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@ -920,28 +920,14 @@
<li class="md-nav__item"> <li class="md-nav__item">
<a href="#611" class="md-nav__link"> <a href="#611" class="md-nav__link">
6.1.1. &nbsp; 双闭区间实现 6.1.1. &nbsp; 区间表示方法
</a> </a>
</li> </li>
<li class="md-nav__item"> <li class="md-nav__item">
<a href="#612" class="md-nav__link"> <a href="#612" class="md-nav__link">
6.1.2. &nbsp; 左闭右开实现 6.1.2. &nbsp; 优点与局限性
</a>
</li>
<li class="md-nav__item">
<a href="#613" class="md-nav__link">
6.1.3. &nbsp; 复杂度分析
</a>
</li>
<li class="md-nav__item">
<a href="#614" class="md-nav__link">
6.1.4. &nbsp; 优点与局限性
</a> </a>
</li> </li>
@ -1876,28 +1862,14 @@
<li class="md-nav__item"> <li class="md-nav__item">
<a href="#611" class="md-nav__link"> <a href="#611" class="md-nav__link">
6.1.1. &nbsp; 双闭区间实现 6.1.1. &nbsp; 区间表示方法
</a> </a>
</li> </li>
<li class="md-nav__item"> <li class="md-nav__item">
<a href="#612" class="md-nav__link"> <a href="#612" class="md-nav__link">
6.1.2. &nbsp; 左闭右开实现 6.1.2. &nbsp; 优点与局限性
</a>
</li>
<li class="md-nav__item">
<a href="#613" class="md-nav__link">
6.1.3. &nbsp; 复杂度分析
</a>
</li>
<li class="md-nav__item">
<a href="#614" class="md-nav__link">
6.1.4. &nbsp; 优点与局限性
</a> </a>
</li> </li>
@ -1926,22 +1898,31 @@
<h1 id="61">6.1. &nbsp; 二分查找<a class="headerlink" href="#61" title="Permanent link">&para;</a></h1> <h1 id="61">6.1. &nbsp; 二分查找<a class="headerlink" href="#61" title="Permanent link">&para;</a></h1>
<p>「二分查找 Binary Search」利用数据的有序性，通过每轮减少一半搜索范围来定位目标元素。</p> <p>「二分查找 Binary Search」是一种基于分治思想的高效搜索算法。它利用数据的有序性，每轮减少一半搜索范围，实现定位目标元素。</p>
<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> ，元素按从小到大的顺序排列。数组索引的取值范围为：</p> <p>我们先来求解一个简单的二分查找问题。</p>
<div class="arithmatex">\[ <div class="admonition question">
0, 1, 2, \cdots, n-1 <p class="admonition-title">给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> ，元素按从小到大的顺序排列。查找并返回元素 <code>target</code> 在该数组中的索引。若数组中不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。数组中不包含重复元素。</p>
\]</div> </div>
<p>我们通常使用以下两种方法来表示这个取值范围：</p> <p>该数组的索引范围可以使用区间 <span class="arithmatex">\([0, n - 1]\)</span> 来表示。其中，<strong>中括号表示“闭区间”，即包含边界值本身</strong>。在该表示下，区间 <span class="arithmatex">\([i, j]\)</span> 在 <span class="arithmatex">\(i = j\)</span> 时仍包含一个元素，在 <span class="arithmatex">\(i &gt; j\)</span> 时为空区间。</p>
<p>接下来，我们基于上述区间定义实现二分查找。先初始化指针 <span class="arithmatex">\(i = 0\)</span> 和 <span class="arithmatex">\(j = n - 1\)</span> ，分别指向数组首元素和尾元素。之后循环执行以下两个步骤:</p>
<ol> <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；在此方法下，区间 <span class="arithmatex">\([i, i]\)</span> 仍包含 <span class="arithmatex">\(1\)</span> 个元素；</li> <li>计算中点索引 <span class="arithmatex">\(m = \lfloor {(i + j) / 2} \rfloor\)</span> ，其中 <span class="arithmatex">\(\lfloor \space \rfloor\)</span> 表示向下取整操作。</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；在此方法下，区间 <span class="arithmatex">\([i, i)\)</span> 不包含元素；</li> <li>根据 <code>nums[m]</code> 和 <code>target</code> 缩小搜索区间，分为三种情况：<ol>
<li>当 <code>nums[m] &lt; target</code> 时，说明 <code>target</code> 在区间 <span class="arithmatex">\([m + 1, j]\)</span> 中，因此执行 <span class="arithmatex">\(i = m + 1\)</span> ；</li>
<li>当 <code>nums[m] &gt; target</code> 时，说明 <code>target</code> 在区间 <span class="arithmatex">\([i, m - 1]\)</span> 中，因此执行 <span class="arithmatex">\(j = m - 1\)</span> ；</li>
<li>当 <code>nums[m] = target</code> 时，说明找到目标元素，直接返回索引 <span class="arithmatex">\(m\)</span> 即可；</li>
</ol> </ol>
<h2 id="611">6.1.1. &nbsp; 双闭区间实现<a class="headerlink" href="#611" title="Permanent link">&para;</a></h2> </li>
<p>首先，我们采用“双闭区间”表示法，在数组 <code>nums</code> 中查找目标元素 <code>target</code> 的对应索引。</p> </ol>
<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">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label></div> <p><strong>若数组不包含目标元素，搜索区间最终会缩小为空</strong>，即达到 <span class="arithmatex">\(i &gt; j\)</span> 。此时，终止循环并返回 <span class="arithmatex">\(-1\)</span> 即可。</p>
<p>为了更清晰地表示区间，我们在下图中以折线图的形式表示数组。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:8"><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" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;0&gt;</label><label for="__tabbed_1_2">&lt;1&gt;</label><label for="__tabbed_1_3">&lt;2&gt;</label><label for="__tabbed_1_4">&lt;3&gt;</label><label for="__tabbed_1_5">&lt;4&gt;</label><label for="__tabbed_1_6">&lt;5&gt;</label><label for="__tabbed_1_7">&lt;6&gt;</label><label for="__tabbed_1_8">&lt;7&gt;</label></div>
<div class="tabbed-content"> <div class="tabbed-content">
<div class="tabbed-block"> <div class="tabbed-block">
<p><img alt="二分查找步骤" src="../binary_search.assets/binary_search_step1.png" /></p> <p><img alt="二分查找步骤" src="../binary_search.assets/binary_search_step0.png" /></p>
</div>
<div class="tabbed-block">
<p><img alt="binary_search_step1" src="../binary_search.assets/binary_search_step1.png" /></p>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<p><img alt="binary_search_step2" src="../binary_search.assets/binary_search_step2.png" /></p> <p><img alt="binary_search_step2" src="../binary_search.assets/binary_search_step2.png" /></p>
@ -1963,7 +1944,8 @@
</div> </div>
</div> </div>
</div> </div>
<p>二分查找在“双闭区间”表示下的代码如下所示。</p> <p>值得注意的是，<strong>当数组长度 <span class="arithmatex">\(n\)</span> 很大时，加法 <span class="arithmatex">\(i + j\)</span> 的结果可能会超出 <code>int</code> 类型的取值范围</strong>。为了避免大数越界，我们通常采用公式 <span class="arithmatex">\(m = \lfloor {i + (j - i) / 2} \rfloor\)</span> 来计算中点。</p>
<p>有趣的是，理论上 Python 的数字可以无限大（取决于内存大小），因此无需考虑大数越界问题。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" 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><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div> <div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" 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><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div>
<div class="tabbed-content"> <div class="tabbed-content">
<div class="tabbed-block"> <div class="tabbed-block">
@ -1973,7 +1955,7 @@
<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> <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>
<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 &gt; j 时为空）</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 &gt; j 时为空）</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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</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> <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="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><span class="c1">// 计算中点索引 m</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">&lt;</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-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">&lt;</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-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> <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>
<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">&gt;</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-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">&gt;</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>
@ -1993,7 +1975,7 @@
<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> <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>
<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 &gt; j 时为空）</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 &gt; 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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</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-1-7" 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="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><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">&lt;</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-1-8" 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">&lt;</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> <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>
<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">&gt;</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-1-10" 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">&gt;</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>
@ -2011,15 +1993,16 @@
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;二分查找（双闭区间）&quot;&quot;&quot;</span> <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;二分查找（双闭区间）&quot;&quot;&quot;</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-2-3" 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-2-4" 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">&lt;=</span> <span class="n">j</span><span class="p">:</span> <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="c1"># 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</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-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span><span class="p">:</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">&lt;</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-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></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-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-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></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">&lt;</span> <span class="n">target</span><span class="p">:</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">&gt;</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-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j] 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-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></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">&gt;</span> <span class="n">target</span><span class="p">:</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-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span> <span class="c1"># 此情况说明 target 在区间 [i, m-1] 中</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-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="k">else</span><span class="p">:</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> <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></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-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></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> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
