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>
 
 <!-- Source file information -->

chapter_searching/binary_search_edge/index.html

@@ -3425,7 +3425,7 @@
 <h2 id="1031">10.3.1 &nbsp; 查找左边界<a class="headerlink" href="#1031" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> ，数组可能包含重复元素。请返回数组中最左一个元素 <code>target</code> 的索引。若数组中不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。</p>
+<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> ，其中可能包含重复元素。请返回数组中最左一个元素 <code>target</code> 的索引。若数组中不包含该元素，则返回 <span class="arithmatex">\(-1\)</span> 。</p>
 </div>
 <p>回忆二分查找插入点的方法，搜索完成后 <span class="arithmatex">\(i\)</span> 指向最左一个 <code>target</code> ，<strong>因此查找插入点本质上是在查找最左一个 <code>target</code> 的索引</strong>。</p>
 <p>考虑通过查找插入点的函数实现查找左边界。请注意，数组中可能不包含 <code>target</code> ，这种情况可能导致以下两种结果。</p>
@@ -3433,7 +3433,7 @@
 <li>插入点的索引 <span class="arithmatex">\(i\)</span> 越界。</li>
 <li>元素 <code>nums[i]</code> 与 <code>target</code> 不相等。</li>
 </ul>
-<p>当遇到以上两种情况时，直接返回 <span class="arithmatex">\(-1\)</span> 即可。</p>
+<p>当遇到以上两种情况时，直接返回 <span class="arithmatex">\(-1\)</span> 即可。代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><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" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -3595,7 +3595,7 @@
 </div>
 </div>
 <h2 id="1032">10.3.2 &nbsp; 查找右边界<a class="headerlink" href="#1032" title="Permanent link">&para;</a></h2>
-<p>那么如何查找最右一个 <code>target</code> 呢？最直接的方式是修改代码，替换在 <code>nums[m] == target</code> 情况下的指针收缩操作。代码在此省略，有兴趣的同学可以自行实现。</p>
+<p>那么如何查找最右一个 <code>target</code> 呢？最直接的方式是修改代码，替换在 <code>nums[m] == target</code> 情况下的指针收缩操作。代码在此省略，有兴趣的读者可以自行实现。</p>
 <p>下面我们介绍两种更加取巧的方法。</p>
 <h3 id="1">1. &nbsp; 复用查找左边界<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
 <p>实际上，我们可以利用查找最左元素的函数来查找最右元素，具体方法为：<strong>将查找最右一个 <code>target</code> 转化为查找最左一个 <code>target + 1</code></strong>。</p>
@@ -3603,7 +3603,7 @@
 <p><a class="glightbox" href="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="将查找右边界转化为查找左边界" class="animation-figure" src="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" /></a></p>
 <p align="center"> 图 10-7 &nbsp; 将查找右边界转化为查找左边界 </p>
 
-<p>请注意，返回的插入点是 <span class="arithmatex">\(i\)</span> ，因此需要将其减 <span class="arithmatex">\(1\)</span> ，从而获得 <span class="arithmatex">\(j\)</span> 。</p>
+<p>请注意，返回的插入点是 <span class="arithmatex">\(i\)</span> ，因此需要将其减 <span class="arithmatex">\(1\)</span> ，从而获得 <span class="arithmatex">\(j\)</span> ：</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">
@@ -3796,7 +3796,7 @@
 <p><a class="glightbox" href="../binary_search_edge.assets/binary_search_edge_by_element.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="将查找边界转化为查找元素" class="animation-figure" src="../binary_search_edge.assets/binary_search_edge_by_element.png" /></a></p>
 <p align="center"> 图 10-8 &nbsp; 将查找边界转化为查找元素 </p>
 
-<p>代码在此省略，值得注意以下两点。</p>
+<p>代码在此省略，以下两点值得注意。</p>
 <ul>
 <li>给定数组不包含小数，这意味着我们无须关心如何处理相等的情况。</li>
 <li>因为该方法引入了小数，所以需要将函数中的变量 <code>target</code> 改为浮点数类型。</li>

