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Hello 算法
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2.4. 小结
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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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< div class = "md-nav__link md-nav__container" >
< a href = "../" class = "md-nav__link " >
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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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2.4. 小结
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目录
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< a href = "#241-q-a" class = "md-nav__link" >
2.4.1. Q & A
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< / ul >
< / nav >
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3. 数据结构
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3. 数据结构
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< a href = "../../chapter_data_structure/classification_of_data_structure/" class = "md-nav__link" >
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3.1. 数据结构分类
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< a href = "../../chapter_data_structure/basic_data_types/" class = "md-nav__link" >
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3.2. 基本数据类型
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< a href = "../../chapter_data_structure/number_encoding/" class = "md-nav__link" >
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3.3. 数字编码 *
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< a href = "../../chapter_data_structure/character_encoding/" class = "md-nav__link" >
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3.4. 字符编码 *
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3.5. 小结
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< a href = "../../chapter_array_and_linkedlist/" class = "md-nav__link " >
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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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< div class = "md-nav__link md-nav__container" >
< 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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< li class = "md-nav__item" >
< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
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5.1. 栈
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5.2. 队列
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< li class = "md-nav__item" >
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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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< li class = "md-nav__item" >
< a href = "../../chapter_hashing/hash_map/" class = "md-nav__link" >
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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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< a href = "../../chapter_tree/" class = "md-nav__link " >
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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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< 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 >
< / div >
< 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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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_graph/" class = "md-nav__link " >
< 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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9. 图
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9. 图
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9.1. 图
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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 " >
< 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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10. 搜索
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10. 搜索
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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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< 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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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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11.6. 归并排序
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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 " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2h-2m-4-6H5v-2h2v2Z" / > < / svg >
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13. 回溯
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13. 回溯
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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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13.3. 子集和问题
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13.4. N 皇后问题
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13.5. 小结
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< a href = "../../chapter_dynamic_programming/" class = "md-nav__link " >
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14. 动态规划
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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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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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14.5. 完全背包问题
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14.6. 编辑距离问题
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14.7. 小结
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< div class = "md-nav__link md-nav__container" >
< a href = "../../chapter_greedy/" class = "md-nav__link " >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M13 3c3.88 0 7 3.14 7 7 0 2.8-1.63 5.19-4 6.31V21H9v-3H8c-1.11 0-2-.89-2-2v-3H4.5c-.42 0-.66-.5-.42-.81L6 9.66A7.003 7.003 0 0 1 13 3m0-2C8.41 1 4.61 4.42 4.06 8.9L2.5 11h-.03l-.02.03c-.55.76-.62 1.76-.19 2.59.36.69 1 1.17 1.74 1.32V16c0 1.85 1.28 3.42 3 3.87V23h11v-5.5c2.5-1.67 4-4.44 4-7.5 0-4.97-4.04-9-9-9m4 7.83c0 1.54-1.36 2.77-3.42 4.64L13 14l-.58-.53C10.36 11.6 9 10.37 9 8.83c0-1.2.96-2.19 2.16-2.2h.04c.69 0 1.35.31 1.8.83.45-.52 1.11-.83 1.8-.83 1.2-.01 2.2.96 2.2 2.16v.04Z" / > < / svg >
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15. 贪心
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15. 贪心
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15.1. 贪心算法
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2.4.1. Q & A
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< h1 id = "24" > 2.4. 小结< a class = "headerlink" href = "#24" title = "Permanent link" > ¶ < / a > < / h1 >
< p > < strong > 算法效率评估< / strong > < / p >
< ul >
< li > 时间效率和空间效率是评价算法性能的两个关键维度。< / 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 > 复杂度分析可以克服实际测试的弊端，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，即函数渐近上界，反映当 < span class = "arithmatex" > \(n\)< / span > 趋向正无穷时，< span class = "arithmatex" > \(T(n)\)< / span > 的增长级别。< / li >
< li > 推算时间复杂度分为两步，首先统计计算操作数量，然后判断渐近上界。< / 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 > 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 > 类似于时间复杂度，空间复杂度用于衡量算法占用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 > 等。< / li >
< / ul >
< h2 id = "241-q-a" > 2.4.1. Q & A< a class = "headerlink" href = "#241-q-a" title = "Permanent link" > ¶ < / a > < / h2 >
< div class = "admonition question" >
< p class = "admonition-title" > 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被优化至 < span class = "arithmatex" > \(O(1)\)< / span > 。不过绝大多数编程语言（例如 Java, Python, C++, Go, C# 等）
都不支持自动优化尾递归，因此一般来说空间复杂度是 < span class = "arithmatex" > \(O(n)\)< / span > 。< / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 函数和方法这两个术语的区别是什么？< / p >
< p > 函数（ ） ，
方法（ ） ， ， < / p >
< p > 因此， ， < / p >
< / div >
< div class = "admonition question" >
< p class = "admonition-title" > 图片“空间复杂度的常见类型”反映的是否是占用空间的绝对大小？< / p >
< p > 不是，该图片展示的是空间复杂度（即增长趋势），而不是占用空间的绝对大小。每个曲线都包含一个常数项，用来把所有曲线的取值范围压缩到一个视觉舒适的范围内。
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；但相对地 < span class = "arithmatex" > \(n = 8^5\)< / span > 就很好选了，这是复杂度占主导的情况。< / p >
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