@ -2029,7 +2012,7 @@
<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="o">-</span><span class="mi">1</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="o">-</span><span class="mi">1</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 &gt; j 时为空）</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 &gt; j 时为空）</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="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</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="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</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> <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="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="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</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">&lt;</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> <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">&lt;</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>
<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> <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>
<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">&gt;</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-1] 中</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">&gt;</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-1] 中</span>
@ -2052,7 +2035,7 @@
<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span> <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m ，使用 parseInt() 向下取整</span> <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m ，使用 parseInt() 向下取整</span>
<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</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="mf">2</span><span class="p">);</span> <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</span><span class="p">(</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span> <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span> <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span>
<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span> <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@ -2075,7 +2058,7 @@
<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span> <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m</span> <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</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="mf">2</span><span class="p">);</span> <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span> <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span>
<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span> <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@ -2098,7 +2081,7 @@
<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></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">len</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span> <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></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">len</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span> <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></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-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></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="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><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></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">&lt;</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-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></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">&lt;</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-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></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> <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></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>
<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></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">&gt;</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-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></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">&gt;</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>
@ -2118,7 +2101,7 @@
<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></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="m">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">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span> <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></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="m">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">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span> <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></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="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span> <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></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="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="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></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">&lt;</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-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></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">&lt;</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-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></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="m">1</span><span class="p">;</span> <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></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="m">1</span><span class="p">;</span>
<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="nf">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">&gt;</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-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="nf">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">&gt;</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>
@ -2139,7 +2122,7 @@
<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">j</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span> <span class="o">-</span> <span class="mi">1</span> <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a> <span class="kd">var</span> <span class="nv">j</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span> <span class="o">-</span> <span class="mi">1</span>
<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span> <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a> <span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span> <span class="p">{</span> <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span> <span class="p">{</span>
<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</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-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a> <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">i</span> <span class="o">+</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</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-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></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">&lt;</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-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></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">&lt;</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-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="n">i</span> <span class="p">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span> <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a> <span class="n">i</span> <span class="p">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="p">}</span> <span class="k">else</span> <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">&gt;</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-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a> <span class="p">}</span> <span class="k">else</span> <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">&gt;</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>
@ -2161,7 +2144,7 @@
<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">j</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</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">items</span><span class="p">.</span><span class="n">len</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span> <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">j</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</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">items</span><span class="p">.</span><span class="n">len</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w"> </span><span class="c1">// 循环，当搜索区间为空时跳出（当 i &gt; 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="c1">// 循环，当搜索区间为空时跳出（当 i &gt; j 时为空）</span>
