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3.3 数字编码 *
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第 0 章 前言
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第 0 章 前言
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0.1 关于本书
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0.2 如何使用本书
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0.3 小结
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第 1 章 初识算法
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第 1 章 初识算法
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1.1 算法无处不在
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1.2 算法是什么
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1.3 小结
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第 2 章 时空复杂度
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第 2 章 时空复杂度
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2.1 算法效率评估
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2.2 迭代与递归
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2.3 时间复杂度
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2.4 空间复杂度
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2.5 小结
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第 3 章 数据结构
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第 3 章 数据结构
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3.1 数据结构分类
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3.2 基本数据类型
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3.3 数字编码 *
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3.3 数字编码 *
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目录
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3.3.1 原码、反码和补码
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3.3.2 浮点数编码
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3.4 字符编码 *
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3.5 小结
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第 4 章 数组与链表
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4.2 链表
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4.3 列表
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第 5 章 栈与队列
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5.1 栈
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5.2 队列
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5.3 双向队列
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5.4 小结
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第 6 章 哈希表
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第 6 章 哈希表
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6.1 哈希表
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6.2 哈希冲突
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6.3 哈希算法
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6.4 小结
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第 7 章 树
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第 7 章 树
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7.1 二叉树
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7.2 二叉树遍历
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7.3 二叉树数组表示
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< li class = "md-nav__item" >
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7.4 二叉搜索树
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7.5 AVL 树 *
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7.6 小结
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第 8 章 堆
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第 8 章 堆
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< / label >
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< li class = "md-nav__item" >
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8.1 堆
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< li class = "md-nav__item" >
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
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8.2 建堆操作
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8.3 Top-K 问题
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8.4 小结
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< div class = "md-nav__link md-nav__container" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59-.44.06M6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43.43 0 .83.16 1.12.43l4.28-2.53H6.6M12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68.93 0 1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25 5.51-9.64m2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24-4.5-2.58Z" / > < / svg >
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第 9 章 图
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第 9 章 图
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< li class = "md-nav__item" >
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< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
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9.1 图
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9.2 图基础操作
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9.3 图的遍历
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9.4 小结
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< div class = "md-nav__link md-nav__container" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4h18M3 16v-2h6v2H3m0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1H3Z" / > < / svg >
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第 10 章 搜索
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第 10 章 搜索
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< / label >
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10.1 二分查找
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search_insertion/" class = "md-nav__link" >
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10.2 二分查找插入点
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10.3 二分查找边界
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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< span class = "md-ellipsis" >
