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chapter_sorting/bubble_sort/index.html
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<li>同理，对剩余 <span class="arithmatex">\(n - 1\)</span> 个元素执行「冒泡」，可将第二大元素交换至正确位置，因而待排序元素只剩 <span class="arithmatex">\(n - 2\)</span> 个。</li>
<li>以此类推…… <strong>循环 <span class="arithmatex">\(n - 1\)</span> 轮「冒泡」，即可完成整个数组的排序</strong>。</li>
</ol>
<p><img alt="bubble_sort_overview" src="../bubble_sort.assets/bubble_sort_overview.png" /></p>
<p align="center"> Fig. 冒泡排序流程 </p>
<p><img alt="冒泡排序流程" src="../bubble_sort.assets/bubble_sort_overview.png" /></p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div>
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<p>「插入排序 Insertion Sort」是一种基于 <strong>数组插入操作</strong> 的排序算法。</p>
<p>「插入操作」原理：选定某个待排序元素为基准数 <code>base</code>，将 <code>base</code> 与其左侧已排序区间元素依次对比大小，并插入到正确位置。</p>
<p>回忆数组插入操作，我们需要将从目标索引到 <code>base</code> 之间的所有元素向右移动一位，然后再将 <code>base</code> 赋值给目标索引。</p>
<p><img alt="insertion_operation" src="../insertion_sort.assets/insertion_operation.png" /></p>
<p align="center"> Fig. 插入操作 </p>
<p><img alt="单次插入操作" src="../insertion_sort.assets/insertion_operation.png" /></p>
<h2 id="1131">11.3.1. &nbsp; 算法流程<a class="headerlink" href="#1131" title="Permanent link">&para;</a></h2>
<ol>
<li>第 1 轮先选取数组的 <strong>第 2 个元素</strong> 为 <code>base</code> ，执行「插入操作」后，<strong>数组前 2 个元素已完成排序</strong>。</li>
<li>第 2 轮选取 <strong>第 3 个元素</strong> 为 <code>base</code> ，执行「插入操作」后，<strong>数组前 3 个元素已完成排序</strong>。</li>
<li>以此类推……最后一轮选取 <strong>数组尾元素</strong> 为 <code>base</code> ，执行「插入操作」后，<strong>所有元素已完成排序</strong>。</li>
</ol>
<p><img alt="insertion_sort_overview" src="../insertion_sort.assets/insertion_sort_overview.png" /></p>
<p align="center"> Fig. 插入排序流程 </p>
<p><img alt="插入排序流程" src="../insertion_sort.assets/insertion_sort_overview.png" /></p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label></div>
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<li>待排序的列表的 <strong>元素类型</strong> 可以是整数、浮点数、字符、或字符串；</li>
<li>排序算法可以根据需要设定 <strong>判断规则</strong>，例如数字大小、字符 ASCII 码顺序、自定义规则；</li>
</ul>
<p><img alt="sorting_examples" src="../intro_to_sort.assets/sorting_examples.png" /></p>
<p align="center"> Fig. 排序中的不同元素类型和判断规则 </p>
<p><img alt="排序中不同的元素类型和判断规则" src="../intro_to_sort.assets/sorting_examples.png" /></p>
<h2 id="1111">11.1.1. &nbsp; 评价维度<a class="headerlink" href="#1111" title="Permanent link">&para;</a></h2>
<p>排序算法主要可根据 <strong>稳定性 、就地性 、自适应性 、比较类</strong> 来分类。</p>
<h3 id="_1">稳定性<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>

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<li><strong>划分阶段</strong>：通过递归不断 <strong>将数组从中点位置划分开</strong>，将长数组的排序问题转化为短数组的排序问题；</li>
<li><strong>合并阶段</strong>：划分到子数组长度为 1 时，开始向上合并，不断将 <strong>左、右两个短排序数组</strong> 合并为 <strong>一个长排序数组</strong>，直至合并至原数组时完成排序；</li>
</ol>
<p><img alt="merge_sort_overview" src="../merge_sort.assets/merge_sort_overview.png" /></p>
<p align="center"> Fig. 归并排序两阶段：划分与合并 </p>
<p><img alt="归并排序的划分与合并阶段" src="../merge_sort.assets/merge_sort_overview.png" /></p>
<h2 id="1151">11.5.1. &nbsp; 算法流程<a class="headerlink" href="#1151" title="Permanent link">&para;</a></h2>
<p><strong>「递归划分」</strong> 从顶至底递归地 <strong>将数组从中点切为两个子数组</strong>，直至长度为 1 ；</p>
<ol>

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<li>不断循环步骤 <code>2.</code> ，直至 <code>i</code> , <code>j</code> 相遇时跳出，最终把基准数交换至两个子数组的分界线；</li>
</ol>
<p>「哨兵划分」执行完毕后，原数组被划分成两个部分，即 <strong>左子数组</strong> 和 <strong>右子数组</strong>，且满足 <strong>左子数组任意元素 &lt; 基准数 &lt; 右子数组任意元素</strong>。因此，接下来我们只需要排序两个子数组即可。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:9"><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" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