chapter_searching/binary_search_insertion/index.html

@@ -3382,21 +3382,21 @@
 
 <!-- Page content -->
 <h1 id="102">10.2 &nbsp; 二分查找插入点<a class="headerlink" href="#102" title="Permanent link">&para;</a></h1>
-<p>二分查找不仅可用于搜索目标元素，还具有许多变种问题，比如搜索目标元素的插入位置。</p>
+<p>二分查找不仅可用于搜索目标元素，还可用于解决许多变种问题，比如搜索目标元素的插入位置。</p>
 <h2 id="1021">10.2.1 &nbsp; 无重复元素的情况<a class="headerlink" href="#1021" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> 和一个元素 <code>target</code> ，数组不存在重复元素。现将 <code>target</code> 插入到数组 <code>nums</code> 中，并保持其有序性。若数组中已存在元素 <code>target</code> ，则插入到其左方。请返回插入后 <code>target</code> 在数组中的索引。</p>
+<p>给定一个长度为 <span class="arithmatex">\(n\)</span> 的有序数组 <code>nums</code> 和一个元素 <code>target</code> ，数组不存在重复元素。现将 <code>target</code> 插入数组 <code>nums</code> 中，并保持其有序性。若数组中已存在元素 <code>target</code> ，则插入到其左方。请返回插入后 <code>target</code> 在数组中的索引。</p>
 </div>
 <p><a class="glightbox" href="../binary_search_insertion.assets/binary_search_insertion_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="二分查找插入点示例数据" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_example.png" /></a></p>
 <p align="center"> 图 10-4 &nbsp; 二分查找插入点示例数据 </p>
 
-<p>如果想要复用上节的二分查找代码，则需要回答以下两个问题。</p>
+<p>如果想复用上一节的二分查找代码，则需要回答以下两个问题。</p>
 <p><strong>问题一</strong>：当数组中包含 <code>target</code> 时，插入点的索引是否是该元素的索引？</p>
 <p>题目要求将 <code>target</code> 插入到相等元素的左边，这意味着新插入的 <code>target</code> 替换了原来 <code>target</code> 的位置。也就是说，<strong>当数组包含 <code>target</code> 时，插入点的索引就是该 <code>target</code> 的索引</strong>。</p>
 <p><strong>问题二</strong>：当数组中不存在 <code>target</code> 时，插入点是哪个元素的索引？</p>
 <p>进一步思考二分查找过程：当 <code>nums[m] &lt; target</code> 时 <span class="arithmatex">\(i\)</span> 移动，这意味着指针 <span class="arithmatex">\(i\)</span> 在向大于等于 <code>target</code> 的元素靠近。同理，指针 <span class="arithmatex">\(j\)</span> 始终在向小于等于 <code>target</code> 的元素靠近。</p>
-<p>因此二分结束时一定有：<span class="arithmatex">\(i\)</span> 指向首个大于 <code>target</code> 的元素，<span class="arithmatex">\(j\)</span> 指向首个小于 <code>target</code> 的元素。<strong>易得当数组不包含 <code>target</code> 时，插入索引为 <span class="arithmatex">\(i\)</span></strong> 。</p>
+<p>因此二分结束时一定有：<span class="arithmatex">\(i\)</span> 指向首个大于 <code>target</code> 的元素，<span class="arithmatex">\(j\)</span> 指向首个小于 <code>target</code> 的元素。<strong>易得当数组不包含 <code>target</code> 时，插入索引为 <span class="arithmatex">\(i\)</span></strong> 。代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><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" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -3636,7 +3636,7 @@
 <p align="center"> 图 10-5 &nbsp; 线性查找重复元素的插入点 </p>
 
 <p>此方法虽然可用，但其包含线性查找，因此时间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。当数组中存在很多重复的 <code>target</code> 时，该方法效率很低。</p>
-<p>现考虑拓展二分查找代码。如图 10-6 所示，整体流程保持不变，每轮先计算中点索引 <span class="arithmatex">\(m\)</span> ，再判断 <code>target</code> 和 <code>nums[m]</code> 大小关系，分为以下几种情况。</p>
+<p>现考虑拓展二分查找代码。如图 10-6 所示，整体流程保持不变，每轮先计算中点索引 <span class="arithmatex">\(m\)</span> ，再判断 <code>target</code> 和 <code>nums[m]</code> 的大小关系，分为以下几种情况。</p>
 <ul>
 <li>当 <code>nums[m] &lt; target</code> 或 <code>nums[m] &gt; target</code> 时，说明还没有找到 <code>target</code> ，因此采用普通二分查找的缩小区间操作，<strong>从而使指针 <span class="arithmatex">\(i\)</span> 和 <span class="arithmatex">\(j\)</span> 向 <code>target</code> 靠近</strong>。</li>
 <li>当 <code>nums[m] == target</code> 时，说明小于 <code>target</code> 的元素在区间 <span class="arithmatex">\([i, m - 1]\)</span> 中，因此采用 <span class="arithmatex">\(j = m - 1\)</span> 来缩小区间，<strong>从而使指针 <span class="arithmatex">\(j\)</span> 向小于 <code>target</code> 的元素靠近</strong>。</li>

