1958 lines
102 KiB
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1958 lines
102 KiB
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Hello 算法
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二分查找
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正在初始化搜索引擎
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.2.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2022 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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|
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|
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<li class="md-nav__item">
|
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编程环境安装
|
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|
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|
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|
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|
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<li class="md-nav__item">
|
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一起参与创作
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|
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--section md-nav__item--nested">
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<label class="md-nav__link" for="__nav_2">
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<nav class="md-nav" aria-label="引言" data-md-level="1">
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数组（Array）
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链表（LinkedList）
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栈与队列
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<nav class="md-nav" aria-label="栈与队列" data-md-level="1">
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栈与队列
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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栈（Stack）
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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队列（Queue）
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<li class="md-nav__item">
|
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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双向队列（Deque）
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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小结
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</ul>
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</nav>
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<label class="md-nav__link" for="__nav_7">
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散列表
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<span class="md-nav__icon md-icon"></span>
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<nav class="md-nav" aria-label="散列表" data-md-level="1">
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散列表
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|
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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哈希表（HashMap）
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</a>
|
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</li>
|
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<li class="md-nav__item">
|
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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小结
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</ul>
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</nav>
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<label class="md-nav__link" for="__nav_8">
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二叉树
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<nav class="md-nav" aria-label="二叉树" data-md-level="1">
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二叉树
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二叉树（Binary Tree）
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
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AVL 树 *
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</a>
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</li>
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<li class="md-nav__item">
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小结
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</nav>
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<li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
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<label class="md-nav__link" for="__nav_9">
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查找算法
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查找算法
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线性查找
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归并排序
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<a href="#_5" class="md-nav__link">
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<article class="md-content__inner md-typeset">
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|
||
<a href="https://github.com/krahets/hello-algo/tree/master/docs/chapter_searching/binary_search.md" title="编辑此页" class="md-content__button md-icon">
|
||
|
||
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25Z"/></svg>
|
||
</a>
|
||
|
||
|
||
<h1 id="_1">二分查找<a class="headerlink" href="#_1" title="Permanent link">¶</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="_2">算法实现<a class="headerlink" href="#_2" title="Permanent link">¶</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">\[
|
||
0, 1, 2, \cdots, n-1
|
||
\]</div>
|
||
<p>使用「区间」来表示这个取值范围的方法主要有两种：</p>
|
||
<ol>
|
||
<li><strong>双闭区间 <span class="arithmatex">\([0, n-1]\)</span></strong> ，即两个边界都包含自身；此方法下，区间 <span class="arithmatex">\([0, 0]\)</span> 仍包含一个元素；</li>
|
||
<li><strong>左闭右开 <span class="arithmatex">\([0, n)\)</span></strong> ，即左边界包含自身、右边界不包含自身；此方法下，区间 <span class="arithmatex">\([0, 0)\)</span> 为空；</li>
|
||
</ol>
|
||
<h3 id="_3">“双闭区间” 实现<a class="headerlink" href="#_3" title="Permanent link">¶</a></h3>
|
||