<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span> <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w"> </span><span class="kr">var</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="kr">var</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><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></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">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</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><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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</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><span class="c1">// 此情况说明 target 在区间 [m+1, j] 中</span>
<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></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> <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></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>
<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></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="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</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><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span> <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></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="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</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><span class="c1">// 此情况说明 target 在区间 [i, m-1] 中</span>
@ -2177,325 +2160,232 @@
</div> </div>
</div> </div>
</div> </div>
<p>需要注意的是，<strong>当数组长度非常大时，加法 <span class="arithmatex">\(i + j\)</span> 的结果可能会超出 <code>int</code> 类型的取值范围</strong>。在这种情况下，我们需要采用一种更安全的计算中点的方法。</p> <p>时间复杂度为 <span class="arithmatex">\(O(\log n)\)</span> 。每轮缩小一半区间，因此二分循环次数为 <span class="arithmatex">\(\log_2 n\)</span> 。</p>
<p>空间复杂度为 <span class="arithmatex">\(O(1)\)</span> 。指针 <code>i</code> , <code>j</code> 使用常数大小空间。</p>
<h2 id="611">6.1.1. &nbsp; 区间表示方法<a class="headerlink" href="#611" title="Permanent link">&para;</a></h2>
<p>除了上述的双闭区间外，常见的区间表示还有“左闭右开”区间，定义为 <span class="arithmatex">\([0, n)\)</span> ，即左边界包含自身，右边界不包含自身。在该表示下，区间 <span class="arithmatex">\([i, j]\)</span> 在 <span class="arithmatex">\(i = j\)</span> 时为空。</p>
<p>我们可以基于该表示实现具有相同功能的二分查找算法。</p>
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<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span> <div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 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="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> <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</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>
<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="c1">// 更换为此写法则不会越界</span> <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></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-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></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> <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></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>
<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></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-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></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="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><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="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">&lt;</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-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></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>
<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></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">&gt;</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-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></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>
<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></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-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></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>
<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span> <div class="highlight"><span class="filename">binary_search.cpp</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>
<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></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> <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</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>
<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="c1">// 更换为此写法则不会越界</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>
<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></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> <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></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>
<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>
<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></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="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><span class="c1">// 计算中点索引 m</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="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">&lt;</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-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></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>
<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></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">&gt;</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-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></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>
<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></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-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="n">m</span><span class="p">;</span>
<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></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> <div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="k">def</span> <span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># 因此无需考虑大数越界问题</span> <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;二分查找（左闭右开）&quot;&quot;&quot;</span>
<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></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-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></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-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></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-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></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-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></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">&lt;</span> <span class="n">target</span><span class="p">:</span>
<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a> <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span> <span class="c1"># 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></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">&gt;</span> <span class="n">target</span><span class="p">:</span>
<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="c1"># 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a> <span class="k">else</span><span class="p">:</span>
<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></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-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></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> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span> <div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></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> <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchLCRO</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>
<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="c1">// 更换为此写法则不会越界</span> <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></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-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></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> <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></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>
<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></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-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></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">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="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="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></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">&lt;</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>
<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></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>
<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></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">&gt;</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>
<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></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>
<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></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>