chapter_searching/replace_linear_by_hashing/index.html

@@ -3388,16 +3388,17 @@
 <p>给定一个整数数组 <code>nums</code> 和一个目标元素 <code>target</code> ，请在数组中搜索“和”为 <code>target</code> 的两个元素，并返回它们的数组索引。返回任意一个解即可。</p>
 </div>
 <h2 id="1041">10.4.1 &nbsp; 线性查找：以时间换空间<a class="headerlink" href="#1041" title="Permanent link">&para;</a></h2>
-<p>考虑直接遍历所有可能的组合。如图 10-9 所示，我们开启一个两层循环，在每轮中判断两个整数的和是否为 <code>target</code> ，若是则返回它们的索引。</p>
+<p>考虑直接遍历所有可能的组合。如图 10-9 所示，我们开启一个两层循环，在每轮中判断两个整数的和是否为 <code>target</code> ，若是，则返回它们的索引。</p>
 <p><a class="glightbox" href="../replace_linear_by_hashing.assets/two_sum_brute_force.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="线性查找求解两数之和" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_brute_force.png" /></a></p>
 <p align="center"> 图 10-9 &nbsp; 线性查找求解两数之和 </p>
 
+<p>代码如下所示：</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><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" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span> <span class="nf">two_sum_brute_force</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">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;方法一：暴力枚举&quot;&quot;&quot;</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</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><span class="p">):</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</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-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
@@ -3409,7 +3410,7 @@
 <div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><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="n">twoSumBruteForce</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-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</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-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</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">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -3424,7 +3425,7 @@
 <div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">twoSumBruteForce</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-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</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-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></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">i</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -3439,7 +3440,7 @@
 <div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">TwoSumBruteForce</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-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</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-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</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="p">(</span><span class="kt">int</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">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -3454,7 +3455,7 @@
 <div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">twoSumBruteForce</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="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">size</span><span class="w"> </span><span class="o">:=</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-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></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">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</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">size</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</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">size</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</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">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="o">+</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
@@ -3469,7 +3470,7 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span> <span class="nf">twoSumBruteForce</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="p">[</span><span class="nb">Int</span><span class="p">]</span> <span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>    <span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>    <span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropLast</span><span class="p">()</span> <span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>        <span class="k">for</span> <span class="n">j</span> <span class="k">in</span> <span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">{</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="p">==</span> <span class="n">target</span> <span class="p">{</span>
@@ -3485,7 +3486,7 @@
 <div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumBruteForce</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-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</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-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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><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">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</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">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -3501,7 +3502,7 @@
 <div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumBruteForce</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="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</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-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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><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">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</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">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </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="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -3517,7 +3518,7 @@
 <div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 方法一： 暴力枚举 */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><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">twoSumBruteForce</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-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</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-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</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">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></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">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">];</span>
@@ -3531,7 +3532,7 @@
 <div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 方法一：暴力枚举 */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">two_sum_brute_force</span><span class="p">(</span><span class="n">nums</span>: <span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</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="nb">Option</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">size</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>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </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="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</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">1</span><span class="o">..</span><span class="n">size</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">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</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">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
@@ -3566,7 +3567,7 @@
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="k">fn</span><span class="w"> </span><span class="n">twoSumBruteForce</span><span class="p">(</span><span class="n">nums</span><span class="o">:</span><span class="w"> </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="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="kt">i32</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="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">size</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">len</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="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-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度 O(n^2)</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// 两层循环，时间复杂度为 O(n^2)</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">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><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="mi">1</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="kr">var</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">i</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-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">)</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="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -3585,7 +3586,7 @@
 <h2 id="1042">10.4.2 &nbsp; 哈希查找：以空间换时间<a class="headerlink" href="#1042" title="Permanent link">&para;</a></h2>
 <p>考虑借助一个哈希表，键值对分别为数组元素和元素索引。循环遍历数组，每轮执行图 10-10 所示的步骤。</p>
 <ol>
-<li>判断数字 <code>target - nums[i]</code> 是否在哈希表中，若是则直接返回这两个元素的索引。</li>
+<li>判断数字 <code>target - nums[i]</code> 是否在哈希表中，若是，则直接返回这两个元素的索引。</li>
 <li>将键值对 <code>nums[i]</code> 和索引 <code>i</code> 添加进哈希表。</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:3"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label></div>
@@ -3603,15 +3604,15 @@
 </div>
 <p align="center"> 图 10-10 &nbsp; 辅助哈希表求解两数之和 </p>
 