<div class="tabbed-set tabbed-alternate" data-tabs="1:19"><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" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><input id="__tabbed_1_14" name="__tabbed_1" type="radio" /><input id="__tabbed_1_15" name="__tabbed_1" type="radio" /><input id="__tabbed_1_16" name="__tabbed_1" type="radio" /><input id="__tabbed_1_17" name="__tabbed_1" type="radio" /><input id="__tabbed_1_18" name="__tabbed_1" type="radio" /><input id="__tabbed_1_19" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label><label for="__tabbed_1_10">Java</label><label for="__tabbed_1_11">C++</label><label for="__tabbed_1_12">Python</label><label for="__tabbed_1_13">Go</label><label for="__tabbed_1_14">JavaScript</label><label for="__tabbed_1_15">TypeScript</label><label for="__tabbed_1_16">C</label><label for="__tabbed_1_17">C#</label><label for="__tabbed_1_18">Swift</label><label for="__tabbed_1_19">Zig</label></div>
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<p><img alt="pivot_division_step1" src="../quick_sort.assets/pivot_division_step1.png" /></p>
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<p><img alt="pivot_division_step9" src="../quick_sort.assets/pivot_division_step9.png" /></p>
</div>
</div>
</div>
<p align="center"> Fig. 哨兵划分 </p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div>
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<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 元素交换 */</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">swap</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">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<li>直至子数组长度为 1 时 <strong>终止递归</strong>，即可完成对整个数组的排序；</li>
</ol>
<p>观察发现，快速排序和「二分查找」的原理类似，都是以对数阶的时间复杂度来缩小处理区间。</p>
<p><img alt="quick_sort_overview" src="../quick_sort.assets/quick_sort_overview.png" /></p>
<p align="center"> Fig. 快速排序流程 </p>
<div class="tabbed-set tabbed-alternate" data-tabs="3:10"><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" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label></div>
<p><img alt="快速排序流程" src="../quick_sort.assets/quick_sort_overview.png" /></p>
<div class="tabbed-set tabbed-alternate" data-tabs="2:10"><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" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label></div>
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<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 快速排序 */</span>
@@ -2047,7 +2039,7 @@
<p><strong>普通快速排序在某些输入下的时间效率变差</strong>。举个极端例子，假设输入数组是完全倒序的，由于我们选取最左端元素为基准数，那么在哨兵划分完成后，基准数被交换至数组最右端，从而 <strong>左子数组长度为 <span class="arithmatex">\(n - 1\)</span>、右子数组长度为 <span class="arithmatex">\(0\)</span></strong> 。这样进一步递归下去，<strong>每轮哨兵划分后的右子数组长度都为 <span class="arithmatex">\(0\)</span></strong> ，分治策略失效，快速排序退化为「冒泡排序」了。</p>
<p>为了尽量避免这种情况发生，我们可以优化一下基准数的选取策略。首先，在哨兵划分中，我们可以 <strong>随机选取一个元素作为基准数</strong>。但如果运气很差，每次都选择到比较差的基准数，那么效率依然不好。</p>
<p>进一步地，我们可以在数组中选取 3 个候选元素（一般为数组的首、尾、中点元素），<strong>并将三个候选元素的中位数作为基准数</strong>，这样基准数“既不大也不小”的概率就大大提升了。当然，如果数组很长的话，我们也可以选取更多候选元素，来进一步提升算法的稳健性。采取该方法后，时间复杂度劣化至 <span class="arithmatex">\(O(n^2)\)</span> 的概率极低。</p>
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<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 选取三个元素的中位数 */</span>
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<h2 id="1145">11.4.5. &nbsp; 尾递归优化<a class="headerlink" href="#1145" title="Permanent link">&para;</a></h2>
<p><strong>普通快速排序在某些输入下的空间效率变差</strong>。仍然以完全倒序的输入数组为例，由于每轮哨兵划分后右子数组长度为 0 ，那么将形成一个高度为 <span class="arithmatex">\(n - 1\)</span> 的递归树，此时使用的栈帧空间大小劣化至 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p>为了避免栈帧空间的累积，我们可以在每轮哨兵排序完成后，判断两个子数组的长度大小，仅递归排序较短的子数组。由于较短的子数组长度不会超过 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此这样做能保证递归深度不超过 <span class="arithmatex">\(\log n\)</span> ，即最差空间复杂度被优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
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<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* 快速排序（尾递归优化） */</span>