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<title>10.2. 二分查找 - Hello 算法</title>
<title>10.2.   二分查找 - Hello 算法</title>
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<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
10.2. 二分查找
10.2. &nbsp; 二分查找
</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
<span class="md-nav__icon md-icon"></span>
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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
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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
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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
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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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1. 引言
1. &nbsp; &nbsp; 引言
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<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
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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
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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
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</li>
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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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<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
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<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
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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
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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
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4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
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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
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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>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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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>
@ -898,7 +898,7 @@
<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">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
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>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
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">
<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
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="true">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
10. 查找算法
10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../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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<label class="md-nav__link md-nav__link--active" for="__toc">
10.2. 二分查找
10.2. &nbsp; 二分查找
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
10.2. 二分查找
10.2. &nbsp; 二分查找
</a>
@ -1270,10 +1270,10 @@
<li class="md-nav__item">
<a href="#1021" class="md-nav__link">
10.2.1. 算法实现
10.2.1. &nbsp; 算法实现
</a>
<nav class="md-nav" aria-label="10.2.1. 算法实现">
<nav class="md-nav" aria-label="10.2.1. &nbsp; 算法实现">
<ul class="md-nav__list">
<li class="md-nav__item">
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<li class="md-nav__item">
<a href="#1022" class="md-nav__link">
10.2.2. 复杂度分析
10.2.2. &nbsp; 复杂度分析
</a>
</li>
<li class="md-nav__item">
<a href="#1023" class="md-nav__link">
10.2.3. 优点与缺点
10.2.3. &nbsp; 优点与缺点
</a>
</li>
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<li class="md-nav__item">
<a href="../hashing_search/" class="md-nav__link">
10.3. 哈希查找
10.3. &nbsp; 哈希查找
</a>
</li>
@ -1353,7 +1353,7 @@
<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
10.4. 小结
10.4. &nbsp; 小结
</a>
</li>
@ -1400,14 +1400,14 @@
<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="false">
<label class="md-nav__title" for="__nav_12">
<span class="md-nav__icon md-icon"></span>
11. 排序算法
11. &nbsp; &nbsp; 排序算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
11.1. 排序简介
11.1. &nbsp; 排序简介
</a>
</li>
@ -1432,7 +1432,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
</a>
</li>
@ -1446,7 +1446,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
11.3. 插入排序
11.3. &nbsp; 插入排序
</a>
</li>
@ -1460,7 +1460,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
11.4. 快速排序
11.4. &nbsp; 快速排序
</a>
</li>
@ -1474,7 +1474,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
11.5. 归并排序
11.5. &nbsp; 归并排序
</a>
</li>
@ -1488,7 +1488,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.6. 小结
11.6. &nbsp; 小结
</a>
</li>
@ -1576,10 +1576,10 @@
<li class="md-nav__item">
<a href="#1021" class="md-nav__link">
10.2.1. 算法实现
10.2.1. &nbsp; 算法实现
</a>
<nav class="md-nav" aria-label="10.2.1. 算法实现">
<nav class="md-nav" aria-label="10.2.1. &nbsp; 算法实现">
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<li class="md-nav__item">
@ -1617,14 +1617,14 @@
<li class="md-nav__item">
<a href="#1022" class="md-nav__link">
10.2.2. 复杂度分析
10.2.2. &nbsp; 复杂度分析
</a>
</li>
<li class="md-nav__item">
<a href="#1023" class="md-nav__link">
10.2.3. 优点与缺点
10.2.3. &nbsp; 优点与缺点
</a>
</li>
@ -1653,14 +1653,14 @@
<h1 id="102">10.2. 二分查找<a class="headerlink" href="#102" title="Permanent link">&para;</a></h1>
<h1 id="102">10.2. &nbsp; 二分查找<a class="headerlink" href="#102" title="Permanent link">&para;</a></h1>
<p>「二分查找 Binary Search」利用数据的有序性，通过每轮缩小一半搜索区间来查找目标元素。</p>
<p>使用二分查找有两个前置条件：</p>
<ul>
<li><strong>要求输入数据是有序的</strong>，这样才能通过判断大小关系来排除一半的搜索区间；</li>
<li><strong>二分查找仅适用于数组</strong>，而在链表中使用效率很低，因为其在循环中需要跳跃式（非连续地）访问元素。</li>
</ul>
<h2 id="1021">10.2.1. 算法实现<a class="headerlink" href="#1021" title="Permanent link">&para;</a></h2>
<h2 id="1021">10.2.1. &nbsp; 算法实现<a class="headerlink" href="#1021" title="Permanent link">&para;</a></h2>
<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的排序数组 <code>nums</code> ，元素从小到大排列。数组的索引取值范围为</p>
<div class="arithmatex">\[
0, 1, 2, \cdots, n-1
@ -2180,10 +2180,10 @@
</div>
</div>
</div>
<h2 id="1022">10.2.2. 复杂度分析<a class="headerlink" href="#1022" title="Permanent link">&para;</a></h2>
<h2 id="1022">10.2.2. &nbsp; 复杂度分析<a class="headerlink" href="#1022" title="Permanent link">&para;</a></h2>
<p><strong>时间复杂度 <span class="arithmatex">\(O(\log n)\)</span></strong> ：其中 <span class="arithmatex">\(n\)</span> 为数组或链表长度；每轮排除一半的区间，因此循环轮数为 <span class="arithmatex">\(\log_2 n\)</span> ，使用 <span class="arithmatex">\(O(\log n)\)</span> 时间。</p>
<p><strong>空间复杂度 <span class="arithmatex">\(O(1)\)</span></strong> ：指针 <code>i</code> , <code>j</code> 使用常数大小空间。</p>
<h2 id="1023">10.2.3. 优点与缺点<a class="headerlink" href="#1023" title="Permanent link">&para;</a></h2>
<h2 id="1023">10.2.3. &nbsp; 优点与缺点<a class="headerlink" href="#1023" title="Permanent link">&para;</a></h2>
<p>二分查找效率很高，体现在：</p>
<ul>
<li><strong>二分查找时间复杂度低</strong>。对数阶在数据量很大时具有巨大优势，例如，当数据大小 <span class="arithmatex">\(n = 2^{20}\)</span> 时，线性查找需要 <span class="arithmatex">\(2^{20} = 1048576\)</span> 轮循环，而二分查找仅需要 <span class="arithmatex">\(\log_2 2^{20} = 20\)</span> 轮循环。</li>
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