-<p>实现代码如下所示，仅需单层循环即可。</p>
+<p>实现代码如下所示，仅需单层循环即可：</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">two_sum.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">two_sum_hash_table</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">list</span><span class="p">[</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;方法二：辅助哈希表&quot;&quot;&quot;</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a>    <span class="c1"># 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a>    <span class="c1"># 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>    <span class="n">dic</span> <span class="o">=</span> <span class="p">{}</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>    <span class="c1"># 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>    <span class="c1"># 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</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-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a>        <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">in</span> <span class="n">dic</span><span class="p">:</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>            <span class="k">return</span> <span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span> <span class="o">-</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span> <span class="n">i</span><span class="p">]</span>
@@ -3623,9 +3624,9 @@
 <div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="cm">/* 方法二：辅助哈希表 */</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><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="n">twoSumHashTable</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="kt">int</span><span class="w"> </span><span class="n">size</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-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">    </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="p">;</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</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="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">find</span><span class="p">(</span><span class="n">target</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">i</span><span class="p">])</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">end</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">{</span><span class="n">dic</span><span class="p">[</span><span class="n">target</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">i</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">};</span>
@@ -3640,9 +3641,9 @@
 <div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 方法二：辅助哈希表 */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">twoSumHashTable</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="kt">int</span><span class="w"> </span><span class="n">size</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-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;&gt;</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">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">};</span>
@@ -3657,9 +3658,9 @@
 <div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 方法二：辅助哈希表 */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">TwoSumHashTable</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="kt">int</span><span class="w"> </span><span class="n">size</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-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </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">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">target</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">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</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">i</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
@@ -3673,9 +3674,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 方法二：辅助哈希表 */</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">twoSumHashTable</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="p">[]</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">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="nx">hashTable</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kd">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">int</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">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</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">idx</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</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="k">if</span><span class="w"> </span><span class="nx">preIdx</span><span class="p">,</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">hashTable</span><span class="p">[</span><span class="nx">target</span><span class="o">-</span><span class="nx">val</span><span class="p">];</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="nx">preIdx</span><span class="p">,</span><span class="w"> </span><span class="nx">idx</span><span class="p">}</span>
@@ -3689,9 +3690,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 方法二：辅助哈希表 */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span> <span class="nf">twoSumHashTable</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="p">[</span><span class="nb">Int</span><span class="p">]</span> <span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>    <span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a>    <span class="kd">var</span> <span class="nv">dic</span><span class="p">:</span> <span class="p">[</span><span class="nb">Int</span><span class="p">:</span> <span class="nb">Int</span><span class="p">]</span> <span class="p">=</span> <span class="p">[:]</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a>    <span class="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a>    <span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="k">in</span> <span class="n">nums</span><span class="p">.</span><span class="bp">indices</span> <span class="p">{</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a>        <span class="k">if</span> <span class="kd">let</span> <span class="nv">j</span> <span class="p">=</span> <span class="n">dic</span><span class="p">[</span><span class="n">target</span> <span class="o">-</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span> <span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a>            <span class="k">return</span> <span class="p">[</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">]</span>
