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    时间复杂度
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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">算法效率评估<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<ul>
<li>时间效率和空间效率是算法性能的两个重要的评价维度。</li>
<li>我们可以通过实际测试来评估算法效率，但难以排除测试环境的干扰，并且非常耗费计算资源。</li>
<li>复杂度分析克服了实际测试的弊端，分析结果适用于所有运行平台，并且可以体现不同数据大小下的算法效率。</li>
</ul>
<h3 id="_2">时间复杂度<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<ul>
<li>时间复杂度统计算法运行时间随着数据量变大时的增长趋势，可以有效评估算法效率，但在某些情况下可能失效，比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性。</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，其反映当 <span class="arithmatex">\(n\)</span> 趋于正无穷时，<span class="arithmatex">\(T(n)\)</span> 处于何种增长级别。</li>
<li>推算时间复杂度分为两步，首先统计计算操作数量，再判断渐近上界。</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> 。</li>
<li>某些算法的时间复杂度不是恒定的，而是与输入数据的分布有关。时间复杂度分为最差时间复杂度和最佳时间复杂度，后者几乎不用，因为输入数据需要满足苛刻的条件才能达到最佳情况。</li>
<li>平均时间复杂度可以反映在随机数据输入下的算法效率，最贴合实际使用情况下的算法性能。计算平均时间复杂度需要统计输入数据的分布，以及综合后的数学期望。</li>
</ul>
<h3 id="_3">空间复杂度<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
<ul>
<li>与时间复杂度的定义类似，空间复杂度统计算法占用空间随着数据量变大时的增长趋势。</li>
<li>算法运行中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，其中栈帧空间一般在递归函数中才会影响到空间复杂度。</li>
<li>我们一般只关心最差空间复杂度，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> 。</li>
</ul>


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