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<title>11.2. 冒泡排序 - Hello 算法</title>
<title>11.2.   冒泡排序 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
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11.2. 冒泡排序
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2.5. 小结
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3.1. &nbsp; 数据与内存
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4.1. 数组（Array）
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4.4. 小结
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5.1. 栈（Stack）
5.1. &nbsp; 栈（Stack）
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5.2. 队列（Queue）
5.2. &nbsp; 队列（Queue）
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5.3. 双向队列（Deque）
5.3. &nbsp; 双向队列（Deque）
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5.4. 小结
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6. 散列表
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6.1. 哈希表（HashMap）
6.1. &nbsp; 哈希表（HashMap）
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6.2. 哈希冲突处理
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6.3. 小结
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7. 二叉树
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7. 二叉树
7. &nbsp; &nbsp; 二叉树
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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）
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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
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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>
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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
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<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. 堆
8. &nbsp; &nbsp; 堆
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8. 堆
8. &nbsp; &nbsp; 堆
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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>
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<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. 图
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<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. 图（Graph）
9.1. &nbsp; 图（Graph）
</a>
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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>
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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>
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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
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10. 查找算法
10. &nbsp; &nbsp; 查找算法
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<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>
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<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
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<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>
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<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. 小结
10.4. &nbsp; 小结
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<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
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11. 排序算法
11. &nbsp; &nbsp; 排序算法
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11.1. 排序简介
11.1. &nbsp; 排序简介
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<label class="md-nav__link md-nav__link--active" for="__toc">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
<span class="md-nav__icon md-icon"></span>
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<a href="./" class="md-nav__link md-nav__link--active">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
</a>
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<li class="md-nav__item">
<a href="#1121" class="md-nav__link">
11.2.1. 算法流程
11.2.1. &nbsp; 算法流程
</a>
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<li class="md-nav__item">
<a href="#1122" class="md-nav__link">
11.2.2. 算法特性
11.2.2. &nbsp; 算法特性
</a>
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<li class="md-nav__item">
<a href="#1123" class="md-nav__link">
11.2.3. 效率优化
11.2.3. &nbsp; 效率优化
</a>
</li>
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<li class="md-nav__item">
<a href="../insertion_sort/" class="md-nav__link">
11.3. 插入排序
11.3. &nbsp; 插入排序
</a>
</li>
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<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>
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<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>
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
11.6. 小结
11.6. &nbsp; 小结
</a>
</li>
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<li class="md-nav__item">
<a href="#1121" class="md-nav__link">
11.2.1. 算法流程
11.2.1. &nbsp; 算法流程
</a>
</li>
<li class="md-nav__item">
<a href="#1122" class="md-nav__link">
11.2.2. 算法特性
11.2.2. &nbsp; 算法特性
</a>
</li>
<li class="md-nav__item">
<a href="#1123" class="md-nav__link">
11.2.3. 效率优化
11.2.3. &nbsp; 效率优化
</a>
</li>
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<h1 id="112">11.2. 冒泡排序<a class="headerlink" href="#112" title="Permanent link">&para;</a></h1>
<h1 id="112">11.2. &nbsp; 冒泡排序<a class="headerlink" href="#112" title="Permanent link">&para;</a></h1>
<p>「冒泡排序 Bubble Sort」是一种最基础的排序算法，非常适合作为第一个学习的排序算法。顾名思义，「冒泡」是该算法的核心操作。</p>
<div class="admonition question">
<p class="admonition-title">为什么叫“冒泡”</p>
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</div>
<p align="center"> Fig. 冒泡操作 </p>
<h2 id="1121">11.2.1. 算法流程<a class="headerlink" href="#1121" title="Permanent link">&para;</a></h2>
<h2 id="1121">11.2.1. &nbsp; 算法流程<a class="headerlink" href="#1121" title="Permanent link">&para;</a></h2>
<ol>
<li>设数组长度为 <span class="arithmatex">\(n\)</span> ，完成第一轮「冒泡」后，数组最大元素已在正确位置，接下来只需排序剩余 <span class="arithmatex">\(n - 1\)</span> 个元素。</li>
<li>同理，对剩余 <span class="arithmatex">\(n - 1\)</span> 个元素执行「冒泡」，可将第二大元素交换至正确位置，因而待排序元素只剩 <span class="arithmatex">\(n - 2\)</span> 个。</li>
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</div>
</div>
</div>
<h2 id="1122">11.2.2. 算法特性<a class="headerlink" href="#1122" title="Permanent link">&para;</a></h2>
<h2 id="1122">11.2.2. &nbsp; 算法特性<a class="headerlink" href="#1122" title="Permanent link">&para;</a></h2>
<p><strong>时间复杂度 <span class="arithmatex">\(O(n^2)\)</span></strong> ：各轮「冒泡」遍历的数组长度为 <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> ，因此使用 <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>：引入 <code>flag</code> 优化后（见下文），最佳时间复杂度为 <span class="arithmatex">\(O(N)\)</span> 。</p>
<h2 id="1123">11.2.3. 效率优化<a class="headerlink" href="#1123" title="Permanent link">&para;</a></h2>
<h2 id="1123">11.2.3. &nbsp; 效率优化<a class="headerlink" href="#1123" title="Permanent link">&para;</a></h2>
<p>我们发现，若在某轮「冒泡」中未执行任何交换操作，则说明数组已经完成排序，可直接返回结果。考虑可以增加一个标志位 <code>flag</code> 来监听该情况，若出现则直接返回。</p>
<p>优化后，冒泡排序的最差和平均时间复杂度仍为 <span class="arithmatex">\(O(n^2)\)</span> ；而在输入数组 <strong>已排序</strong> 时，达到 <strong>最佳时间复杂度</strong> <span class="arithmatex">\(O(n)\)</span> 。 </p>
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