deploy

chapter_sorting/bucket_sort/index.html

@@ -3396,18 +3396,19 @@
 
 <!-- Page content -->
 <h1 id="118">11.8 &nbsp; 桶排序<a class="headerlink" href="#118" title="Permanent link">&para;</a></h1>
-<p>前述的几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>
+<p>前述几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>
 <p>「桶排序 bucket sort」是分治策略的一个典型应用。它通过设置一些具有大小顺序的桶，每个桶对应一个数据范围，将数据平均分配到各个桶中；然后，在每个桶内部分别执行排序；最终按照桶的顺序将所有数据合并。</p>
 <h2 id="1181">11.8.1 &nbsp; 算法流程<a class="headerlink" href="#1181" title="Permanent link">&para;</a></h2>
-<p>考虑一个长度为 <span class="arithmatex">\(n\)</span> 的数组，元素是范围 <span class="arithmatex">\([0, 1)\)</span> 的浮点数。桶排序的流程如图 11-13 所示。</p>
+<p>考虑一个长度为 <span class="arithmatex">\(n\)</span> 的数组，其元素是范围 <span class="arithmatex">\([0, 1)\)</span> 内的浮点数。桶排序的流程如图 11-13 所示。</p>
 <ol>
 <li>初始化 <span class="arithmatex">\(k\)</span> 个桶，将 <span class="arithmatex">\(n\)</span> 个元素分配到 <span class="arithmatex">\(k\)</span> 个桶中。</li>
-<li>对每个桶分别执行排序（本文采用编程语言的内置排序函数）。</li>
-<li>按照桶的从小到大的顺序，合并结果。</li>
+<li>对每个桶分别执行排序（这里采用编程语言的内置排序函数）。</li>
+<li>按照桶从小到大的顺序合并结果。</li>
 </ol>
 <p><a class="glightbox" href="../bucket_sort.assets/bucket_sort_overview.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="桶排序算法流程" class="animation-figure" src="../bucket_sort.assets/bucket_sort_overview.png" /></a></p>
 <p align="center"> 图 11-13 &nbsp; 桶排序算法流程 </p>
 