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10.4 哈希优化策略
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< / span >
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< li class = "md-nav__item" >
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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10.5 重识搜索算法
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/summary/" class = "md-nav__link" >
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10.6 小结
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第 11 章 排序
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第 11 章 排序
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11.1 排序算法
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11.2 选择排序
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11.3 冒泡排序
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11.4 插入排序
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11.5 快速排序
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11.6 归并排序
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11.7 堆排序
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11.8 桶排序
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11.9 计数排序
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11.10 基数排序
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11.11 小结
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第 12 章 分治
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第 12 章 分治
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12.2 分治搜索策略
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12.3 构建树问题
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12.4 汉诺塔问题
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12.5 小结
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第 13 章 回溯
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第 13 章 回溯
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13.1 回溯算法
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13.2 全排列问题
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13.3 子集和问题
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13.4 N 皇后问题
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3.3.2 浮点数编码
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< p > 在本书中，标题带有的 * 符号的是选读章节。如果你时间有限或感到理解困难，可以先跳过，等学完必读章节后再单独攻克。< / p >
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< h2 id = "331" > 3.3.1 原码、反码和补码< a class = "headerlink" href = "#331" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 在上一节的表格中我们发现，所有整数类型能够表示的负数都比正数多一个，例如 < code > byte< / code > 的取值范围是 < span class = "arithmatex" > \([-128, 127]\)< / span > 。这个现象比较反直觉，它的内在原因涉及到原码、反码、补码的相关知识。< / p >
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< p > 首先需要指出，< strong > 数字是以“补码”的形式存储在计算机中的< / strong > 。在分析这样做的原因之前，我们首先给出三者的定义。< / p >
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< ul >
< li > < strong > 原码< / strong > ：我们将数字的二进制表示的最高位视为符号位，其中 < span class = "arithmatex" > \(0\)< / span > 表示正数，< span class = "arithmatex" > \(1\)< / span > 表示负数，其余位表示数字的值。< / li >
< li > < strong > 反码< / strong > ：正数的反码与其原码相同，负数的反码是对其原码除符号位外的所有位取反。< / li >
< li > < strong > 补码< / strong > ：正数的补码与其原码相同，负数的补码是在其反码的基础上加 < span class = "arithmatex" > \(1\)< / span > 。< / li >
< / ul >
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< p > 图 3-4 展示了原吗、反码和补码之间的转换方法。< / p >
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< p > < img alt = "原码、反码与补码之间的相互转换" src = "../number_encoding.assets/1s_2s_complement.png" / > < / p >
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< p align = "center" > 图 3-4 原码、反码与补码之间的相互转换 < / p >
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< p > 「原码 true form」虽然最直观， ， < strong > 负数的原码不能直接用于运算< / strong > 。例如在原码下计算 < span class = "arithmatex" > \(1 + (-2)\)< / span > ，得到的结果是 < span class = "arithmatex" > \(-3\)< / span > ，这显然是不对的。< / p >
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< div class = "arithmatex" > \[
\begin{aligned}
& 1 + (-2) \newline
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& \rightarrow 0000 \space 0001 + 1000 \space 0010 \newline
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& = 1000 \space 0011 \newline
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& \rightarrow -3
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\end{aligned}
\]< / div >
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< p > 为了解决此问题，计算机引入了「反码 1's complement code」。如果我们先将原码转换为反码， < span class = "arithmatex" > \(1 + (-2)\)< / span > ，最后将结果从反码转化回原码，则可得到正确结果 < span class = "arithmatex" > \(-1\)< / span > 。< / p >
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< div class = "arithmatex" > \[
\begin{aligned}
& 1 + (-2) \newline
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& \rightarrow 0000 \space 0001 \space \text{(原码)} + 1000 \space 0010 \space \text{(原码)} \newline
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& = 0000 \space 0001 \space \text{(反码)} + 1111 \space 1101 \space \text{(反码)} \newline
& = 1111 \space 1110 \space \text{(反码)} \newline
& = 1000 \space 0001 \space \text{(原码)} \newline
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& \rightarrow -1
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\end{aligned}
\]< / div >
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< p > 另一方面，< strong > 数字零的原码有 < span class = "arithmatex" > \(+0\)< / span > 和 < span class = "arithmatex" > \(-0\)< / span > 两种表示方式< / strong > 。这意味着数字零对应着两个不同的二进制编码，其可能会带来歧义。比如在条件判断中，如果没有区分正零和负零，则可能会导致判断结果出错。而如果我们想要处理正零和负零歧义，则需要引入额外的判断操作，其可能会降低计算机的运算效率。< / p >
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< div class = "arithmatex" > \[
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\begin{aligned}
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+0 & \rightarrow 0000 \space 0000 \newline
-0 & \rightarrow 1000 \space 0000
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\end{aligned}