<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="c1">// (i + j) 有可能超出 int 的取值范围</span> <div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</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="mf">2</span><span class="p">);</span> <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="c1">// 更换为此写法则不会越界</span> <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></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-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</span><span class="p">(</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="mf">2</span><span class="p">);</span> <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></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-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m ，使用 parseInt() 向下取整</span>
<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</span><span class="p">(</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></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="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
<a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <div class="tabbed-block">
<div class="highlight"><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="c1">// (i + j) 有可能超出 Number 的取值范围</span> <div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</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="mf">2</span><span class="p">);</span> <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="c1">// 更换为此写法则不会越界</span> <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></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-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</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="mf">2</span><span class="p">);</span> <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></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-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></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="p">(</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="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></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>
<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <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> <div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 二分查找（左闭右开） */</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> <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="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</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">len</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>
<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> <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></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-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> <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></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">len</span><span class="p">;</span>
<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></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-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></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="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><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></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">&lt;</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-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></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>
<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></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">&gt;</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-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></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>
<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></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-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
<a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <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> <div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 二分查找（左闭右开） */</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="m">2</span><span class="p">;</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="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</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>
<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-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></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-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="m">2</span><span class="p">;</span> <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></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="m">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">Length</span><span class="p">;</span>
<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></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-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></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="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="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></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">&lt;</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-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></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="m">1</span><span class="p">;</span>
<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="nf">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">&gt;</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-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></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>
<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></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-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <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">// (i + j) 有可能超出 int 的取值范围</span> <div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">let</span> <span class="nv">m</span> <span class="p">=</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> <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span> <span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="c1">// 更换为此写法则不会越界</span> <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></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-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">i</span> <span class="o">+</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span> <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a> <span class="kd">var</span> <span class="nv">i</span> <span class="p">=</span> <span class="mi">0</span>
<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a> <span class="kd">var</span> <span class="nv">j</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></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-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="p">{</span>
<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a> <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</span> <span class="n">i</span> <span class="o">+</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</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-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></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">&lt;</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-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a> <span class="n">i</span> <span class="p">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a> <span class="p">}</span> <span class="k">else</span> <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">&gt;</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-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a> <span class="n">j</span> <span class="p">=</span> <span class="n">m</span>
<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a> <span class="p">}</span> <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-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a> <span class="k">return</span> <span class="n">m</span>
<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a> <span class="p">}</span>
<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a> <span class="p">}</span>
<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a> <span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
<div class="tabbed-block"> <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> <div class="highlight"><span class="filename">binary_search.zig</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="c1">// 二分查找（左闭右开）</span>
<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">binarySearchLCRO</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">T</span><span class="o">:</span><span class="w"> </span><span class="kt">type</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">:</span><span class="w"> </span><span class="n">std</span><span class="p">.</span><span class="n">ArrayList</span><span class="p">(</span><span class="n">T</span><span class="p">),</span><span class="w"> </span><span class="n">target</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="p">)</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></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-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">j</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</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">items</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></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-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w"> </span><span class="kr">var</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><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></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">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</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><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></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>
