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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<!-- Page content -->
<h1 id="2">第 2 章 &nbsp; 复杂度分析<a class="headerlink" href="#2" title="Permanent link">&para;</a></h1>
<div class="center-table">
<p><img alt="复杂度分析" src="../assets/covers/chapter_complexity_analysis.jpg" width="600" /></p>
<p><a class="glightbox" href="../assets/covers/chapter_complexity_analysis.jpg" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="复杂度分析" src="../assets/covers/chapter_complexity_analysis.jpg" width="600" /></a></p>
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<div class="admonition abstract">
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</head>
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.6 &nbsp; 编辑距离问题
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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</div>
</div>
<p>图 2-1 展示了该求和函数的流程框图。</p>
<p><img alt="求和函数的流程框图" src="../iteration_and_recursion.assets/iteration.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="求和函数的流程框图" src="../iteration_and_recursion.assets/iteration.png" /></a></p>
<p align="center"> 图 2-1 &nbsp; 求和函数的流程框图 </p>
<p>此求和函数的操作数量与输入数据大小 <span class="arithmatex">\(n\)</span> 成正比，或者说成“线性关系”。实际上，<strong>时间复杂度描述的就是这个“线性关系”</strong>。相关内容将会在下一节中详细介绍。</p>
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</div>
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<p>图 2-2 给出了该嵌套循环的流程框图。</p>
<p><img alt="嵌套循环的流程框图" src="../iteration_and_recursion.assets/nested_iteration.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/nested_iteration.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="嵌套循环的流程框图" src="../iteration_and_recursion.assets/nested_iteration.png" /></a></p>
<p align="center"> 图 2-2 &nbsp; 嵌套循环的流程框图 </p>
<p>在这种情况下，函数的操作数量与 <span class="arithmatex">\(n^2\)</span> 成正比，或者说算法运行时间和输入数据大小 <span class="arithmatex">\(n\)</span> 成“平方关系”。</p>
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</div>
</div>
<p>图 2-3 展示了该函数的递归过程。</p>
<p><img alt="求和函数的递归过程" src="../iteration_and_recursion.assets/recursion_sum.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="求和函数的递归过程" src="../iteration_and_recursion.assets/recursion_sum.png" /></a></p>
<p align="center"> 图 2-3 &nbsp; 求和函数的递归过程 </p>
<p>虽然从计算角度看，迭代与递归可以得到相同的结果，<strong>但它们代表了两种完全不同的思考和解决问题的范式</strong>。</p>
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<li>递归调用函数会产生额外的开销。<strong>因此递归通常比循环的时间效率更低</strong>。</li>
</ul>
<p>如图 2-4 所示，在触发终止条件前，同时存在 <span class="arithmatex">\(n\)</span> 个未返回的递归函数，<strong>递归深度为 <span class="arithmatex">\(n\)</span></strong> 。</p>
<p><img alt="递归调用深度" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_sum_depth.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="递归调用深度" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></a></p>
<p align="center"> 图 2-4 &nbsp; 递归调用深度 </p>
<p>在实际中，编程语言允许的递归深度通常是有限的，过深的递归可能导致栈溢出报错。</p>
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<li><strong>普通递归</strong>：求和操作是在“归”的过程中执行的，每层返回后都要再执行一次求和操作。</li>
<li><strong>尾递归</strong>：求和操作是在“递”的过程中执行的，“归”的过程只需层层返回。</li>
</ul>
<p><img alt="尾递归过程" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/tail_recursion_sum.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="尾递归过程" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></a></p>
<p align="center"> 图 2-5 &nbsp; 尾递归过程 </p>
<div class="admonition tip">
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</div>
</div>
<p>观察以上代码，我们在函数内递归调用了两个函数，<strong>这意味着从一个调用产生了两个调用分支</strong>。如图 2-6 所示，这样不断递归调用下去，最终将产生一个层数为 <span class="arithmatex">\(n\)</span> 的「递归树 recursion tree」。</p>
<p><img alt="斐波那契数列的递归树" src="../iteration_and_recursion.assets/recursion_tree.png" /></p>
<p><a class="glightbox" href="../iteration_and_recursion.assets/recursion_tree.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="斐波那契数列的递归树" src="../iteration_and_recursion.assets/recursion_tree.png" /></a></p>