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\]< / div >
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< p > 与原码一样，反码也存在正负零歧义问题，因此计算机进一步引入了「补码 2's complement code」。我们先来观察一下负零的原码、反码、补码的转换过程： < / p >
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< div class = "arithmatex" > \[
\begin{aligned}
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-0 \rightarrow \space & 1000 \space 0000 \space \text{(原码)} \newline
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= \space & 1111 \space 1111 \space \text{(反码)} \newline
= 1 \space & 0000 \space 0000 \space \text{(补码)} \newline
\end{aligned}
\]< / div >
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< p > 在负零的反码基础上加 < span class = "arithmatex" > \(1\)< / span > 会产生进位，但 < code > byte< / code > 类型的长度只有 8 位，因此溢出到第 9 位的 < span class = "arithmatex" > \(1\)< / span > 会被舍弃。也就是说，< strong > 负零的补码为 < span class = "arithmatex" > \(0000 \space 0000\)< / span > ，与正零的补码相同< / strong > 。这意味着在补码表示中只存在一个零，正负零歧义从而得到解决。< / p >
< p > 还剩余最后一个疑惑：< code > byte< / code > 类型的取值范围是 < span class = "arithmatex" > \([-128, 127]\)< / span > ，多出来的一个负数 < span class = "arithmatex" > \(-128\)< / span > 是如何得到的呢？我们注意到，区间 < span class = "arithmatex" > \([-127, +127]\)< / span > 内的所有整数都有对应的原码、反码和补码，并且原码和补码之间是可以互相转换的。< / p >
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< p > 然而，< strong > 补码 < span class = "arithmatex" > \(1000 \space 0000\)< / span > 是一个例外，它并没有对应的原码< / strong > 。根据转换方法，我们得到该补码的原码为 < span class = "arithmatex" > \(0000 \space 0000\)< / span > 。这显然是矛盾的，因为该原码表示数字 < span class = "arithmatex" > \(0\)< / span > ，它的补码应该是自身。计算机规定这个特殊的补码 < span class = "arithmatex" > \(1000 \space 0000\)< / span > 代表 < span class = "arithmatex" > \(-128\)< / span > 。实际上，< span class = "arithmatex" > \((-1) + (-127)\)< / span > 在补码下的计算结果就是 < span class = "arithmatex" > \(-128\)< / span > 。< / p >
< div class = "arithmatex" > \[
\begin{aligned}
& (-127) + (-1) \newline
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& \rightarrow 1111 \space 1111 \space \text{(原码)} + 1000 \space 0001 \space \text{(原码)} \newline
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& = 1000 \space 0000 \space \text{(反码)} + 1111 \space 1110 \space \text{(反码)} \newline
& = 1000 \space 0001 \space \text{(补码)} + 1111 \space 1111 \space \text{(补码)} \newline
& = 1000 \space 0000 \space \text{(补码)} \newline
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& \rightarrow -128
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\end{aligned}
\]< / div >
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< p > 你可能已经发现，上述的所有计算都是加法运算。这暗示着一个重要事实：< strong > 计算机内部的硬件电路主要是基于加法运算设计的< / strong > 。这是因为加法运算相对于其他运算（比如乘法、除法和减法）来说，硬件实现起来更简单，更容易进行并行化处理，运算速度更快。< / p >
< p > 请注意，这并不意味着计算机只能做加法。< strong > 通过将加法与一些基本逻辑运算结合，计算机能够实现各种其他的数学运算< / strong > 。例如，计算减法 < span class = "arithmatex" > \(a - b\)< / span > 可以转换为计算加法 < span class = "arithmatex" > \(a + (-b)\)< / span > ；计算乘法和除法可以转换为计算多次加法或减法。< / p >
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< p > 现在我们可以总结出计算机使用补码的原因：基于补码表示，计算机可以用同样的电路和操作来处理正数和负数的加法，不需要设计特殊的硬件电路来处理减法，并且无须特别处理正负零的歧义问题。这大大简化了硬件设计，提高了运算效率。< / p >
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< p > 补码的设计非常精妙，因篇幅关系我们就先介绍到这里，建议有兴趣的读者进一步深度了解。< / p >
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< h2 id = "332" > 3.3.2 浮点数编码< a class = "headerlink" href = "#332" title = "Permanent link" > ¶ < / a > < / h2 >
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< p > 细心的你可能会发现：< code > int< / code > 和 < code > float< / code > 长度相同，都是 4 bytes， < code > float< / code > 的取值范围远大于 < code > int< / code > ？这非常反直觉，因为按理说 < code > float< / code > 需要表示小数，取值范围应该变小才对。< / p >
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< p > 实际上，< strong > 这是因为浮点数 < code > float< / code > 采用了不同的表示方式< / strong > 。记一个 32-bit 长度的二进制数为：< / p >
< div class = "arithmatex" > \[
b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
\]< / div >
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< p > 根据 IEEE 754 标准， < code > float< / code > 由以下三个部分构成。< / p >
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< ul >
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< li > 符号位 < span class = "arithmatex" > \(\mathrm{S}\)< / span > ：占 1 bit ，对应 < span class = "arithmatex" > \(b_{31}\)< / span > 。< / li >
< li > 指数位 < span class = "arithmatex" > \(\mathrm{E}\)< / span > ：占 8 bits ，对应 < span class = "arithmatex" > \(b_{30} b_{29} \ldots b_{23}\)< / span > 。< / li >
< li > 分数位 < span class = "arithmatex" > \(\mathrm{N}\)< / span > ：占 23 bits ，对应 < span class = "arithmatex" > \(b_{22} b_{21} \ldots b_0\)< / span > 。< / li >
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< / ul >
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< p > 二进制数 < code > float< / code > 对应的值的计算方法：< / p >
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< div class = "arithmatex" > \[
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\text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
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\]< / div >
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< p > 转化到十进制下的计算公式：< / p >
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< div class = "arithmatex" > \[
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\text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
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\]< / div >
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< p > 其中各项的取值范围：< / p >
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< div class = "arithmatex" > \[
\begin{aligned}
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\mathrm{S} \in & \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
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(1 + \mathrm{N}) = & (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
