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<h1 id="25">2.5 &nbsp; 小结<a class="headerlink" href="#25" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p><strong>算法效率评估</strong></p>
<ul>
<li>时间效率和空间效率是衡量算法优劣的两个主要评价指标。</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>复杂度分析可以消除实际测试的弊端，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，如在输入的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，对应函数渐近上界，反映当 <span class="arithmatex">\(n\)</span> 趋向正无穷时，操作数量 <span class="arithmatex">\(T(n)\)</span> 的增长级别。</li>
<li>推算时间复杂度分为两步，首先统计操作数量，然后判断渐近上界。</li>
<li>常见时间复杂度从低到高排列有 <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> 和 <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>空间复杂度的作用类似于时间复杂度，用于衡量算法占用内存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，输入空间不纳入空间复杂度计算。暂存空间可分为暂存数据、栈帧空间和指令空间，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刻下的空间复杂度。</li>
<li>常见空间复杂度从低到高排列有 <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> 和 <span class="arithmatex">\(O(2^n)\)</span> 等。</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>：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）不支持自动优化尾递归，因此通常认为空间复杂度是 <span class="arithmatex">\(O(n)\)</span> 。</p>
<p><strong>Q</strong>：函数和方法这两个术语的区别是什么？</p>
<p>「函数 function」可以被独立执行，所有参数都以显式传递。「方法 method」与一个对象关联，被隐式传递给调用它的对象，能够对类的实例中包含的数据进行操作。</p>
<p>下面以几种常见的编程语言为例来说明。</p>
<ul>
<li>C 语言是过程式编程语言，没有面向对象的概念，所以只有函数。但我们可以通过创建结构体（struct）来模拟面向对象编程，与结构体相关联的函数就相当于其他编程语言中的方法。</li>
<li>Java 和 C# 是面向对象的编程语言，代码块（方法）通常作为某个类的一部分。静态方法的行为类似于函数，因为它被绑定在类上，不能访问特定的实例变量。</li>
<li>C++ 和 Python 既支持过程式编程（函数），也支持面向对象编程（方法）。</li>
</ul>
<p><strong>Q</strong>：图解“常见的空间复杂度类型”反映的是否是占用空间的绝对大小？</p>
<p>不是，该图展示的是空间复杂度，其反映的是增长趋势，而不是占用空间的绝对大小。</p>
<p>假设取 <span class="arithmatex">\(n = 8\)</span> ，你可能会发现每条曲线的值与函数对应不上。这是因为每条曲线都包含一个常数项，用于将取值范围压缩到一个视觉舒适的范围内。</p>
<p>在实际中，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 <span class="arithmatex">\(n = 8\)</span> 之下的最优解法。但对于 <span class="arithmatex">\(n = 8^5\)</span> 就很好选了，这时增长趋势已经占主导了。</p>

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