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< / label >
< / div >
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< label class = "md-nav__title" for = "__nav_4" >
< span class = "md-nav__icon md-icon" > < / span >
3. 数据结构
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< a href = "../../chapter_data_structure/classification_of_data_structure/" class = "md-nav__link" >
3.1. 数据结构分类
< / a >
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< a href = "../../chapter_data_structure/basic_data_types/" class = "md-nav__link" >
3.2. 基本数据类型
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< a href = "../../chapter_data_structure/number_encoding/" class = "md-nav__link" >
3.3. 数字编码 *
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< a href = "../../chapter_data_structure/character_encoding/" class = "md-nav__link" >
3.4. 字符编码 *
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< a href = "../../chapter_data_structure/summary/" class = "md-nav__link" >
3.5. 小结
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< a href = "../../chapter_array_and_linkedlist/" > 4. 数组与链表< / a >
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< / div >
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< span class = "md-nav__icon md-icon" > < / span >
4. 数组与链表
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< a href = "../../chapter_array_and_linkedlist/array/" class = "md-nav__link" >
4.1. 数组
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< / li >
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< a href = "../../chapter_array_and_linkedlist/linked_list/" class = "md-nav__link" >
4.2. 链表
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< a href = "../../chapter_array_and_linkedlist/list/" class = "md-nav__link" >
4.3. 列表
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4.4. 小结
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< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_stack_and_queue/" > 5. 栈与队列< / a >
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< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_6_label" aria-expanded = "false" >
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< span class = "md-nav__icon md-icon" > < / span >
5. 栈与队列
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< a href = "../../chapter_stack_and_queue/stack/" class = "md-nav__link" >
5.1. 栈
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< a href = "../../chapter_stack_and_queue/queue/" class = "md-nav__link" >
5.2. 队列
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< a href = "../../chapter_stack_and_queue/deque/" class = "md-nav__link" >
5.3. 双向队列
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5.4. 小结
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< a href = "../../chapter_hashing/" > 6. 散列表< / a >
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< / label >
< / div >
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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" >
6.1. 哈希表（ ）
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< a href = "../../chapter_hashing/hash_collision/" class = "md-nav__link" >
6.2. 哈希冲突（ ）
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< a href = "../../chapter_hashing/hash_algorithm/" class = "md-nav__link" >
6.3. 哈希算法（ ）
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6.4. 小结
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_8" >
< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_tree/" > 7. 树< / a >
< label for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_8_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_8" >
< span class = "md-nav__icon md-icon" > < / span >
7. 树
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< li class = "md-nav__item" >
< a href = "../../chapter_tree/binary_tree/" class = "md-nav__link" >
7.1. 二叉树
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< / li >
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< a href = "../../chapter_tree/binary_tree_traversal/" class = "md-nav__link" >
7.2. 二叉树遍历
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< a href = "../../chapter_tree/array_representation_of_tree/" class = "md-nav__link" >
7.3. 二叉树数组表示
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< / li >
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< a href = "../../chapter_tree/binary_search_tree/" class = "md-nav__link" >
7.4. 二叉搜索树
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< a href = "../../chapter_tree/avl_tree/" class = "md-nav__link" >
7.5. AVL 树 *
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< a href = "../../chapter_tree/summary/" class = "md-nav__link" >
7.6. 小结
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< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_heap/" > 8. 堆< / a >
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< span class = "md-nav__icon md-icon" > < / span >
< / 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" >
< span class = "md-nav__icon md-icon" > < / span >
8. 堆
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< a href = "../../chapter_heap/heap/" class = "md-nav__link" >
8.1. 堆
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< a href = "../../chapter_heap/build_heap/" class = "md-nav__link" >
8.2. 建堆操作
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< a href = "../../chapter_heap/top_k/" class = "md-nav__link" >
