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Hello 算法
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0. 前言
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0. 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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1. 初识算法
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1. 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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2. 复杂度
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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< a href = "../" class = "md-nav__link " >
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3. 数据结构
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3. 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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3.4. 字符编码 *
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< a href = "./" class = "md-nav__link md-nav__link--active" >
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3.4. 字符编码 *
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目录
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< a href = "#341-ascii" class = "md-nav__link" >
3.4.1. ASCII 字符集
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< a href = "#342-gbk" class = "md-nav__link" >
3.4.2. GBK 字符集
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< a href = "#343-unicode" class = "md-nav__link" >
3.4.3. Unicode 字符集
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< a href = "#344-utf-8" class = "md-nav__link" >
3.4.4. UTF-8 编码
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< li class = "md-nav__item" >
< a href = "#345" class = "md-nav__link" >
3.4.5. 编程语言的字符编码
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3.5. 小结
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4. 数组与链表
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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4.3. 列表
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4.4. 小结
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< a href = "../../chapter_stack_and_queue/" class = "md-nav__link " >
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5. 栈与队列
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< / label >
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5. 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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< a href = "../../chapter_hashing/" class = "md-nav__link " >
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6. 散列表
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6. 散列表
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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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7. 树
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
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7. 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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7.4. 二叉搜索树
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< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
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7.5. AVL 树 *
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7.6. 小结
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_9" >
< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_heap/" class = "md-nav__link " >
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8. 堆
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< / label >
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_9_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_9" >
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8. 堆
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8.1. 堆
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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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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_10" >
< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_graph/" class = "md-nav__link " >
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9. 图
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_10_label" aria-expanded = "false" >
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9. 图
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< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
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9.1. 图
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
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9.2. 图基础操作
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< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