+<p>代码如下所示：</p>
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@@ -3418,7 +3419,7 @@
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">buckets</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)]</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">num</span> <span class="o">*</span> <span class="n">k</span><span class="p">)</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># 将 num 添加进桶 i</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
@@ -3442,7 +3443,7 @@
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">(</span><span class="n">k</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="c1">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></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="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 bucket_idx</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">push_back</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
@@ -3473,7 +3474,7 @@
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><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">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </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="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></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="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="na">add</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
@@ -3504,7 +3505,7 @@
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><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">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</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="w"> </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="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></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="p">(</span><span class="kt">int</span><span class="p">)(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">Add</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
@@ -3535,7 +3536,7 @@
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><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">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</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-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nb">float64</span><span class="p">(</span><span class="nx">k</span><span class="p">))</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span>
@@ -3564,7 +3565,7 @@
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>    <span class="kd">var</span> <span class="nv">buckets</span> <span class="p">=</span> <span class="p">(</span><span class="mi">0</span> <span class="p">..</span><span class="o">&lt;</span> <span class="n">k</span><span class="p">).</span><span class="bp">map</span> <span class="p">{</span> <span class="kc">_</span> <span class="k">in</span> <span class="p">[</span><span class="nb">Double</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="c1">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a>    <span class="k">for</span> <span class="n">num</span> <span class="k">in</span> <span class="n">nums</span> <span class="p">{</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>        <span class="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>        <span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>        <span class="kd">let</span> <span class="nv">i</span> <span class="p">=</span> <span class="nb">Int</span><span class="p">(</span><span class="n">num</span> <span class="o">*</span> <span class="nb">Double</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a>        <span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>        <span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">append</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
@@ -3596,7 +3597,7 @@
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><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">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </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="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
@@ -3627,7 +3628,7 @@
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><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">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </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="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
@@ -3656,7 +3657,7 @@
 <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="w">  </span><span class="c1">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">_num</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="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">// 输入数据范围 [0, 1)，使用 _num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// 输入数据范围为 [0, 1)，使用 _num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></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="p">(</span><span class="n">_num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">).</span><span class="n">toInt</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="c1">// 将 _num 添加进桶 bucket_idx</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">add</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
@@ -3683,7 +3684,7 @@
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></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">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[];</span><span class="w"> </span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// 1. 将数组元素分配到各个桶中</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// 将 num 添加进桶 i</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
@@ -3717,7 +3718,7 @@
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 1. 将数组元素分配到各个桶中</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>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// 输入数据范围 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// 输入数据范围为 [0, 1)，使用 num * k 映射到索引范围 [0, k-1]</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">bucket_idx</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">k</span><span class="p">;</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="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="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// 如果桶中有数据且数据小于当前值 nums[i], 要将其放到当前桶的后面，相当于 cpp 中的 push_back</span>
@@ -3764,14 +3765,14 @@
 <p>桶排序适用于处理体量很大的数据。例如，输入数据包含 100 万个元素，由于空间限制，系统内存无法一次性加载所有数据。此时，可以将数据分成 1000 个桶，然后分别对每个桶进行排序，最后将结果合并。</p>
 <ul>
 <li><strong>时间复杂度 <span class="arithmatex">\(O(n + k)\)</span></strong> ：假设元素在各个桶内平均分布，那么每个桶内的元素数量为 <span class="arithmatex">\(\frac{n}{k}\)</span> 。假设排序单个桶使用 <span class="arithmatex">\(O(\frac{n}{k} \log\frac{n}{k})\)</span> 时间，则排序所有桶使用 <span class="arithmatex">\(O(n \log\frac{n}{k})\)</span> 时间。<strong>当桶数量 <span class="arithmatex">\(k\)</span> 比较大时，时间复杂度则趋向于 <span class="arithmatex">\(O(n)\)</span></strong> 。合并结果时需要遍历所有桶和元素，花费 <span class="arithmatex">\(O(n + k)\)</span> 时间。</li>
-<li><strong>自适应排序</strong>：在最坏情况下，所有数据被分配到一个桶中，且排序该桶使用 <span class="arithmatex">\(O(n^2)\)</span> 时间。</li>
+<li><strong>自适应排序</strong>：在最差情况下，所有数据被分配到一个桶中，且排序该桶使用 <span class="arithmatex">\(O(n^2)\)</span> 时间。</li>
 <li><strong>空间复杂度 <span class="arithmatex">\(O(n + k)\)</span>、非原地排序</strong>：需要借助 <span class="arithmatex">\(k\)</span> 个桶和总共 <span class="arithmatex">\(n\)</span> 个元素的额外空间。</li>
 <li>桶排序是否稳定取决于排序桶内元素的算法是否稳定。</li>
 </ul>
 <h2 id="1183">11.8.3 &nbsp; 如何实现平均分配<a class="headerlink" href="#1183" title="Permanent link">&para;</a></h2>
-<p>桶排序的时间复杂度理论上可以达到 <span class="arithmatex">\(O(n)\)</span> ，<strong>关键在于将元素均匀分配到各个桶中</strong>，因为实际数据往往不是均匀分布的。例如，我们想要将淘宝上的所有商品按价格范围平均分配到 10 个桶中，但商品价格分布不均，低于 100 元的非常多，高于 1000 元的非常少。若将价格区间平均划分为 10 份，各个桶中的商品数量差距会非常大。</p>
-<p>为实现平均分配，我们可以先设定一个大致的分界线，将数据粗略地分到 3 个桶中。<strong>分配完毕后，再将商品较多的桶继续划分为 3 个桶，直至所有桶中的元素数量大致相等</strong>。</p>
-<p>如图 11-14 所示，这种方法本质上是创建一个递归树，目标是让叶节点的值尽可能平均。当然，不一定要每轮将数据划分为 3 个桶，具体划分方式可根据数据特点灵活选择。</p>
+<p>桶排序的时间复杂度理论上可以达到 <span class="arithmatex">\(O(n)\)</span> ，<strong>关键在于将元素均匀分配到各个桶中</strong>，因为实际数据往往不是均匀分布的。例如，我们想要将淘宝上的所有商品按价格范围平均分配到 10 个桶中，但商品价格分布不均，低于 100 元的非常多，高于 1000 元的非常少。若将价格区间平均划分为 10 个，各个桶中的商品数量差距会非常大。</p>
+<p>为实现平均分配，我们可以先设定一条大致的分界线，将数据粗略地分到 3 个桶中。<strong>分配完毕后，再将商品较多的桶继续划分为 3 个桶，直至所有桶中的元素数量大致相等</strong>。</p>
+<p>如图 11-14 所示，这种方法本质上是创建一棵递归树，目标是让叶节点的值尽可能平均。当然，不一定要每轮将数据划分为 3 个桶，具体划分方式可根据数据特点灵活选择。</p>
 <p><a class="glightbox" href="../bucket_sort.assets/scatter_in_buckets_recursively.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="递归划分桶" class="animation-figure" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></a></p>
 <p align="center"> 图 11-14 &nbsp; 递归划分桶 </p>