8.3. Top-K 问题（ ）
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< a href = "../../chapter_heap/summary/" class = "md-nav__link" >
8.4. 小结
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< a href = "../../chapter_graph/" > 9. 图< / a >
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< / label >
< / div >
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< span class = "md-nav__icon md-icon" > < / span >
9. 图
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< a href = "../../chapter_graph/graph/" class = "md-nav__link" >
9.1. 图
< / a >
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< a href = "../../chapter_graph/graph_operations/" class = "md-nav__link" >
9.2. 图基础操作
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< a href = "../../chapter_graph/graph_traversal/" class = "md-nav__link" >
9.3. 图的遍历
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< a href = "../../chapter_graph/summary/" class = "md-nav__link" >
9.4. 小结
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< input class = "md-nav__toggle md-toggle " type = "checkbox" id = "__nav_11" >
< div class = "md-nav__link md-nav__link--index " >
< a href = "../../chapter_searching/" > 10. 搜索< / a >
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< span class = "md-nav__icon md-icon" > < / span >
< / label >
< / div >
< nav class = "md-nav" data-md-level = "1" aria-labelledby = "__nav_11_label" aria-expanded = "false" >
< label class = "md-nav__title" for = "__nav_11" >
< span class = "md-nav__icon md-icon" > < / span >
10. 搜索
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< a href = "../../chapter_searching/binary_search/" class = "md-nav__link" >
10.1. 二分查找
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< a href = "../../chapter_searching/binary_search_edge/" class = "md-nav__link" >
10.2. 二分查找边界
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< a href = "../../chapter_searching/replace_linear_by_hashing/" class = "md-nav__link" >
10.3. 哈希优化策略
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< a href = "../../chapter_searching/searching_algorithm_revisited/" class = "md-nav__link" >
10.4. 重识搜索算法
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10.5. 小结
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< a href = "../../chapter_sorting/" > 11. 排序< / a >
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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" >
11.2. 选择排序
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< a href = "../../chapter_sorting/bubble_sort/" class = "md-nav__link" >
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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< a href = "../../chapter_sorting/bucket_sort/" class = "md-nav__link" >
11.8. 桶排序
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< a href = "../../chapter_sorting/counting_sort/" class = "md-nav__link" >
11.9. 计数排序
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< a href = "../../chapter_sorting/radix_sort/" class = "md-nav__link" >
11.10. 基数排序
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11.11. 小结
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< a href = "../../chapter_backtracking/" > 12. 回溯< / a >
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12. 回溯
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12.1. 回溯算法
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12.2. 全排列问题
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12.3. 子集和问题（ ）
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12.4. N 皇后问题
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12.5. 小结
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< a href = "../" > 13. 动态规划< / a >
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< / div >
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< label class = "md-nav__title" for = "__nav_14" >
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13. 动态规划
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< a href = "../intro_to_dynamic_programming/" class = "md-nav__link" >
13.1. 初探动态规划（ ）
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< a href = "../dp_problem_features/" class = "md-nav__link" >
13.2. DP 问题特性（ ）
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13.3. DP 解题思路（ ）
< span class = "md-nav__icon md-icon" > < / span >
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13.3. DP 解题思路（ ）
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目录
< / label >
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13.3.1. 问题判断
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13.3.2. 问题求解
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13.3.3. 方法一：暴力搜索
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< a href = "#1334" class = "md-nav__link" >
13.3.4. 方法二：记忆化搜索
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13.3.5. 方法三：动态规划
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13.4. 0-1 背包问题（ ）
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< label class = "md-nav__link" for = "__nav_15" id = "__nav_15_label" tabindex = "0" >
14. 附录
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14. 附录
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< a href = "../../chapter_appendix/installation/" class = "md-nav__link" >
14.1. 编程环境安装
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14.2. 一起参与创作