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9.3. 图的遍历
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< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
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9.4. 小结
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_searching/" class = "md-nav__link " >
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< span class = "md-ellipsis" >
10. 搜索
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10. 搜索
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< li class = "md-nav__item" >
< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
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10.1. 二分查找
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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
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10.2. 二分查找边界
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
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10.3. 哈希优化策略
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
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10.4. 重识搜索算法
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10.5. 小结
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_sorting/" class = "md-nav__link " >
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11. 排序
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11. 排序
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11.1. 排序算法
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< a href = "../../chapter_sorting/selection_sort/" class = "md-nav__link" >
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11.2. 选择排序
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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11.3. 冒泡排序
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< a href = "../../chapter_sorting/insertion_sort/" class = "md-nav__link" >
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11.4. 插入排序
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< a href = "../../chapter_sorting/quick_sort/" class = "md-nav__link" >
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11.5. 快速排序
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< a href = "../../chapter_sorting/merge_sort/" class = "md-nav__link" >
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11.6. 归并排序
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< a href = "../../chapter_sorting/heap_sort/" class = "md-nav__link" >
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11.7. 堆排序
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< a href = "../../chapter_sorting/bucket_sort/" class = "md-nav__link" >
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11.8. 桶排序
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< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
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11.9. 计数排序
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< a href = "../../chapter_sorting/radix_sort/" class = "md-nav__link" >
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11.10. 基数排序
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11.11. 小结
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_divide_and_conquer/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M17 7v2h5V7h-5M2 9v6h5V9H2m10 0v2H9v2h3v2l3-3-3-3m5 2v2h5v-2h-5m0 4v2h5v-2h-5Z" / > < / svg >
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12. 分治
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12. 分治
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12.1. 分治算法
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< a href = "../../chapter_divide_and_conquer/binary_search_recur/" class = "md-nav__link" >
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12.2. 分治搜索策略
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< a href = "../../chapter_divide_and_conquer/build_binary_tree_problem/" class = "md-nav__link" >
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12.3. 构建树问题
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< a href = "../../chapter_divide_and_conquer/hanota_problem/" class = "md-nav__link" >
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12.4. 汉诺塔问题
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12.5. 小结
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< a href = "../../chapter_backtracking/" class = "md-nav__link " >
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13. 回溯
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13. 回溯
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< a href = "../../chapter_backtracking/backtracking_algorithm/" class = "md-nav__link" >