@@ -3705,9 +3706,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 方法二：辅助哈希表 */</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">twoSumHashTable</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">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</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">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </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="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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><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">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="p">[</span><span class="nx">target</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">i</span><span class="p">]]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">undefined</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="nx">m</span><span class="p">[</span><span class="nx">target</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">i</span><span class="p">]],</span><span class="w"> </span><span class="nx">i</span><span class="p">];</span>
@@ -3722,9 +3723,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 方法二：辅助哈希表 */</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">twoSumHashTable</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="p">[]</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">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</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">m</span><span class="o">:</span><span class="w"> </span><span class="kt">Map</span><span class="o">&lt;</span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="kt">number</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</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="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</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><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">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">target</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">i</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="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">undefined</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -3741,9 +3742,9 @@
 <div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 方法二： 辅助哈希表 */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><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">twoSumHashTable</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="kt">int</span><span class="w"> </span><span class="n">size</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-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">  </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">  </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">();</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">  </span><span class="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">  </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</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">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</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">dic</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">target</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">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</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">i</span><span class="p">]]</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
@@ -3757,9 +3758,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 方法二：辅助哈希表 */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span> <span class="nf">two_sum_hash_table</span><span class="p">(</span><span class="n">nums</span>: <span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</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="nb">Option</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</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">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</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">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span>::<span class="n">new</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">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">enumerate</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="k">match</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">num</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="fm">vec!</span><span class="p">[</span><span class="o">*</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">]),</span>
@@ -3819,11 +3820,11 @@
 <div class="highlight"><span class="filename">two_sum.zig</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// 方法二：辅助哈希表</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">twoSumHashTable</span><span class="p">(</span><span class="n">nums</span><span class="o">:</span><span class="w"> </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="o">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="o">!?</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="kt">i32</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="kr">var</span><span class="w"> </span><span class="n">size</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">len</span><span class="p">;</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度 O(n)</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// 辅助哈希表，空间复杂度为 O(n)</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">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">.</span><span class="n">AutoHashMap</span><span class="p">(</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="kt">i32</span><span class="p">).</span><span class="n">init</span><span class="p">(</span><span class="n">std</span><span class="p">.</span><span class="n">heap</span><span class="p">.</span><span class="n">page_allocator</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="k">defer</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">deinit</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="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-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度 O(n)</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="c1">// 单层循环，时间复杂度为 O(n)</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></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">size</span><span class="p">)</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="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="n">target</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">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">_</span><span class="p">]</span><span class="kt">i32</span><span class="p">{</span><span class="n">dic</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="n">target</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">i</span><span class="p">]).</span><span class="o">?</span><span class="p">,</span><span class="w"> </span><span class="nb">@intCast</span><span class="p">(</span><span class="n">i</span><span class="p">)};</span>
@@ -3836,7 +3837,7 @@
 </div>
 </div>
 </div>
-<p>此方法通过哈希查找将时间复杂度从 <span class="arithmatex">\(O(n^2)\)</span> 降低至 <span class="arithmatex">\(O(n)\)</span> ，大幅提升运行效率。</p>
+<p>此方法通过哈希查找将时间复杂度从 <span class="arithmatex">\(O(n^2)\)</span> 降至 <span class="arithmatex">\(O(n)\)</span> ，大幅提升运行效率。</p>
 <p>由于需要维护一个额外的哈希表，因此空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。<strong>尽管如此，该方法的整体时空效率更为均衡，因此它是本题的最优解法</strong>。</p>
 