<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></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="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</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><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></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>
<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></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>
<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">);</span>
<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></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>
<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="p">}</span>
</code></pre></div> </code></pre></div>
</div> </div>
</div> </div>
</div> </div>
<h2 id="612">6.1.2. &nbsp; 左闭右开实现<a class="headerlink" href="#612" title="Permanent link">&para;</a></h2> <p>如下图所示，在两种区间表示下，二分查找算法的初始化、循环条件和缩小区间操作皆有所不同。</p>
<p>我们可以采用“左闭右开”的表示法，编写具有相同功能的二分查找代码。</p> <p>在“双闭区间”表示法中，由于左右边界都被定义为闭区间，因此指针 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 缩小区间操作也是对称的。这样更不容易出错。因此，<strong>我们通常采用“双闭区间”的写法</strong>。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="4:10"><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" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" 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><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label></div> <p><img alt="两种区间定义" src="../binary_search.assets/binary_search_ranges.png" /></p>
<div class="tabbed-content"> <p align="center"> Fig. 两种区间定义 </p>
<div class="tabbed-block">
<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 二分查找（左闭右开） */</span> <h2 id="612">6.1.2. &nbsp; 优点与局限性<a class="headerlink" href="#612" title="Permanent link">&para;</a></h2>
<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</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>
<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></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-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></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>
<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></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-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></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-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></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">&lt;</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-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></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>
<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></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">&gt;</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-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></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>
<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></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-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></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>
<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="p">}</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-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</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>
<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></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-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></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>
<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></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-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></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-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></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">&lt;</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-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></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>
<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></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">&gt;</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-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></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>
<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></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-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="p">}</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-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="k">def</span> <span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;二分查找（左闭右开）&quot;&quot;&quot;</span>
<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></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-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></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-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></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-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></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-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></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">&lt;</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-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></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-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></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">&gt;</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-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a> <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>
<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></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-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a> <span class="k">return</span> <span class="n">m</span>
<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></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-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchLCRO</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>
<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></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-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></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>
<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></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-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></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">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></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>
<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></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">&lt;</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>
<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></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>
<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></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">&gt;</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>
<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></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>
<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></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>
<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="p">}</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-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></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-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></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-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m ，使用 parseInt() 向下取整</span>
<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></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="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="p">}</span>
</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-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></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-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></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-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w"> </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</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="mf">2</span><span class="p">);</span>
<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</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="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></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="p">(</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="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w"> </span><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></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>
<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w"> </span><span class="c1">// 找到目标元素，返回其索引</span>