<p align="center"> 图 2-6 &nbsp; 斐波那契数列的递归树 </p>
<p>本质上看，递归体现“将问题分解为更小子问题”的思维范式，这种分治策略是至关重要的。</p>
@ -5110,10 +4945,15 @@ aria-label="页脚"
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</head>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
.gslide-title { margin-top: 0px; user-select: text; }
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body[data-md-color-scheme="slate"] .gdesc-inner { background: var(--md-default-bg-color);}
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
</span>
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</span>
</a>
</li>
@ -2983,14 +2874,6 @@
14.7 &nbsp; 小结
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</span>
</a>
</li>
@ -3049,14 +2932,6 @@
第 15 章 &nbsp; 贪心
</span>
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</span>
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@ -3088,14 +2963,6 @@
15.1 &nbsp; 贪心算法
</span>
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</span>
</a>
</li>
@ -3116,14 +2983,6 @@
15.2 &nbsp; 分数背包问题
</span>
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</span>
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@ -3144,14 +3003,6 @@
15.3 &nbsp; 最大容量问题
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</span>
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@ -3172,14 +3023,6 @@
15.4 &nbsp; 最大切分乘积问题
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</span>
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</li>
@ -3200,14 +3043,6 @@
15.5 &nbsp; 小结
</span>
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@ -3673,10 +3508,15 @@ aria-label="页脚"
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</head>
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html.glightbox-open { overflow: initial; height: 100%; }
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@ -1986,14 +1997,6 @@
10.2 &nbsp; 二分查找插入点
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@ -2014,14 +2017,6 @@
10.3 &nbsp; 二分查找边界
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@ -2441,14 +2436,6 @@
第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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@ -2508,14 +2487,6 @@
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@ -2536,14 +2507,6 @@
12.3 &nbsp; 构建树问题
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@ -2564,14 +2527,6 @@
12.4 &nbsp; 汉诺塔问题
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@ -2592,14 +2547,6 @@
12.5 &nbsp; 小结
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@ -2831,14 +2778,6 @@
第 14 章 &nbsp; 动态规划
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@ -2870,14 +2809,6 @@
14.1 &nbsp; 初探动态规划
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@ -2898,14 +2829,6 @@
14.2 &nbsp; DP 问题特性
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@ -2926,14 +2849,6 @@
14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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14.7 &nbsp; 小结
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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@ -3199,14 +3058,6 @@
15.3 &nbsp; 最大容量问题
</span>
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15.4 &nbsp; 最大切分乘积问题
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@ -3255,14 +3098,6 @@
15.5 &nbsp; 小结
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</span>
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</li>
@ -3600,7 +3435,7 @@
<li><strong>指令空间</strong>：用于保存编译后的程序指令，在实际统计中通常忽略不计。</li>
</ul>
<p>在分析一段程序的空间复杂度时，<strong>我们通常统计暂存数据、栈帧空间和输出数据三部分</strong>。</p>
<p><img alt="算法使用的相关空间" src="../space_complexity.assets/space_types.png" /></p>
<p><a class="glightbox" href="../space_complexity.assets/space_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="算法使用的相关空间" src="../space_complexity.assets/space_types.png" /></a></p>
