deploy

chapter_sorting/radix_sort/index.html

@@ -3382,7 +3382,7 @@
 
 <!-- Page content -->
 <h1 id="1110">11.10 &nbsp; 基数排序<a class="headerlink" href="#1110" title="Permanent link">&para;</a></h1>
-<p>上一节我们介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>
+<p>上一节介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>
 <p>「基数排序 radix sort」的核心思想与计数排序一致，也通过统计个数来实现排序。在此基础上，基数排序利用数字各位之间的递进关系，依次对每一位进行排序，从而得到最终的排序结果。</p>
 <h2 id="11101">11.10.1 &nbsp; 算法流程<a class="headerlink" href="#11101" title="Permanent link">&para;</a></h2>
 <p>以学号数据为例，假设数字的最低位是第 <span class="arithmatex">\(1\)</span> 位，最高位是第 <span class="arithmatex">\(8\)</span> 位，基数排序的流程如图 11-18 所示。</p>
@@ -3394,12 +3394,12 @@
 <p><a class="glightbox" href="../radix_sort.assets/radix_sort_overview.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="基数排序算法流程" class="animation-figure" src="../radix_sort.assets/radix_sort_overview.png" /></a></p>
 <p align="center"> 图 11-18 &nbsp; 基数排序算法流程 </p>
 
-<p>下面来剖析代码实现。对于一个 <span class="arithmatex">\(d\)</span> 进制的数字 <span class="arithmatex">\(x\)</span> ，要获取其第 <span class="arithmatex">\(k\)</span> 位 <span class="arithmatex">\(x_k\)</span> ，可以使用以下计算公式：</p>
+<p>下面剖析代码实现。对于一个 <span class="arithmatex">\(d\)</span> 进制的数字 <span class="arithmatex">\(x\)</span> ，要获取其第 <span class="arithmatex">\(k\)</span> 位 <span class="arithmatex">\(x_k\)</span> ，可以使用以下计算公式：</p>
 <div class="arithmatex">\[
 x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 \]</div>
 <p>其中 <span class="arithmatex">\(\lfloor a \rfloor\)</span> 表示对浮点数 <span class="arithmatex">\(a\)</span> 向下取整，而 <span class="arithmatex">\(\bmod \: d\)</span> 表示对 <span class="arithmatex">\(d\)</span> 取余。对于学号数据，<span class="arithmatex">\(d = 10\)</span> 且 <span class="arithmatex">\(k \in [1, 8]\)</span> 。</p>
-<p>此外，我们需要小幅改动计数排序代码，使之可以根据数字的第 <span class="arithmatex">\(k\)</span> 位进行排序。</p>
+<p>此外，我们需要小幅改动计数排序代码，使之可以根据数字的第 <span class="arithmatex">\(k\)</span> 位进行排序：</p>
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@@ -3410,7 +3410,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="k">def</span> <span class="nf">counting_sort_digit</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">exp</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w">    </span><span class="sd">&quot;&quot;&quot;计数排序（根据 nums 第 k 位排序）&quot;&quot;&quot;</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="n">counter</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># 统计 0~9 各数字的出现次数</span>
@@ -3455,7 +3455,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</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">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </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="n">counter</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</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-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3503,7 +3503,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</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">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</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="mi">10</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</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-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3554,7 +3554,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="k">void</span><span class="w"> </span><span class="nf">CountingSortDigit</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">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="m">10</span><span class="p">];</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</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-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3607,7 +3607,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">countingSortDigit</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">exp</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">counter</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="nx">n</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-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3662,7 +3662,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="kd">func</span> <span class="nf">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="kr">inout</span> <span class="p">[</span><span class="nb">Int</span><span class="p">],</span> <span class="n">exp</span><span class="p">:</span> <span class="nb">Int</span><span class="p">)</span> <span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>    <span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>    <span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a>    <span class="kd">var</span> <span class="nv">counter</span> <span class="p">=</span> <span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span> <span class="mi">0</span><span class="p">,</span> <span class="bp">count</span><span class="p">:</span> <span class="mi">10</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>    <span class="kd">let</span> <span class="nv">n</span> <span class="p">=</span> <span class="n">nums</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a>    <span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3717,7 +3717,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</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">Array</span><span class="p">(</span><span class="mf">10</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></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-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3772,7 +3772,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortDigit</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">exp</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="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</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">Array</span><span class="p">(</span><span class="mf">10</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></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-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3827,7 +3827,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="kt">void</span><span class="w"> </span><span class="n">countingSortDigit</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">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </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">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</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-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3876,7 +3876,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="k">fn</span> <span class="nf">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span>: <span class="kp">&amp;</span><span class="nc">mut</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">exp</span>: <span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></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">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">10</span><span class="p">];</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</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-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
@@ -3924,7 +3924,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* 计数排序（根据 nums 第 k 位排序） */</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="kt">int</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">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">((</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="p">));</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 统计 0~9 各数字的出现次数</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></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>
@@ -3979,7 +3979,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="c1">// 计数排序（根据 nums 第 k 位排序）</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="k">fn</span><span class="w"> </span><span class="n">countingSortDigit</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">exp</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="kt">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 十进制的位范围为 0~9 ，因此需要长度为 10 的桶数组</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="kr">var</span><span class="w"> </span><span class="n">mem_arena</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">heap</span><span class="p">.</span><span class="n">ArenaAllocator</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-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// defer mem_arena.deinit();</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="kr">const</span><span class="w"> </span><span class="n">mem_allocator</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mem_arena</span><span class="p">.</span><span class="n">allocator</span><span class="p">();</span>
@@ -4036,7 +4036,7 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 </div>
 <div class="admonition question">
 <p class="admonition-title">为什么从最低位开始排序？</p>
-<p>在连续的排序轮次中，后一轮排序会覆盖前一轮排序的结果。举例来说，如果第一轮排序结果 <span class="arithmatex">\(a &lt; b\)</span> ，而第二轮排序结果 <span class="arithmatex">\(a &gt; b\)</span> ，那么第二轮的结果将取代第一轮的结果。由于数字的高位优先级高于低位，我们应该先排序低位再排序高位。</p>
+<p>在连续的排序轮次中，后一轮排序会覆盖前一轮排序的结果。举例来说，如果第一轮排序结果 <span class="arithmatex">\(a &lt; b\)</span> ，而第二轮排序结果 <span class="arithmatex">\(a &gt; b\)</span> ，那么第二轮的结果将取代第一轮的结果。由于数字的高位优先级高于低位，因此应该先排序低位再排序高位。</p>
 </div>
 <h2 id="11102">11.10.2 &nbsp; 算法特性<a class="headerlink" href="#11102" title="Permanent link">&para;</a></h2>
 <p>相较于计数排序，基数排序适用于数值范围较大的情况，<strong>但前提是数据必须可以表示为固定位数的格式，且位数不能过大</strong>。例如，浮点数不适合使用基数排序，因为其位数 <span class="arithmatex">\(k\)</span> 过大，可能导致时间复杂度 <span class="arithmatex">\(O(nk) \gg O(n^2)\)</span> 。</p>