<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="p">}</span>
</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-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</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">len</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>
<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></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-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></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">len</span><span class="p">;</span>
<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></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-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></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-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></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">&lt;</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-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></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>
<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></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">&gt;</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-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></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>
<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></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-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a><span class="p">}</span>
</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-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</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>
<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></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-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></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="m">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">Length</span><span class="p">;</span>
<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></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-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></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">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></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="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 计算中点索引 m</span>
<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></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">&lt;</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-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></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="m">1</span><span class="p">;</span>
<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="nf">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">&gt;</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-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></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>
<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></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-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
<a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
<a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kd">func</span> <span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">-&gt;</span> <span class="nb">Int</span> <span class="p">{</span>
<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></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-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a> <span class="kd">var</span> <span class="nv">i</span> <span class="p">=</span> <span class="mi">0</span>
<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a> <span class="kd">var</span> <span class="nv">j</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span>
<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></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-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a> <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="p">{</span>
<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a> <span class="kd">let</span> <span class="nv">m</span> <span class="p">=</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-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></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">&lt;</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-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a> <span class="n">i</span> <span class="p">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>
<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a> <span class="p">}</span> <span class="k">else</span> <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">&gt;</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-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a> <span class="n">j</span> <span class="p">=</span> <span class="n">m</span>
<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a> <span class="p">}</span> <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-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a> <span class="k">return</span> <span class="n">m</span>
<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a> <span class="p">}</span>
<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a> <span class="p">}</span>
<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a> <span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a> <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
<a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">binary_search.zig</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="c1">// 二分查找（左闭右开）</span>
<a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">binarySearchLCRO</span><span class="p">(</span><span class="kr">comptime</span><span class="w"> </span><span class="n">T</span><span class="o">:</span><span class="w"> </span><span class="kt">type</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">:</span><span class="w"> </span><span class="n">std</span><span class="p">.</span><span class="n">ArrayList</span><span class="p">(</span><span class="n">T</span><span class="p">),</span><span class="w"> </span><span class="n">target</span><span class="o">:</span><span class="w"> </span><span class="n">T</span><span class="p">)</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></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-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">i</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w"> </span><span class="kr">var</span><span class="w"> </span><span class="n">j</span><span class="o">:</span><span class="w"> </span><span class="kt">usize</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">items</span><span class="p">.</span><span class="n">len</span><span class="p">;</span>
<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></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-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></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">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w"> </span><span class="kr">var</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-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></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">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</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><span class="c1">// 此情况说明 target 在区间 [m+1, j) 中</span>
<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></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>
<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></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="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">items</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</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><span class="c1">// 此情况说明 target 在区间 [i, m) 中</span>
<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></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>
<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></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>
<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">T</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">);</span>
<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w"> </span><span class="c1">// 未找到目标元素，返回 -1</span>
<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></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>
<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<p>对比这两种代码写法，我们可以发现以下不同点：</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">\(i &gt; j\)</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>在“双闭区间”表示法中，由于对左右两边界的定义相同，因此缩小区间的 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 的处理方法也是对称的，这样更不容易出错。因此，<strong>建议采用“双闭区间”的写法</strong>。</p>
<h2 id="613">6.1.3. &nbsp; 复杂度分析<a class="headerlink" href="#613" 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="614">6.1.4. &nbsp; 优点与局限性<a class="headerlink" href="#614" title="Permanent link">&para;</a></h2>
<p>二分查找效率很高，主要体现在：</p> <p>二分查找效率很高，主要体现在：</p>
<ul> <ul>
<li><strong>二分查找的时间复杂度较低</strong>。对数阶在大数据量情况下具有显著优势。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 <span class="arithmatex">\(\log_2 2^{20} = 20\)</span> 轮循环。</li> <li><strong>二分查找的时间复杂度较低</strong>。对数阶在大数据量情况下具有显著优势。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 <span class="arithmatex">\(\log_2 2^{20} = 20\)</span> 轮循环。</li>

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