deploy

chapter_searching/binary_search/index.html

@@ -3382,10 +3382,10 @@
 
 <!-- Page content -->
 <h1 id="101">10.1 &nbsp; 二分查找<a class="headerlink" href="#101" title="Permanent link">&para;</a></h1>
-<p>「二分查找 binary search」是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮减少一半搜索范围，直至找到目标元素或搜索区间为空为止。</p>
+<p>「二分查找 binary search」是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮缩小一半搜索范围，直至找到目标元素或搜索区间为空为止。</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，元素按从小到大的顺序排列，数组不包含重复元素。请查找并返回元素 <code>target</code> 在该数组中的索引。若数组不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。</p>
+<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的数组 <code>nums</code> ，元素按从小到大的顺序排列且不重复。请查找并返回元素 <code>target</code> 在该数组中的索引。若数组不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。示例如图 10-1 所示。</p>
 </div>
 <p><a class="glightbox" href="../binary_search.assets/binary_search_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二分查找示例数据" class="animation-figure" src="../binary_search.assets/binary_search_example.png" /></a></p>
 <p align="center"> 图 10-1 &nbsp; 二分查找示例数据 </p>
@@ -3430,6 +3430,7 @@
 <p align="center"> 图 10-2 &nbsp; 二分查找流程 </p>
 
 <p>值得注意的是，由于 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 都是 <code>int</code> 类型，<strong>因此 <span class="arithmatex">\(i + j\)</span> 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>代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:12"><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" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -3694,14 +3695,14 @@
 <p><strong>时间复杂度 <span class="arithmatex">\(O(\log n)\)</span></strong> ：在二分循环中，区间每轮缩小一半，循环次数为 <span class="arithmatex">\(\log_2 n\)</span> 。</p>
 <p><strong>空间复杂度 <span class="arithmatex">\(O(1)\)</span></strong> ：指针 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 使用常数大小空间。</p>
 <h2 id="1011">10.1.1 &nbsp; 区间表示方法<a class="headerlink" href="#1011" 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>
+<p>除了上述双闭区间外，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>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <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="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-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;二分查找（左闭右开区间）&quot;&quot;&quot;</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a>    <span class="c1"># 初始化左闭右开区间 [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>
@@ -3716,9 +3717,9 @@
 </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-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.cpp</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（左闭右开区间） */</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></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-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-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-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">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>
@@ -3736,9 +3737,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.java</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>
+<div class="highlight"><span class="filename">binary_search.java</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（左闭右开区间） */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></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-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-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></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-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></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>
@@ -3756,9 +3757,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 二分查找（左闭右开区间） */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></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-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-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></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-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></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>
@@ -3776,9 +3777,9 @@
 </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-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.go</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（左闭右开区间） */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></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-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-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-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">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>
@@ -3797,9 +3798,9 @@
 </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-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.swift</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（左闭右开区间） */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></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-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></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-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="c1">// 初始化左闭右开区间 [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="kd">var</span> <span class="nv">i</span> <span class="p">=</span> <span class="mi">0</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></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-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>    <span class="c1">// 循环，当搜索区间为空时跳出（当 i = j 时为空）</span>
@@ -3819,9 +3820,9 @@
 </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-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.js</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（左闭右开区间） */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></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-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></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-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></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-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></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>
@@ -3843,9 +3844,9 @@
 </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-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 二分查找（左闭右开区间） */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></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-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-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></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-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>
@@ -3870,7 +3871,7 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search.dart</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（左闭右开区间） */</span>
 <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="n">binarySearchLCRO</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</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-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">  </span><span class="c1">// 初始化左闭右开区间 [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="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-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>
@@ -3892,9 +3893,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.rs</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>
+<div class="highlight"><span class="filename">binary_search.rs</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（左闭右开区间） */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="k">fn</span> <span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span>: <span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span>-&gt; <span class="kt">i32</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-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="kd">let</span><span class="w"> </span><span class="k">mut</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>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</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">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></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>
@@ -3914,9 +3915,9 @@
 </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-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 二分查找（左闭右开） */</span>
+<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 二分查找（左闭右开区间） */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></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-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></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-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></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-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></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>
@@ -3934,9 +3935,9 @@
 </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-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 二分查找（左闭右开）</span>
+<div class="highlight"><span class="filename">binary_search.zig</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 二分查找（左闭右开区间）</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></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-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-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// 初始化左闭右开区间 [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="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-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></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-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></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>
@@ -3958,7 +3959,7 @@
 </div>
 </div>
 <p>如图 10-3 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>由于“双闭区间”表示中的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，<strong>因此一般建议采用“双闭区间”的写法</strong>。</p>
+<p>由于“双闭区间”表示中的左右边界都被定义为闭区间，因此通过指针 <span class="arithmatex">\(i\)</span> 和指针 <span class="arithmatex">\(j\)</span> 缩小区间的操作也是对称的。这样更不容易出错，<strong>因此一般建议采用“双闭区间”的写法</strong>。</p>
 <p><a class="glightbox" href="../binary_search.assets/binary_search_ranges.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="两种区间定义" class="animation-figure" src="../binary_search.assets/binary_search_ranges.png" /></a></p>
 <p align="center"> 图 10-3 &nbsp; 两种区间定义 </p>
 
@@ -3972,7 +3973,7 @@
 <ul>
 <li>二分查找仅适用于有序数据。若输入数据无序，为了使用二分查找而专门进行排序，得不偿失。因为排序算法的时间复杂度通常为 <span class="arithmatex">\(O(n \log n)\)</span> ，比线性查找和二分查找都更高。对于频繁插入元素的场景，为保持数组有序性，需要将元素插入到特定位置，时间复杂度为 <span class="arithmatex">\(O(n)\)</span> ，也是非常昂贵的。</li>
 <li>二分查找仅适用于数组。二分查找需要跳跃式（非连续地）访问元素，而在链表中执行跳跃式访问的效率较低，因此不适合应用在链表或基于链表实现的数据结构。</li>
-<li>小数据量下，线性查找性能更佳。在线性查找中，每轮只需要 1 次判断操作；而在二分查找中，需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法（减法），共 4 ~ 6 个单元操作；因此，当数据量 <span class="arithmatex">\(n\)</span> 较小时，线性查找反而比二分查找更快。</li>
+<li>小数据量下，线性查找性能更佳。在线性查找中，每轮只需 1 次判断操作；而在二分查找中，需要 1 次加法、1 次除法、1 ~ 3 次判断操作、1 次加法（减法），共 4 ~ 6 个单元操作；因此，当数据量 <span class="arithmatex">\(n\)</span> 较小时，线性查找反而比二分查找更快。</li>
 </ul>
 
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