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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>
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</ol>
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<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 < 基准数 < 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>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<p><img alt="哨兵划分步骤" src="../quick_sort.assets/pivot_division_step1.png" /></p>
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@ -1754,6 +1754,14 @@
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<div class="tabbed-block">
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<p><img alt="pivot_division_step9" src="../quick_sort.assets/pivot_division_step9.png" /></p>
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</div>
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</div>
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</div>
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<div class="admonition note">
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<p class="admonition-title">快速排序的分治思想</p>
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<p>哨兵划分的实质是将 <strong>一个长数组的排序问题</strong> 简化为 <strong>两个短数组的排序问题</strong>。</p>
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</div>
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<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>
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<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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</div>
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</div>
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</div>
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<div class="admonition note">
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<p class="admonition-title">快速排序的分治思想</p>
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<p>哨兵划分的实质是将 <strong>一个长数组的排序问题</strong> 简化为 <strong>两个短数组的排序问题</strong>。</p>
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</div>
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<h2 id="1141">11.4.1. 算法流程<a class="headerlink" href="#1141" title="Permanent link">¶</a></h2>
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<ol>
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<li>首先，对数组执行一次「哨兵划分」，得到待排序的 <strong>左子数组</strong> 和 <strong>右子数组</strong>；</li>
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<p><img alt="快速排序流程" src="../quick_sort.assets/quick_sort_overview.png" /></p>
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<p align="center"> Fig. 快速排序流程 </p>
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<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="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>
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<div class="tabbed-content">
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<div class="tabbed-block">
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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>
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<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>
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<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>
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<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. 尾递归优化<a class="headerlink" href="#1145" title="Permanent link">¶</a></h2>
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<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>
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<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>
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BIN
sitemap.xml.gz
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