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<li><strong>复杂度的常数系数小</strong>：在上述三种算法中，快速排序的比较、赋值、交换等操作的总数量最少。这与“插入排序”比“冒泡排序”更快的原因类似。</li>
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<h2 id="1154">11.5.4 &nbsp; 基准数优化<a class="headerlink" href="#1154" title="Permanent link">&para;</a></h2>
<p><strong>快速排序在某些输入下的时间效率可能降低</strong>。举一个极端例子，假设输入数组是完全倒序的，由于我们选择最左端元素作为基准数，那么在哨兵划分完成后，基准数被交换至数组最右端，导致左子数组长度为 <span class="arithmatex">\(n - 1\)</span>、右子数组长度为 <span class="arithmatex">\(0\)</span> 。如此递归下去，每轮哨兵划分后的右子数组长度都为 <span class="arithmatex">\(0\)</span> ，分治策略失效，快速排序退化为“冒泡排序”。</p>
<p><strong>快速排序在某些输入下的时间效率可能降低</strong>。举一个极端例子，假设输入数组是完全倒序的，由于我们选择最左端元素作为基准数，那么在哨兵划分完成后，基准数被交换至数组最右端，导致左子数组长度为 <span class="arithmatex">\(n - 1\)</span>、右子数组长度为 <span class="arithmatex">\(0\)</span> 。如此递归下去，每轮哨兵划分后都有一个子数组的长度为 <span class="arithmatex">\(0\)</span> ，分治策略失效，快速排序退化为“冒泡排序”的近似形式。</p>
<p>为了尽量避免这种情况发生，<strong>我们可以优化哨兵划分中的基准数的选取策略</strong>。例如，我们可以随机选取一个元素作为基准数。然而，如果运气不佳，每次都选到不理想的基准数，效率仍然不尽如人意。</p>
<p>需要注意的是，编程语言通常生成的是“伪随机数”。如果我们针对伪随机数序列构建一个特定的测试样例，那么快速排序的效率仍然可能劣化。</p>
<p>为了进一步改进，我们可以在数组中选取三个候选元素（通常为数组的首、尾、中点元素），<strong>并将这三个候选元素的中位数作为基准数</strong>。这样一来，基准数“既不太小也不太大”的概率将大幅提升。当然，我们还可以选取更多候选元素，以进一步提高算法的稳健性。采用这种方法后，时间复杂度劣化至 <span class="arithmatex">\(O(n^2)\)</span> 的概率大大降低。</p>

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@ -112,6 +112,11 @@
text-transform: none;
}
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color: var(--md-typeset-a-color);
text-decoration: underline;
}
.md-typeset code {
border-radius: 0.2rem;
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@ -150,34 +155,6 @@ body {
width: 100%; /* Default to full width */
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/* rounded button */
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display: inline-flex; /* Use flexbox for alignment */
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justify-content: center; /* Center items horizontally */
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text-decoration: none; /* Remove underline from anchor text */
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width: 1.2em;
height: auto;
margin-right: 0.5em; /* Add some space between the SVG icon and the text */
}
/* Admonition for python tutor */
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.md-typeset details.pythontutor {
@ -237,10 +214,69 @@ body {
max-width: 70vw;
}
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margin: 0 auto;
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}
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}
a:hover .hero-caption {
text-decoration: underline;
color: var(--md-typeset-btn-color);
font-weight: bold;
}
/* code badge */

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