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<h1 id="24">2.4. &nbsp; 小结<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
<p><strong>算法效率评估</strong></p>
<ul>
<li>时间效率和空间效率是评价算法性能的两个关键维度。</li>
<li>我们可以通过实际测试来评估算法效率，但难以消除测试环境的影响，且会耗费大量计算资源。</li>
<li>复杂度分析可以克服实际测试的弊端，分析结果适用于所有运行平台，并且能够揭示算法在不同数据规模下的效率。</li>
</ul>
<p><strong>时间复杂度</strong></p>
<ul>
<li>时间复杂度用于衡量算法运行时间随数据量增长的趋势，可以有效评估算法效率，但在某些情况下可能失效，如在输入数据量较小或时间复杂度相同时，无法精确对比算法效率的优劣。</li>
<li>最差时间复杂度使用大 <span class="arithmatex">\(O\)</span> 符号表示，即函数渐近上界，反映当 <span class="arithmatex">\(n\)</span> 趋向正无穷时，<span class="arithmatex">\(T(n)\)</span> 的增长级别。</li>
<li>推算时间复杂度分为两步，首先统计计算操作数量，然后判断渐近上界。</li>
<li>常见时间复杂度从小到大排列有 <span class="arithmatex">\(O(1)\)</span> , <span class="arithmatex">\(O(\log n)\)</span> , <span class="arithmatex">\(O(n)\)</span> , <span class="arithmatex">\(O(n \log n)\)</span> , <span class="arithmatex">\(O(n^2)\)</span> , <span class="arithmatex">\(O(2^n)\)</span> , <span class="arithmatex">\(O(n!)\)</span> 等。</li>
<li>某些算法的时间复杂度非固定，而是与输入数据的分布有关。时间复杂度分为最差、最佳、平均时间复杂度，最佳时间复杂度几乎不用，因为输入数据一般需要满足严格条件才能达到最佳情况。</li>
<li>平均时间复杂度反映算法在随机数据输入下的运行效率，最接近实际应用中的算法性能。计算平均时间复杂度需要统计输入数据分布以及综合后的数学期望。</li>
</ul>
<p><strong>空间复杂度</strong></p>
<ul>
<li>类似于时间复杂度，空间复杂度用于衡量算法占用空间随数据量增长的趋势。</li>
<li>算法运行过程中的相关内存空间可分为输入空间、暂存空间、输出空间。通常情况下，输入空间不计入空间复杂度计算。暂存空间可分为指令空间、数据空间、栈帧空间，其中栈帧空间通常仅在递归函数中影响空间复杂度。</li>
<li>我们通常只关注最差空间复杂度，即统计算法在最差输入数据和最差运行时间点下的空间复杂度。</li>
<li>常见空间复杂度从小到大排列有 <span class="arithmatex">\(O(1)\)</span> , <span class="arithmatex">\(O(\log n)\)</span> , <span class="arithmatex">\(O(n)\)</span> , <span class="arithmatex">\(O(n^2)\)</span> , <span class="arithmatex">\(O(2^n)\)</span> 等。</li>
</ul>
<h2 id="241-q-a">2.4.1. &nbsp; Q &amp; A<a class="headerlink" href="#241-q-a" title="Permanent link">&para;</a></h2>
<div class="admonition question">
<p class="admonition-title">尾递归的空间复杂度是 <span class="arithmatex">\(O(1)\)</span> 吗？</p>
<p>理论上，尾递归函数的空间复杂度可以被优化至 <span class="arithmatex">\(O(1)\)</span> 。不过绝大多数编程语言（例如 Java, Python, C++, Go, C# 等）
都不支持自动优化尾递归，因此一般来说空间复杂度是 <span class="arithmatex">\(O(n)\)</span> 。</p>
</div>
<div class="admonition question">
<p class="admonition-title">函数和方法这两个术语的区别是什么？</p>
<p>函数（function）可以独立被执行，所有参数都以显式传递。
方法（method）与一个对象关联，方法被隐式传递给调用它的对象，方法能够对类的实例中包含的数据进行操作。</p>
</div>
<div class="admonition question">
<p class="admonition-title">图片“空间复杂度的常见类型”反映的是否是占用空间的绝对大小？</p>
<p>不是，该图片展示的是空间复杂度（即增长趋势），而不是占用空间的绝对大小。每个曲线都包含一个常数项，用来把所有曲线的取值范围压缩到一个视觉舒适的范围内。
实际中，因为我们通常不知道每个方法的“常数项”复杂度是多少，所以一般无法仅凭复杂度来选择 $n = 8 之下的最优解法；但相对地 <span class="arithmatex">\(n = 8^5\)</span> 就很好选了，这是复杂度占主导的情况。</p>
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