<p align="center"> 图 2-15 &nbsp; 算法使用的相关空间 </p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Zig</label></div>
@ -4226,7 +4061,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
\text{常数阶} &lt; \text{对数阶} &lt; \text{线性阶} &lt; \text{平方阶} &lt; \text{指数阶}
\end{aligned}
\]</div>
<p><img alt="常见的空间复杂度类型" src="../space_complexity.assets/space_complexity_common_types.png" /></p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="常见的空间复杂度类型" src="../space_complexity.assets/space_complexity_common_types.png" /></a></p>
<p align="center"> 图 2-16 &nbsp; 常见的空间复杂度类型 </p>
<h3 id="1-o1">1. &nbsp; 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">&para;</a></h3>
@ -4903,7 +4738,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
</div>
</div>
</div>
<p><img alt="递归函数产生的线性阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="递归函数产生的线性阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></a></p>
<p align="center"> 图 2-17 &nbsp; 递归函数产生的线性阶空间复杂度 </p>
<h3 id="3-on2">3. &nbsp; 平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-on2" title="Permanent link">&para;</a></h3>
@ -5238,7 +5073,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
</div>
</div>
</div>
<p><img alt="递归函数产生的平方阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_recursive_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="递归函数产生的平方阶空间复杂度" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></a></p>
<p align="center"> 图 2-18 &nbsp; 递归函数产生的平方阶空间复杂度 </p>
<h3 id="4-o2n">4. &nbsp; 指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-o2n" title="Permanent link">&para;</a></h3>
@ -5387,7 +5222,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
</div>
</div>
</div>
<p><img alt="满二叉树产生的指数阶空间复杂度" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
<p><a class="glightbox" href="../space_complexity.assets/space_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="满二叉树产生的指数阶空间复杂度" src="../space_complexity.assets/space_complexity_exponential.png" /></a></p>
<p align="center"> 图 2-19 &nbsp; 满二叉树产生的指数阶空间复杂度 </p>
<h3 id="5-olog-n">5. &nbsp; 对数阶 <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-olog-n" title="Permanent link">&para;</a></h3>
@ -5560,10 +5395,15 @@ aria-label="页脚"
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@ -1931,14 +1942,6 @@
10.2 &nbsp; 二分查找插入点
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@ -1959,14 +1962,6 @@
10.3 &nbsp; 二分查找边界
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@ -2386,14 +2381,6 @@
第 12 章 &nbsp; 分治
</span>
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@ -2425,14 +2412,6 @@
12.1 &nbsp; 分治算法
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</span>
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@ -2453,14 +2432,6 @@
12.2 &nbsp; 分治搜索策略
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</span>
</a>
</li>
@ -2481,14 +2452,6 @@
12.3 &nbsp; 构建树问题
</span>
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</span>
</a>
</li>
@ -2509,14 +2472,6 @@
12.4 &nbsp; 汉诺塔问题
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@ -2537,14 +2492,6 @@
12.5 &nbsp; 小结
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</span>
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@ -2776,14 +2723,6 @@
第 14 章 &nbsp; 动态规划
</span>
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@ -2815,14 +2754,6 @@
14.1 &nbsp; 初探动态规划
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</span>
</a>
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@ -2843,14 +2774,6 @@
14.2 &nbsp; DP 问题特性
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -2871,14 +2794,6 @@
14.3 &nbsp; DP 解题思路
</span>
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</span>
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</li>
@ -2899,14 +2814,6 @@
14.4 &nbsp; 0-1 背包问题
</span>
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</span>
</a>
</li>
@ -2927,14 +2834,6 @@
14.5 &nbsp; 完全背包问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -2955,14 +2854,6 @@