<p>首先，我们先采用 “双闭区间” 的表示，在数组 <code>nums</code> 中查找目标元素 <code>target</code> 的对应索引。</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="1:7"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Step 1</label><label for="__tabbed_1_2">Step 2</label><label for="__tabbed_1_3">Step 3</label><label for="__tabbed_1_4">Step 4</label><label for="__tabbed_1_5">Step 5</label><label for="__tabbed_1_6">Step 6</label><label for="__tabbed_1_7">Step 7</label></div>
|
||
<div class="tabbed-content">
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step1" src="../binary_search.assets/binary_search_step1.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step2" src="../binary_search.assets/binary_search_step2.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step3" src="../binary_search.assets/binary_search_step3.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step4" src="../binary_search.assets/binary_search_step4.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step5" src="../binary_search.assets/binary_search_step5.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step6" src="../binary_search.assets/binary_search_step6.png" /></p>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<p><img alt="binary_search_step7" src="../binary_search.assets/binary_search_step7.png" /></p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>二分查找 “双闭区间” 表示下的代码如下所示。</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label></div>
|
||
<div class="tabbed-content">
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 二分查找（双闭区间） */</span><span class="w"></span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="c1">// 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span><span class="w"></span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i > j 时为空）</span><span class="w"></span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span><span class="w"></span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span><span class="w"></span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 二分查找（双闭区间） */</span><span class="w"></span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w"> </span><span class="c1">// 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i > j 时为空）</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="sd">""" 二分查找（双闭区间） """</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="k">def</span> <span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a> <span class="c1"># 初始化双闭区间 [0, n-1] ，即 i, j 分别指向数组首元素、尾元素</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="k">while</span> <span class="n">i</span> <span class="o"><=</span> <span class="n">j</span><span class="p">:</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># 计算中点索引 m</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o"><</span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j] 中</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">></span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [i, m-1] 中</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="k">else</span><span class="p">:</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">return</span> <span class="n">m</span> <span class="c1"># 找到目标元素，返回其索引</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="c1"># 未找到目标元素，返回 -1</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearch1</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"></span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p">></span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
||
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"></span>
|
||
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<h3 id="_4">“左闭右开” 实现<a class="headerlink" href="#_4" title="Permanent link">¶</a></h3>
|
||
<p>当然，我们也可以使用 “左闭右开” 的表示方法，写出相同功能的二分查找代码。</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="3:8"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label></div>
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||
<div class="tabbed-content">
|
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|
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<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
||
<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch1</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
||
<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
||
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
||
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
||
<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
||
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
||
<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
||
<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
||
<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch1</span><span class="p">(</span><span class="n">vector</span><span class="o"><</span><span class="kt">int</span><span class="o">>&</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span>
|
||
<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"></span>
|
||
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span>
|
||
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
|
||
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
|
||
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
|
||
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
|
||
<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
|
||
<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="sd">""" 二分查找（左闭右开） """</span>
|
||
<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">def</span> <span class="nf">binary_search1</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">):</span>
|
||
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a> <span class="c1"># 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span>
|
||
<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
|
||
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a> <span class="c1"># 循环，当搜索区间为空时跳出（当 i = j 时为空）</span>
|
||
<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a> <span class="k">while</span> <span class="n">i</span> <span class="o"><</span> <span class="n">j</span><span class="p">:</span>
|
||
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a> <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># 计算中点索引 m</span>
|
||