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13.1. 回溯算法
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< a href = "../../chapter_backtracking/permutations_problem/" class = "md-nav__link" >
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13.2. 全排列问题
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< a href = "../../chapter_backtracking/subset_sum_problem/" class = "md-nav__link" >
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13.3. 子集和问题
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< a href = "../../chapter_backtracking/n_queens_problem/" class = "md-nav__link" >
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13.4. N 皇后问题
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13.5. 小结
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_dynamic_programming/" class = "md-nav__link " >
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14. 动态规划
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< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_15_label" aria-expanded = "false" >
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< span class = "md-nav__icon md-icon" > < / span >
14. 动态规划
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< a href = "../../chapter_dynamic_programming/intro_to_dynamic_programming/" class = "md-nav__link" >
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14.1. 初探动态规划
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< a href = "../../chapter_dynamic_programming/dp_problem_features/" class = "md-nav__link" >
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14.2. DP 问题特性
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14.3. DP 解题思路
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14.4. 0-1 背包问题
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< a href = "../../chapter_dynamic_programming/unbounded_knapsack_problem/" class = "md-nav__link" >
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14.5. 完全背包问题
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< a href = "../../chapter_dynamic_programming/edit_distance_problem/" class = "md-nav__link" >
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14.6. 编辑距离问题
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14.7. 小结
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< a href = "../../chapter_appendix/" class = "md-nav__link " >
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15. 附录
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15. 附录
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15.1. 编程环境安装
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15.2. 一起参与创作
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参考文献
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3.4.1. ASCII 字符集
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3.4.2. GBK 字符集
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3.4.3. Unicode 字符集
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3.4.4. UTF-8 编码
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3.4.5. 编程语言的字符编码
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< h1 id = "34" > 3.4. 字符编码 *< a class = "headerlink" href = "#34" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 在计算机中，所有数据都是以二进制数的形式存储的，字符 < code > char< / code > 也不例外。为了表示字符，我们需要建立一套「字符集」，规定每个字符和二进制数之间的一一对应关系。有了字符集之后，计算机就可以通过查表完成二进制数到字符的转换。< / p >
< h2 id = "341-ascii" > 3.4.1. ASCII 字符集< a class = "headerlink" href = "#341-ascii" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 「ASCII 码」是最早出现的字符集，全称为“美国标准信息交换代码”。它使用 7 位二进制数（即一个字节的低 7 位）表示一个字符，最多能够表示 128 个不同的字符。这包括英文字母的大小写、数字 0-9 、一些标点符号，以及一些控制字符（如换行符和制表符）。< / p >
< p > < img alt = "ASCII 码" src = "../character_encoding.assets/ascii_table.png" / > < / p >
< p align = "center" > Fig. ASCII 码 < / p >
< p > 然而，< strong > ASCII 码仅能够表示英文< / strong > 。随着计算机的全球化， < / p >
< h2 id = "342-gbk" > 3.4.2. GBK 字符集< a class = "headerlink" href = "#342-gbk" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 后来人们发现，< strong > EASCII 码仍然无法满足许多语言的字符数量要求< / strong > 。例如，汉字大约有近十万个，光日常使用的就有几千个。中国国家标准总局于 1980 年发布了「GB2312」字符集， < / p >
< p > 然而， ， ， ， < / p >
< h2 id = "343-unicode" > 3.4.3. Unicode 字符集< a class = "headerlink" href = "#343-unicode" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 随着计算机的蓬勃发展，字符集与编码标准百花齐放，而这带来了许多问题。一方面，这些字符集一般只定义了特定语言的字符，无法在多语言环境下正常工作；另一方面，同一种语言也存在多种字符集标准，如果两台电脑安装的是不同的编码标准，则在信息传递时就会出现乱码。< / p >
< p > 那个时代的人们就在想：< strong > 如果推出一个足够完整的字符集，将世界范围内的所有语言和符号都收录其中，不就可以解决跨语言环境和乱码问题了吗< / strong > ？在这种想法的驱动下，一个大而全的字符集 Unicode 应运而生。< / p >
< p > 「Unicode」的全称为“统一字符编码”， ， ， < / p >
< p > 自 1991 年发布以来， ， < / p >
