> ，但实际运行速度却有很大差别，这是为什么呢？</p>
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