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<h1 id="_1">桶排序<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p>「桶排序 Bucket Sort」考虑设置 <span class="arithmatex">\(k\)</span> 个桶，并将 <span class="arithmatex">\(n\)</span> 个元素根据大小分配到 <span class="arithmatex">\(k\)</span> 个桶中，<strong>并在每个桶内部分别执行排序</strong>，由于桶之间的大小关系的确定的，因此最后按照桶之间的顺序将元素依次展开即可。</p>
<p>假设元素平均分布在各个桶内，则每个桶内元素数量为 <span class="arithmatex">\(\frac{n}{k}\)</span> ；如果使用「快速排序」来实现桶内排序，则排序单个桶使用 <span class="arithmatex">\(O(\frac{n}{k} \log\frac{n}{k})\)</span> 时间，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> 。</p>
<div class="admonition note 计数排序与桶排序的关系">
<p class="admonition-title">Note</p>
<p><strong>计数排序可以看作是桶排序的一种特例</strong>。我们可以把计数排序中 <code>counter</code> 的每个索引想象成一个桶，将统计数量的过程想象成把 <span class="arithmatex">\(n\)</span> 个元素分配到对应的桶中，再根据桶之间的有序性输出结果，从而实现排序。</p>
<h1 id="117">11.7. &nbsp; 桶排序<a class="headerlink" href="#117" title="Permanent link">&para;</a></h1>
<p>「桶排序 Bucket Sort」是分治思想的典型体现，其通过设置一些桶，将数据平均分配到各个桶中，并在每个桶内部分别执行排序，最终根据桶之间天然的大小顺序将各个桶内元素合并，从而得到排序结果。</p>
<h2 id="1171">11.7.1. &nbsp; 算法流程<a class="headerlink" href="#1171" title="Permanent link">&para;</a></h2>
<p>输入一个长度为 <span class="arithmatex">\(n\)</span> 的数组，元素是范围 <span class="arithmatex">\([0, 1)\)</span> 的浮点数，桶排序流程为：</p>
<ol>
<li>初始化 <span class="arithmatex">\(k\)</span> 个桶，将 <span class="arithmatex">\(n\)</span> 个元素分配至 <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>
</ol>
<p><img alt="桶排序算法流程" src="../bucket_sort.assets/bucket_sort_overview.png" /></p>
<p align="center"> Fig. 桶排序算法流程 </p>
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<div class="highlight"><span class="filename">bucket_sort.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 桶排序 */</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="kt">float</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-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w"> </span><span class="c1">// 初始化 k = n/3 个桶，预期向每个桶分配 3 个元素</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</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">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w"> </span><span class="k">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">k</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-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">());</span>
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w"> </span><span class="c1">// 1. 将数组元素分配到各个桶中</span>
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></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-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></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-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></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="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-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w"> </span><span class="c1">// 将 num 添加进桶 i</span>
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></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>
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w"> </span><span class="c1">// 2. 对各个桶执行排序</span>
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="w"> </span><span class="c1">// 使用内置排序函数，也可以替换成其它排序算法</span>
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w"> </span><span class="n">Collections</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">bucket</span><span class="p">);</span>
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w"> </span><span class="c1">// 3. 遍历桶合并结果</span>
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></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>
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></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">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="w"> </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">num</span><span class="p">;</span>
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a><span class="w"> </span><span class="p">}</span>
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="p">}</span>
</code></pre></div>
</div>
<p>（图）</p>
<p>理论上桶排序的时间复杂度是 <span class="arithmatex">\(O(n)\)</span> ，<strong>但前提是需要将元素均匀分配到各个桶中</strong>，而这并不容易做到。假设想要把淘宝的 <span class="arithmatex">\(100\)</span> 万件商品根据价格范围平均分配到 <span class="arithmatex">\(100\)</span> 个桶中，而商品价格不是均匀分布的，例如 <span class="arithmatex">\(100\)</span> 元以下的商品非常多、<span class="arithmatex">\(1000\)</span> 元以上的商品非常少等。如果我们将价格区间平均划分为 <span class="arithmatex">\(100\)</span> 份，那么各个桶内的商品数量差距会非常大。为了实现平均分配，我们一般这样做：</p>
<ul>
<li>先粗略设置分界线，将元素分配完后，<strong>把元素较多的桶继续划分为多个桶</strong>，直至所有桶内元素数量合理为止；该做法本质上是一个递归树；</li>
<li>如果我们提前知道商品价格的概率分布，<strong>则可以根据已知分布来设置每个桶的价格分界线</strong>；值得说明的是，数据分布不一定需要特意统计，也可以根据数据特点采用某种常见概率模型来近似，例如自然界的正态分布等；</li>
</ul>
<p>（图）</p>