\end{aligned}
\]< / div >
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< p > < img alt = "IEEE 754 标准下的 float 的计算示例" src = "../number_encoding.assets/ieee_754_float.png" / > < / p >
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< p align = "center" > 图 3-5 IEEE 754 标准下的 float 的计算示例 < / p >
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< p > 观察图 3-5 ，给定一个示例数据 < span class = "arithmatex" > \(\mathrm{S} = 0\)< / span > ， < span class = "arithmatex" > \(\mathrm{E} = 124\)< / span > ， < span class = "arithmatex" > \(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)< / span > ，则有：< / p >
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< div class = "arithmatex" > \[
\text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
\]< / div >
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< p > 现在我们可以回答最初的问题：< strong > < code > float< / code > 的表示方式包含指数位，导致其取值范围远大于 < code > int< / code > < / strong > 。根据以上计算，< code > float< / code > 可表示的最大正数为 < span class = "arithmatex" > \(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)< / span > ，切换符号位便可得到最小负数。< / p >
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< p > < strong > 尽管浮点数 < code > float< / code > 扩展了取值范围，但其副作用是牺牲了精度< / strong > 。整数类型 < code > int< / code > 将全部 32 位用于表示数字，数字是均匀分布的；而由于指数位的存在，浮点数 < code > float< / code > 的数值越大，相邻两个数字之间的差值就会趋向越大。< / p >
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< p > 如表 3-2 所示，指数位 < span class = "arithmatex" > \(E = 0\)< / span > 和 < span class = "arithmatex" > \(E = 255\)< / span > 具有特殊含义，< strong > 用于表示零、无穷大、< span class = "arithmatex" > \(\mathrm{NaN}\)< / span > 等< / strong > 。< / p >
< p align = "center" > 表 3-2 指数位含义 < / p >
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< div class = "center-table" >
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< table >
< thead >
< tr >
< th > 指数位 E< / th >
< th > 分数位 < span class = "arithmatex" > \(\mathrm{N} = 0\)< / span > < / th >
< th > 分数位 < span class = "arithmatex" > \(\mathrm{N} \ne 0\)< / span > < / th >
< th > 计算公式< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td > < span class = "arithmatex" > \(0\)< / span > < / td >
< td > < span class = "arithmatex" > \(\pm 0\)< / span > < / td >
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< td > 次正规数< / td >
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< td > < span class = "arithmatex" > \((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)< / span > < / td >
< / tr >
< tr >
< td > < span class = "arithmatex" > \(1, 2, \dots, 254\)< / span > < / td >
< td > 正规数< / td >
< td > 正规数< / td >
< td > < span class = "arithmatex" > \((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)< / span > < / td >
< / tr >
< tr >
< td > < span class = "arithmatex" > \(255\)< / span > < / td >
< td > < span class = "arithmatex" > \(\pm \infty\)< / span > < / td >
< td > < span class = "arithmatex" > \(\mathrm{NaN}\)< / span > < / td >
< td > < / td >
< / tr >
< / tbody >
< / table >
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< / div >
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< p > 值得说明的是，次正规数显著提升了浮点数的精度。最小正正规数为 < span class = "arithmatex" > \(2^{-126}\)< / span > ，最小正次正规数为 < span class = "arithmatex" > \(2^{-126} \times 2^{-23}\)< / span > 。< / p >
< p > 双精度 < code > double< / code > 也采用类似 < code > float< / code > 的表示方法，在此不做赘述。< / p >
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3.4 字符编码 *
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< div class = "md-footer__button md-icon" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4Z" / > < / svg >
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< div class = "md-footer-meta md-typeset" >
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Copyright © 2023 Krahets
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< div class = "md-social" >
< a href = "https://github.com/krahets" target = "_blank" rel = "noopener" title = "github.com" class = "md-social__link" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 496 512" > <!-- ! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z" / > < / svg >
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< / a >
< a href = "https://twitter.com/krahets" target = "_blank" rel = "noopener" title = "twitter.com" class = "md-social__link" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > <!-- ! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z" / > < / svg >
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< / a >
< a href = "https://leetcode.cn/u/jyd/" target = "_blank" rel = "noopener" title = "leetcode.cn" class = "md-social__link" >
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< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 640 512" > <!-- ! Font Awesome Free 6.4.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc. --> < path d = "M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z" / > < / svg >
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< / a >
< / div >
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< / footer >
< / div >
< div class = "md-dialog" data-md-component = "dialog" >
< div class = "md-dialog__inner md-typeset" > < / div >
< / div >
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< script src = "../../assets/javascripts/bundle.f11ae8b1.min.js" > < / script >
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< script src = "../../javascripts/mathjax.js" > < / script >
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< script src = "https://polyfill.io/v3/polyfill.min.js?features=es6" > < / script >
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< script src = "https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" > < / script >
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< / body >
< / html >