14.6 &nbsp; 编辑距离问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -2983,14 +2874,6 @@
14.7 &nbsp; 小结
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
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</li>
@ -3049,14 +2932,6 @@
第 15 章 &nbsp; 贪心
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
@ -3088,14 +2963,6 @@
15.1 &nbsp; 贪心算法
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -3116,14 +2983,6 @@
15.2 &nbsp; 分数背包问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -3144,14 +3003,6 @@
15.3 &nbsp; 最大容量问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -3172,14 +3023,6 @@
15.4 &nbsp; 最大切分乘积问题
</span>
<span class="md-status md-status--new" title="最近添加">
</span>
</a>
</li>
@ -3200,14 +3043,6 @@
15.5 &nbsp; 小结
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@ -3681,10 +3516,15 @@ aria-label="页脚"
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@ -60,7 +60,18 @@
</head>
<link href="../../assets/stylesheets/glightbox.min.css" rel="stylesheet"/><style>
html.glightbox-open { overflow: initial; height: 100%; }
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@ -2027,14 +2038,6 @@
10.2 &nbsp; 二分查找插入点
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@ -2055,14 +2058,6 @@
10.3 &nbsp; 二分查找边界
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@ -2482,14 +2477,6 @@
第 12 章 &nbsp; 分治
</span>
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@ -2521,14 +2508,6 @@
12.1 &nbsp; 分治算法
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@ -2549,14 +2528,6 @@
12.2 &nbsp; 分治搜索策略
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@ -2577,14 +2548,6 @@
12.3 &nbsp; 构建树问题
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@ -2605,14 +2568,6 @@
12.4 &nbsp; 汉诺塔问题
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</a>
</li>
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12.5 &nbsp; 小结
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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14.4 &nbsp; 0-1 背包问题
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14.5 &nbsp; 完全背包问题
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14.6 &nbsp; 编辑距离问题
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14.7 &nbsp; 小结
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第 15 章 &nbsp; 贪心
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15.1 &nbsp; 贪心算法
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15.2 &nbsp; 分数背包问题
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15.3 &nbsp; 最大容量问题
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15.4 &nbsp; 最大切分乘积问题
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15.5 &nbsp; 小结
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<li>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大呈线性增长。此算法的时间复杂度被称为“线性阶”。</li>
<li>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，虽然运行时间很长，但它与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为“常数阶”。</li>
</ul>
<p><img alt="算法 A、B 和 C 的时间增长趋势" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_simple_example.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="算法 A、B 和 C 的时间增长趋势" src="../time_complexity.assets/time_complexity_simple_example.png" /></a></p>
<p align="center"> 图 2-7 &nbsp; 算法 A、B 和 C 的时间增长趋势 </p>
<p>相较于直接统计算法运行时间，时间复杂度分析有哪些特点呢？</p>
@ -4212,7 +4047,7 @@ T(n) = 3 + 2n
<p>若存在正实数 <span class="arithmatex">\(c\)</span> 和实数 <span class="arithmatex">\(n_0\)</span> ，使得对于所有的 <span class="arithmatex">\(n &gt; n_0\)</span> ，均有 <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span> ，则可认为 <span class="arithmatex">\(f(n)\)</span> 给出了 <span class="arithmatex">\(T(n)\)</span> 的一个渐近上界，记为 <span class="arithmatex">\(T(n) = O(f(n))\)</span> 。</p>
</div>
<p>如图 2-8 所示，计算渐近上界就是寻找一个函数 <span class="arithmatex">\(f(n)\)</span> ，使得当 <span class="arithmatex">\(n\)</span> 趋向于无穷大时，<span class="arithmatex">\(T(n)\)</span> 和 <span class="arithmatex">\(f(n)\)</span> 处于相同的增长级别，仅相差一个常数项 <span class="arithmatex">\(c\)</span> 的倍数。</p>