<p>另外，排序桶内元素需要选择一种合适的排序算法，比如快速排序。</p>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="p">[</span><span class="k">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.py</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bucket_sort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.go</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="p">[</span><span class="nx">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="nx">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.js</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.ts</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="nx">func</span><span class="p">]{</span><span class="nx">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.c</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.cs</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="na">[class]</span><span class="p">{</span><span class="n">bucket_sort</span><span class="p">}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.swift</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="p">[</span><span class="kd">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="kd">func</span><span class="p">]{</span><span class="n">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
<div class="tabbed-block">
<div class="highlight"><span class="filename">bucket_sort.zig</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="p">[</span><span class="n">class</span><span class="p">]{}</span><span class="o">-</span><span class="p">[</span><span class="n">func</span><span class="p">]{</span><span class="n">bucketSort</span><span class="p">}</span>
</code></pre></div>
</div>
</div>
</div>
<div class="admonition note">
<p class="admonition-title">桶排序是计数排序的一种推广</p>
<p>从桶排序的角度，我们可以把计数排序中计数数组 <code>counter</code> 的每个索引想象成一个桶，将统计数量的过程想象成把各个元素分配到对应的桶中，再根据桶之间的有序性输出结果，从而实现排序。</p>
</div>
<h2 id="1172">11.7.2. &nbsp; 算法特性<a class="headerlink" href="#1172" title="Permanent link">&para;</a></h2>
<p><strong>时间复杂度 <span class="arithmatex">\(O(n + k)\)</span></strong> ：假设元素平均分布在各个桶内，则每个桶内元素数量为 <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> 。最后合并结果需要遍历 <span class="arithmatex">\(n\)</span> 个桶，使用 <span class="arithmatex">\(O(k)\)</span> 时间。</p>
<p>最差情况下，所有数据被分配到一个桶中，且排序算法退化至 <span class="arithmatex">\(O(n^2)\)</span> ，此时使用 <span class="arithmatex">\(O(n^2)\)</span> 时间，因此是“自适应排序”。</p>
<p><strong>空间复杂度 <span class="arithmatex">\(O(n + k)\)</span></strong> ：需要借助 <span class="arithmatex">\(k\)</span> 个桶和共 <span class="arithmatex">\(n\)</span> 个元素的额外空间，是“非原地排序”。</p>
<p>桶排序是否稳定取决于排序桶内元素的算法是否稳定。</p>
<h2 id="1173">11.7.3. &nbsp; 如何实现平均分配<a class="headerlink" href="#1173" 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>难点是需要将元素均匀分配到各个桶中</strong>，因为现实中的数据往往都不是均匀分布的。举个例子，假设我们想要把淘宝的所有商品根据价格范围平均分配到 10 个桶中，然而商品价格不是均匀分布的，100 元以下非常多、1000 元以上非常少；如果我们将价格区间平均划为 10 份，那么各个桶内的商品数量差距会非常大。</p>
<p>为了实现平均分配，我们可以先大致设置一个分界线，将数据粗略分到 3 个桶，分配完后，<strong>再把商品较多的桶继续划分为 3 个桶，直至所有桶内元素数量大致平均为止</strong>。此方法本质上是生成一个递归树，让叶结点的值尽量平均。当然，不一定非要划分为 3 个桶，可以根据数据特点灵活选取。</p>
<p><img alt="递归划分桶" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></p>
<p align="center"> Fig. 递归划分桶 </p>
<p>如果我们提前知道商品价格的概率分布，<strong>那么也可以根据数据概率分布来设置每个桶的价格分界线</strong>。注意，数据分布不一定需要特意去统计，也可以根据数据特点采用某种概率模型来近似。如下图所示，我们假设商品价格服从正态分布，就可以合理设置价格区间，将商品平均分配到各个桶中。</p>
<p><img alt="根据概率分布划分桶" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
<p align="center"> Fig. 根据概率分布划分桶 </p>
<h2 id="__comments">评论</h2>
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<h2 id="1142">11.4.2. &nbsp; 算法特性<a class="headerlink" href="#1142" title="Permanent link">&para;</a></h2>
<p><strong>时间复杂度 <span class="arithmatex">\(O(n \log n)\)</span></strong> ：平均情况下，哨兵划分的递归层数为 <span class="arithmatex">\(\log n\)</span> ，每层中的总循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n \log n)\)</span> 时间。最差情况下，每轮哨兵划分操作都将长度为 <span class="arithmatex">\(n\)</span> 的数组划分为长度为 <span class="arithmatex">\(0\)</span> 和 <span class="arithmatex">\(n - 1\)</span> 的两个子数组，此时递归层数达到 <span class="arithmatex">\(n\)</span> 层，每层中的循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n^2)\)</span> 时间，因此是“非稳定排序”。</p>
<p><strong>时间复杂度 <span class="arithmatex">\(O(n \log n)\)</span></strong> ：平均情况下，哨兵划分的递归层数为 <span class="arithmatex">\(\log n\)</span> ，每层中的总循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n \log n)\)</span> 时间。</p>
<p>最差情况下，每轮哨兵划分操作都将长度为 <span class="arithmatex">\(n\)</span> 的数组划分为长度为 <span class="arithmatex">\(0\)</span> 和 <span class="arithmatex">\(n - 1\)</span> 的两个子数组，此时递归层数达到 <span class="arithmatex">\(n\)</span> 层，每层中的循环数为 <span class="arithmatex">\(n\)</span> ，总体使用 <span class="arithmatex">\(O(n^2)\)</span> 时间，因此是“非稳定排序”。</p>
<p><strong>空间复杂度 <span class="arithmatex">\(O(n)\)</span></strong> ：输入数组完全倒序下，达到最差递归深度 <span class="arithmatex">\(n\)</span> 。由于未借助辅助数组空间，因此是“原地排序”。</p>
<p><strong>非稳定排序</strong>：哨兵划分最后一步可能会将基准数交换至相等元素的右边。</p>
<h2 id="1143">11.4.3. &nbsp; 快排为什么快？<a class="headerlink" href="#1143" title="Permanent link">&para;</a></h2>