<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a> <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o"><</span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j) 中</span>
|
||
<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
|
||
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a> <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">></span> <span class="n">target</span><span class="p">:</span> <span class="c1"># 此情况说明 target 在区间 [i, m) 中</span>
|
||
<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>
|
||
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a> <span class="k">else</span><span class="p">:</span> <span class="c1"># 找到目标元素，返回其索引</span>
|
||
<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a> <span class="k">return</span> <span class="n">m</span>
|
||
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span> <span class="c1"># 未找到目标元素，返回 -1</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 二分查找（左闭右开） */</span><span class="w"></span>
|
||
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearch1</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w"> </span><span class="c1">// 初始化左闭右开 [0, n) ，即 i, j 分别指向数组首元素、尾元素+1</span><span class="w"></span>
|
||
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"></span>
|
||
<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span><span class="w"></span>
|
||
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
|
||
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span><span class="w"></span>
|
||
<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p"><</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span><span class="w"></span>
|
||
<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
|
||
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="p">></span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span><span class="w"></span>
|
||
<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
||
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span><span class="w"></span>
|
||
<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="w"></span>
|
||
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="p">}</span><span class="w"></span>
|
||
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span><span class="w"></span>
|
||
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"></span>
|
||
<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="p">}</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<h3 id="_5">两种表示对比<a class="headerlink" href="#_5" title="Permanent link">¶</a></h3>
|
||
<p>对比下来，两种表示的代码写法有以下不同点：</p>
|
||
<div class="center-table">
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>表示方法</th>
|
||
<th>初始化指针</th>
|
||
<th>缩小区间</th>
|
||
<th>循环终止条件</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>双闭区间 <span class="arithmatex">\([0, n-1]\)</span></td>
|
||
<td><span class="arithmatex">\(i = 0\)</span> , <span class="arithmatex">\(j = n-1\)</span></td>
|
||
<td><span class="arithmatex">\(i = m + 1\)</span> , <span class="arithmatex">\(j = m - 1\)</span></td>
|
||
<td><span class="arithmatex">\(i > j\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>左闭右开 <span class="arithmatex">\([0, n)\)</span></td>
|
||
<td><span class="arithmatex">\(i = 0\)</span> , <span class="arithmatex">\(j = n\)</span></td>
|
||
<td><span class="arithmatex">\(i = m + 1\)</span> , <span class="arithmatex">\(j = m\)</span></td>
|
||
<td><span class="arithmatex">\(i = j\)</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
<p>观察发现，在 “双闭区间” 表示中，由于对左右两边界的定义是相同的，因此缩小区间的 <span class="arithmatex">\(i\)</span> , <span class="arithmatex">\(j\)</span> 处理方法也是对称的，这样更不容易出错。综上所述，<strong>建议你采用 “双闭区间” 的写法。</strong></p>
|
||
<h3 id="_6">大数越界处理<a class="headerlink" href="#_6" title="Permanent link">¶</a></h3>
|
||
<p>当数组长度很大时，加法 <span class="arithmatex">\(i + j\)</span> 的结果有可能会超出 <code>int</code> 类型的取值范围。在此情况下，我们需要换一种计算中点的写法。</p>
|
||
<div class="tabbed-set tabbed-alternate" data-tabs="4:8"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label></div>
|
||
<div class="tabbed-content">
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span><span class="w"></span>
|
||
<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="c1">// 更换为此写法则不会越界</span><span class="w"></span>
|
||
<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span>
|
||
<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
||
<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="c1">// 更换为此写法则不会越界</span>
|
||
<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1"># Python 中的数字理论上可以无限大（取决于内存大小）</span>
|
||
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1"># 因此无需考虑大数越界问题</span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span><span class="w"></span>
|
||
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
|
||
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="c1">// 更换为此写法则不会越界</span><span class="w"></span>
|
||
<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
<div class="tabbed-block">
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a>
|
||
</code></pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<h2 id="_7">复杂度分析<a class="headerlink" href="#_7" title="Permanent link">¶</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="_8">优缺点<a class="headerlink" href="#_8" title="Permanent link">¶</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>
|
||
<li><strong>二分查找不需要额外空间。</strong> 相对于借助额外数据结构来实现查找的算法来说，其更加节约空间使用。</li>
|
||
</ul>
|
||
<p>但并不意味着所有情况下都应使用二分查找，这是因为：</p>
|
||
<ul>
|
||
<li><strong>二分查找仅适用于有序数据。</strong> 如果输入数据是无序的，为了使用二分查找而专门执行数据排序，那么是得不偿失的，因为排序算法的时间复杂度一般为 <span class="arithmatex">\(O(n \log n)\)</span> ，比线性查找和二分查找都更差。再例如，对于频繁插入元素的场景，为了保持数组的有序性，需要将元素插入到特定位置，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，也是非常昂贵的。</li>
|
||
<li><strong>二分查找仅适用于数组。</strong> 由于在二分查找中，访问索引是 ”非连续“ 的，因此链表或者基于链表实现的数据结构都无法使用。</li>
|
||
<li><strong>在小数据量下，线性查找的性能更好。</strong> 在线性查找中，每轮只需要 1 次判断操作；而在二分查找中，需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法（减法），共 4 ~ 6 个单元操作；因此，在数据量 <span class="arithmatex">\(n\)</span> 较小时，线性查找反而比二分查找更快。</li>
|
||
</ul>
|
||
|
||
|
||
|
||
|
||
|
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