 <!-- Source file information -->

chapter_searching/searching_algorithm_revisited/index.html

@@ -3407,7 +3407,7 @@
 <p>暴力搜索通过遍历数据结构的每个元素来定位目标元素。</p>
 <ul>
 <li>“线性搜索”适用于数组和链表等线性数据结构。它从数据结构的一端开始，逐个访问元素，直到找到目标元素或到达另一端仍没有找到目标元素为止。</li>
-<li>“广度优先搜索”和“深度优先搜索”是图和树的两种遍历策略。广度优先搜索从初始节点开始逐层搜索，由近及远地访问各个节点。深度优先搜索是从初始节点开始，沿着一条路径走到头为止，再回溯并尝试其他路径，直到遍历完整个数据结构。</li>
+<li>“广度优先搜索”和“深度优先搜索”是图和树的两种遍历策略。广度优先搜索从初始节点开始逐层搜索，由近及远地访问各个节点。深度优先搜索从初始节点开始，沿着一条路径走到头，再回溯并尝试其他路径，直到遍历完整个数据结构。</li>
 </ul>
 <p>暴力搜索的优点是简单且通用性好，<strong>无须对数据做预处理和借助额外的数据结构</strong>。</p>
 <p>然而，<strong>此类算法的时间复杂度为 <span class="arithmatex">\(O(n)\)</span></strong> ，其中 <span class="arithmatex">\(n\)</span> 为元素数量，因此在数据量较大的情况下性能较差。</p>
@@ -3419,13 +3419,13 @@
 <li>“树查找”在特定的树结构（例如二叉搜索树）中，基于比较节点值来快速排除节点，从而定位目标元素。</li>
 </ul>
 <p>此类算法的优点是效率高，<strong>时间复杂度可达到 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span></strong> 。</p>
-<p>然而，<strong>使用这些算法往往需要对数据进行预处理</strong>。例如，二分查找需要预先对数组进行排序，哈希查找和树查找都需要借助额外的数据结构，维护这些数据结构也需要额外的时间和空间开支。</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>自适应搜索算法常被称为查找算法，<strong>主要关注在特定数据结构中快速检索目标元素</strong>。</p>
+<p>然而，<strong>使用这些算法往往需要对数据进行预处理</strong>。例如，二分查找需要预先对数组进行排序，哈希查找和树查找都需要借助额外的数据结构，维护这些数据结构也需要额外的时间和空间开销。</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>自适应搜索算法常被称为查找算法，<strong>主要用于在特定数据结构中快速检索目标元素</strong>。</p>
 </div>
 <h2 id="1053">10.5.3 &nbsp; 搜索方法选取<a class="headerlink" href="#1053" title="Permanent link">&para;</a></h2>
-<p>给定大小为 <span class="arithmatex">\(n\)</span> 的一组数据，我们可以使用线性搜索、二分查找、树查找、哈希查找等多种方法在该数据中搜索目标元素。各个方法的工作原理如图 10-11 所示。</p>
+<p>给定大小为 <span class="arithmatex">\(n\)</span> 的一组数据，我们可以使用线性搜索、二分查找、树查找、哈希查找等多种方法从中搜索目标元素。各个方法的工作原理如图 10-11 所示。</p>
 <p><a class="glightbox" href="../searching_algorithm_revisited.assets/searching_algorithms.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="多种搜索策略" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></a></p>
 <p align="center"> 图 10-11 &nbsp; 多种搜索策略 </p>
 

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@@ -3317,12 +3317,12 @@
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 <h1 id="106">10.6 &nbsp; 小结<a class="headerlink" href="#106" title="Permanent link">&para;</a></h1>
 <ul>
-<li>二分查找依赖于数据的有序性，通过循环逐步缩减一半搜索区间来实现查找。它要求输入数据有序，且仅适用于数组或基于数组实现的数据结构。</li>
-<li>暴力搜索通过遍历数据结构来定位数据。线性搜索适用于数组和链表，广度优先搜索和深度优先搜索适用于图和树。此类算法通用性好，无须对数据预处理，但时间复杂度 <span class="arithmatex">\(O(n)\)</span> 较高。</li>
+<li>二分查找依赖数据的有序性，通过循环逐步缩减一半搜索区间来进行查找。它要求输入数据有序，且仅适用于数组或基于数组实现的数据结构。</li>
+<li>暴力搜索通过遍历数据结构来定位数据。线性搜索适用于数组和链表，广度优先搜索和深度优先搜索适用于图和树。此类算法通用性好，无须对数据进行预处理，但时间复杂度 <span class="arithmatex">\(O(n)\)</span> 较高。</li>
 <li>哈希查找、树查找和二分查找属于高效搜索方法，可在特定数据结构中快速定位目标元素。此类算法效率高，时间复杂度可达 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span> ，但通常需要借助额外数据结构。</li>
 <li>实际中，我们需要对数据体量、搜索性能要求、数据查询和更新频率等因素进行具体分析，从而选择合适的搜索方法。</li>
-<li>线性搜索适用于小型或频繁更新的数据；二分查找适用于大型、排序的数据；哈希查找适合对查询效率要求较高且无须范围查询的数据；树查找适用于需要维护顺序和支持范围查询的大型动态数据。</li>
-<li>用哈希查找替换线性查找是一种常用的优化运行时间的策略，可将时间复杂度从 <span class="arithmatex">\(O(n)\)</span> 降低至 <span class="arithmatex">\(O(1)\)</span> 。</li>
+<li>线性搜索适用于小型或频繁更新的数据；二分查找适用于大型、排序的数据；哈希查找适用于对查询效率要求较高且无须范围查询的数据；树查找适用于需要维护顺序和支持范围查询的大型动态数据。</li>
+<li>用哈希查找替换线性查找是一种常用的优化运行时间的策略，可将时间复杂度从 <span class="arithmatex">\(O(n)\)</span> 降至 <span class="arithmatex">\(O(1)\)</span> 。</li>
 </ul>
 
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