<p><img alt="函数的渐近上界" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/asymptotic_upper_bound.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="函数的渐近上界" src="../time_complexity.assets/asymptotic_upper_bound.png" /></a></p>
<p align="center"> 图 2-8 &nbsp; 函数的渐近上界 </p>
<h2 id="233">2.3.3 &nbsp; 推算方法<a class="headerlink" href="#233" title="Permanent link">&para;</a></h2>
@ -4472,7 +4307,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
\text{常数阶} &lt; \text{对数阶} &lt; \text{线性阶} &lt; \text{线性对数阶} &lt; \text{平方阶} &lt; \text{指数阶} &lt; \text{阶乘阶}
\end{aligned}
\]</div>
<p><img alt="常见的时间复杂度类型" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_common_types.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="常见的时间复杂度类型" src="../time_complexity.assets/time_complexity_common_types.png" /></a></p>
<p align="center"> 图 2-9 &nbsp; 常见的时间复杂度类型 </p>
<h3 id="1-o1">1. &nbsp; 常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-o1" title="Permanent link">&para;</a></h3>
@ -5073,7 +4908,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
</div>
</div>
<p>图 2-10 对比了常数阶、线性阶和平方阶三种时间复杂度。</p>
<p><img alt="常数阶、线性阶和平方阶的时间复杂度" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="常数阶、线性阶和平方阶的时间复杂度" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></a></p>
<p align="center"> 图 2-10 &nbsp; 常数阶、线性阶和平方阶的时间复杂度 </p>
<p>以冒泡排序为例，外层循环执行 <span class="arithmatex">\(n - 1\)</span> 次，内层循环执行 <span class="arithmatex">\(n-1\)</span>、<span class="arithmatex">\(n-2\)</span>、<span class="arithmatex">\(\dots\)</span>、<span class="arithmatex">\(2\)</span>、<span class="arithmatex">\(1\)</span> 次，平均为 <span class="arithmatex">\(n / 2\)</span> 次，因此时间复杂度为 <span class="arithmatex">\(O((n - 1) n / 2) = O(n^2)\)</span> 。</p>
@ -5534,7 +5369,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
</div>
</div>
</div>
<p><img alt="指数阶的时间复杂度" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_exponential.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="指数阶的时间复杂度" src="../time_complexity.assets/time_complexity_exponential.png" /></a></p>
<p align="center"> 图 2-11 &nbsp; 指数阶的时间复杂度 </p>
<p>在实际算法中，指数阶常出现于递归函数中。例如在以下代码中，其递归地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止：</p>
@ -5800,7 +5635,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!
</div>
</div>
</div>
<p><img alt="对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic.png" /></a></p>
<p align="center"> 图 2-12 &nbsp; 对数阶的时间复杂度 </p>
<p>与指数阶类似，对数阶也常出现于递归函数中。以下代码形成了一个高度为 <span class="arithmatex">\(\log_2 n\)</span> 的递归树：</p>
@ -6087,7 +5922,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
</div>
</div>
<p>图 2-13 展示了线性对数阶的生成方式。二叉树的每一层的操作总数都为 <span class="arithmatex">\(n\)</span> ，树共有 <span class="arithmatex">\(\log_2 n + 1\)</span> 层，因此时间复杂度为 <span class="arithmatex">\(O(n \log n)\)</span> 。</p>
<p><img alt="线性对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_logarithmic_linear.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="线性对数阶的时间复杂度" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></a></p>
<p align="center"> 图 2-13 &nbsp; 线性对数阶的时间复杂度 </p>
<p>主流排序算法的时间复杂度通常为 <span class="arithmatex">\(O(n \log n)\)</span> ，例如快速排序、归并排序、堆排序等。</p>
@ -6265,7 +6100,7 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
</div>
</div>
</div>
<p><img alt="阶乘阶的时间复杂度" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
<p><a class="glightbox" href="../time_complexity.assets/time_complexity_factorial.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="阶乘阶的时间复杂度" src="../time_complexity.assets/time_complexity_factorial.png" /></a></p>
<p align="center"> 图 2-14 &nbsp; 阶乘阶的时间复杂度 </p>
<p>请注意，因为当 <span class="arithmatex">\(n \geq 4\)</span> 时恒有 <span class="arithmatex">\(n! &gt; 2^n\)</span> ，所以阶乘阶比指数阶增长得更快，在 <span class="arithmatex">\(n\)</span> 较大时也是不可接受的。</p>
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