< p > Unicode 是一种字符集标准，本质上是给每个字符分配一个编号（称为“码点”），< strong > 但它并没有规定在计算机中如何存储这些字符码点< / strong > 。我们不禁会问：当多种长度的 Unicode 码点同时出现在同一个文本中时，系统如何解析字符？例如，给定一个长度为 2 字节的编码，系统如何确认它是一个 2 字节的字符还是两个 1 字节的字符？< / p >
< p > 对于以上问题，< strong > 一种直接的解决方案是将所有字符存储为等长的编码< / strong > 。如下图所示， ， < / p >
< p > < img alt = "Unicode 编码示例" src = "../character_encoding.assets/unicode_hello_algo.png" / > < / p >
< p align = "center" > Fig. Unicode 编码示例 < / p >
< p > 然而， < / p >
< h2 id = "344-utf-8" > 3.4.4. UTF-8 编码< a class = "headerlink" href = "#344-utf-8" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 目前， < strong > 它是一种可变长的编码< / strong > ，使用 1 到 4 个字节来表示一个字符， < / p >
< p > UTF-8 的编码规则并不复杂，分为两种情况：< / p >
< ul >
< li > 对于长度为 1 字节的字符，将最高位设置为 < span class = "arithmatex" > \(0\)< / span > 、其余 7 位设置为 Unicode 码点。值得注意的是， < strong > UTF-8 编码可以向下兼容 ASCII 码< / strong > 。这意味着我们可以使用 UTF-8 来解析年代久远的 ASCII 码文本。< / li >
< li > 对于长度为 < span class = "arithmatex" > \(n\)< / span > 字节的字符（其中 < span class = "arithmatex" > \(n > 1\)< / span > ），将首个字节的高 < span class = "arithmatex" > \(n\)< / span > 位都设置为 < span class = "arithmatex" > \(1\)< / span > 、第 < span class = "arithmatex" > \(n + 1\)< / span > 位设置为 < span class = "arithmatex" > \(0\)< / span > ；从第二个字节开始，将每个字节的高 2 位都设置为 < span class = "arithmatex" > \(10\)< / span > ；其余所有位用于填充字符的 Unicode 码点。< / li >
< / ul >
< p > 下图展示了“Hello算法”对应的 UTF-8 编码。将最高 < span class = "arithmatex" > \(n\)< / span > 位设置为 < span class = "arithmatex" > \(1\)< / span > 比较容易理解，可以向系统指出字符的长度为 < span class = "arithmatex" > \(n\)< / span > 。那么，为什么要将其余所有字节的高 2 位都设置为 < span class = "arithmatex" > \(10\)< / span > 呢？实际上，这个 < span class = "arithmatex" > \(10\)< / span > 能够起到校验符的作用，因为在 UTF-8 编码规则下，不可能有字符的最高两位是 < span class = "arithmatex" > \(10\)< / span > 。这是因为长度为 1 字节的字符的最高一位是 < span class = "arithmatex" > \(0\)< / span > 。假设系统从一个错误的字节开始解析文本，字节头部的 < span class = "arithmatex" > \(10\)< / span > 能够帮助系统快速的判断出异常。< / p >
< p > < img alt = "UTF-8 编码示例" src = "../character_encoding.assets/utf-8_hello_algo.png" / > < / p >
< p align = "center" > Fig. UTF-8 编码示例 < / p >
< p > 除了 UTF-8 之外，常见的编码方式还包括 UTF-16 和 UTF-32 。它们为 Unicode 字符集提供了不同的编码方法。< / p >
< ul >
< li > < strong > UTF-16 编码< / strong > ：使用 2 或 4 个字节来表示一个字符。所有的 ASCII 字符和很多常用的非英文字符，都用 2 个字节表示；少数字符需要用到 4 个字节表示。对于 2 字节的字符， < / li >
< li > < strong > UTF-32 编码< / strong > ：每个字符都使用 4 个字节。这意味着 UTF-32 会比 UTF-8 和 UTF-16 更占用空间，特别是对于主要使用 ASCII 字符的文本。< / li >
< / ul >
< p > 从存储空间的角度看，使用 UTF-8 表示英文字符非常高效，因为它仅需 1 个字节；使用 UTF-16 编码某些非英文字符（例如中文）会更加高效，因为它只需要 2 个字节，而 UTF-8 可能需要 3 个字节。从兼容性的角度看， < / p >
< h2 id = "345" > 3.4.5. 编程语言的字符编码< a class = "headerlink" href = "#345" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 对于以往的大多数编程语言，程序运行中的字符串都采用 UTF-16 或 UTF-32 这类等长的编码。这是因为在等长编码下，我们可以将字符串看作数组来处理，具体来说：< / p >
< ul >
< li > < strong > 随机访问< / strong > : UTF-16 编码的字符串可以很容易地进行随机访问。UTF-8 是一种变长编码，要找到第 < span class = "arithmatex" > \(i\)< / span > 个字符，我们需要从字符串的开始处遍历到第 < span class = "arithmatex" > \(i\)< / span > 个字符，这需要 < span class = "arithmatex" > \(O(n)\)< / span > 的时间。< / li >
< li > < strong > 字符计数< / strong > : 与随机访问类似，计算 UTF-16 字符串的长度也是 < span class = "arithmatex" > \(O(1)\)< / span > 的操作。但是，计算 UTF-8 编码的字符串的长度需要遍历整个字符串。< / li >
< li > < strong > 字符串操作< / strong > : 在 UTF-16 编码的字符串中，很多字符串操作（如分割、连接、插入、删除等）都更容易进行。在 UTF-8 编码的字符串上进行这些操作通常需要额外的计算，以确保不会产生无效的 UTF-8 编码。< / li >
< / ul >
< p > 编程语言的字符编码方案设计是一个很有趣的话题，涉及到许多因素：< / p >
< ul >
< li > Java 的 < code > String< / code > 类型使用 UTF-16 编码，每个字符占用 2 字节。这是因为 Java 语言设计之初，人们认为 16 位足以表示所有可能的字符。然而，这是一个不正确的判断。后来 Unicode 规范扩展到了超过 16 位，所以 Java 中的字符现在可能由一对 16 位的值（称为“代理对”）表示。< / li >
< li > JavaScript 和 TypeScript 的字符串使用 UTF-16 编码的原因与 Java 类似。当 JavaScript 语言在 1995 年被 Netscape 公司首次引入时， < / li >
< li > C# 使用 UTF-16 编码，主要因为 .NET 平台是由 Microsoft 设计的，而 Microsoft 的很多技术，包括 Windows 操作系统，都广泛地使用 UTF-16 编码。< / li >
< / ul >
< p > 由于以上编程语言对字符数量的低估，它们不得不采取“代理对”的方式来表示超过 16 位长度的 Unicode 字符。这是一个不得已为之的无奈之举。一方面，包含代理对的字符串中，一个字符可能占用 2 字节或 4 字节，因此丧失了等长编码的优势。另一方面，处理代理对需要增加额外代码，这增加了编程的复杂性和 Debug 难度。< / p >
< p > 出于以上原因，部分编程语言提出了不同的编码方案：< / p >
< ul >
< li > Python 3 使用一种灵活的字符串表示，存储的字符长度取决于字符串中最大的 Unicode 码点。对于全部是 ASCII 字符的字符串，每个字符占用 1 个字节；如果字符串中包含的字符超出了 ASCII 范围， （ ） ， < / li >
< li > Go 语言的 < code > string< / code > 类型在内部使用 UTF-8 编码。Go 语言还提供了 < code > rune< / code > 类型，它用于表示单个 Unicode 码点。< / li >
< li > Rust 语言的 str 和 String 类型在内部使用 UTF-8 编码。Rust 也提供了 char 类型，用于表示单个 Unicode 码点。< / li >
< / ul >
< p > 需要注意的是，以上讨论的都是字符串在编程语言中的存储方式，< strong > 这和字符串如何在文件中存储或在网络中传输是两个不同的问题< / strong > 。在文件存储或网络传输中，我们一般会将字符串编码为 UTF-8 格式，以达到最优的兼容性和空间效率。< / p >
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