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<ul>
<li>冒泡排序通过交换相邻元素来实现排序。通过增加标志位实现提前返回，我们可将冒泡排序的最佳时间复杂度优化至 <span class="arithmatex">\(O(N)\)</span> 。</li>
<li>插入排序每轮将待排序区间内元素插入至已排序区间的正确位置，从而实现排序。插入排序的时间复杂度虽为 <span class="arithmatex">\(O(N^2)\)</span> ，但因为总体操作少而很受欢迎，一般用于小数据量的排序工作。</li>
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<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.7. &nbsp; 桶排序（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.7. &nbsp; 小结
11.8. &nbsp; 小结
</a>
</li>

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@ -1448,6 +1448,8 @@
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@ -1551,9 +1553,23 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.7. &nbsp; 桶排序（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.7. &nbsp; 小结
11.8. &nbsp; 小结
</a>
</li>

18
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@ -1415,6 +1415,8 @@
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
@ -1518,9 +1520,23 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/bucket_sort/" class="md-nav__link">
11.7. &nbsp; 桶排序（New）
</a>
</li>
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.7. &nbsp; 小结
11.8. &nbsp; 小结
</a>
</li>

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@ -1351,6 +1351,8 @@
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
@ -1454,9 +1456,23 @@
<li class="md-nav__item">
<a href="chapter_sorting/bucket_sort/" class="md-nav__link">
11.7. &nbsp; 桶排序（New）
</a>
</li>
<li class="md-nav__item">
<a href="chapter_sorting/summary/" class="md-nav__link">
11.7. &nbsp; 小结
11.8. &nbsp; 小结
</a>
</li>

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