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< a href = "../../chapter_reference/" > 参考文献< / a >
< / div >
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< label class = "md-nav__title" for = "__nav_16" >
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参考文献
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< / ul >
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< / div >
< div class = "md-sidebar md-sidebar--secondary" data-md-component = "sidebar" data-md-type = "toc" >
< div class = "md-sidebar__scrollwrap" >
< div class = "md-sidebar__inner" >
< nav class = "md-nav md-nav--secondary" aria-label = "目录" >
< label class = "md-nav__title" for = "__toc" >
< span class = "md-nav__icon md-icon" > < / span >
目录
< / label >
< ul class = "md-nav__list" data-md-component = "toc" data-md-scrollfix >
< li class = "md-nav__item" >
< a href = "#1331" class = "md-nav__link" >
13.3.1. 问题判断
< / a >
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< li class = "md-nav__item" >
< a href = "#1332" class = "md-nav__link" >
13.3.2. 问题求解
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#1333" class = "md-nav__link" >
13.3.3. 方法一：暴力搜索
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#1334" class = "md-nav__link" >
13.3.4. 方法二：记忆化搜索
< / a >
< / li >
< li class = "md-nav__item" >
< a href = "#1335" class = "md-nav__link" >
13.3.5. 方法三：动态规划
< / a >
< / li >
< / ul >
< / nav >
< / div >
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< div class = "md-content" data-md-component = "content" >
< article class = "md-content__inner md-typeset" >
< a href = "https://github.com/krahets/hello-algo/tree/main/docs/chapter_dynamic_programming/dp_solution_pipeline.md" title = "编辑此页" class = "md-content__button md-icon" >
< svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 24 24" > < path d = "M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z" / > < / svg >
< / a >
< h1 id = "133" > 13.3. 动态规划解题思路< a class = "headerlink" href = "#133" title = "Permanent link" > ¶ < / a > < / h1 >
< p > 上两节介绍了动态规划问题的主要特征，接下来我们一起探究两个更加实用的问题：< / p >
< ol >
< li > 如何判断一个问题是不是动态规划问题？< / li >
< li > 求解动态规划问题该从何处入手，完整步骤是什么？< / li >
< / ol >
< h2 id = "1331" > 13.3.1. 问题判断< a class = "headerlink" href = "#1331" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 总的来说，如果一个问题包含重叠子问题、最优子结构，并满足无后效性，那么它通常就适合用动态规划求解，但我们很难从问题描述上直接提取出这些特性。因此我们通常会放宽条件，< strong > 先观察问题是否适合使用回溯（穷举）解决< / strong > 。< / p >
< p > < strong > 适合用回溯解决的问题通常满足“决策树模型”< / strong > ，这种问题可以使用树形结构来描述，其中每一个节点代表一个决策，每一条路径代表一个决策序列。< / p >
< p > 换句话说，如果问题包含明确的决策概念，并且解是通过一系列决策产生的，那么它就满足决策树模型，通常可以使用回溯来解决。< / p >
< p > 在此基础上，还有一些判断问题是动态规划问题的“加分项”，包括：< / p >
< ul >
< li > 问题包含最大（小）或最多（少）等最优化描述；< / li >
< li > 问题的状态能够使用一个列表、多维矩阵或树来表示，并且一个状态与其周围的状态存在某种递推关系；< / li >
< / ul >
< p > 而相应的“减分项”包括：< / p >
< ul >
< li > 问题的目标是找出所有可能的解决方案，而不是找出最优解。< / li >
< li > 问题描述中有明显的排列组合的特征，需要返回具体的多个方案。< / li >
< / ul >
< p > 如果一个问题满足决策树模型，并具有较为明显的“加分项“，我们就可以假设它是一个动态规划问题，并尝试求解它。< / p >
< h2 id = "1332" > 13.3.2. 问题求解< a class = "headerlink" href = "#1332" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 动态规划的解题流程可能会因问题的性质和难度而有所不同，但通常遵循以下步骤：描述决策，定义状态，建立 < span class = "arithmatex" > \(dp\)< / span > 表，推导状态转移方程，确定边界条件等。< / p >
< p > 为了更形象地展示解题步骤，我们使用一个经典问题「最小路径和」来举例。< / p >
< div class = "admonition question" >
< p class = "admonition-title" > Question< / p >
< p > 给定一个 < span class = "arithmatex" > \(n \times m\)< / span > 的二维网格 < code > grid< / code > ，网格中的每个单元格包含一个非负整数，表示该单元格的代价。机器人以左上角单元格为起始点，每次只能向下或者向右移动一步，直至到达右下角单元格。请返回从左上角到右下角的最小路径和。< / p >
< / div >
< p > 例如以下示例数据，给定网格的最小路径和为 < span class = "arithmatex" > \(13\)< / span > 。< / p >
< p > < img alt = "最小路径和示例数据" src = "../dp_solution_pipeline.assets/min_path_sum_example.png" / > < / p >
< p align = "center" > Fig. 最小路径和示例数据 < / p >
< p > < strong > 第一步：思考每轮的决策，定义状态，从而得到 < span class = "arithmatex" > \(dp\)< / span > 表< / strong > < / p >
< p > 本题的每一轮的决策就是从当前格子向下或向右一步。设当前格子的行列索引为 < span class = "arithmatex" > \([i, j]\)< / span > ，则向下或向右走一步后，索引变为 < span class = "arithmatex" > \([i+1, j]\)< / span > 或 < span class = "arithmatex" > \([i, j+1]\)< / span > 。因此，状态应包含行索引和列索引两个变量，记为 < span class = "arithmatex" > \([i, j]\)< / span > 。< / p >
< p > 状态 < span class = "arithmatex" > \([i, j]\)< / span > 对应的子问题为：从起始点 < span class = "arithmatex" > \([0, 0]\)< / span > 走到 < span class = "arithmatex" > \([i, j]\)< / span > 的最小路径和，解记为 < span class = "arithmatex" > \(dp[i, j]\)< / span > 。< / p >
< p > 至此，我们就得到了一个二维 < span class = "arithmatex" > \(dp\)< / span > 矩阵，其尺寸与输入网格 < span class = "arithmatex" > \(grid\)< / span > 相同。< / p >
< p > < img alt = "状态定义与 dp 表" src = "../dp_solution_pipeline.assets/min_path_sum_solution_step1.png" / > < / p >
< p align = "center" > Fig. 状态定义与 dp 表 < / p >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 动态规划和回溯通常都会被描述为一个决策序列，而状态通常由所有决策变量构成。它应当包含描述解题进度的所有变量，其包含了足够的信息，能够用来推导出下一个状态。< / p >
< p > 每个状态都对应一个子问题，我们会定义一个 < span class = "arithmatex" > \(dp\)< / span > 表来存储所有子问题的解，状态的每个独立变量都是 < span class = "arithmatex" > \(dp\)< / span > 表的一个维度。本质上看，< span class = "arithmatex" > \(dp\)< / span > 表是子问题的解和状态之间的映射。< / p >
< / div >
< p > < strong > 第二步：找出最优子结构，进而推导出状态转移方程< / strong > < / p >
< p > 对于状态 < span class = "arithmatex" > \([i, j]\)< / span > ，它只能从上边格子 < span class = "arithmatex" > \([i-1, j]\)< / span > 和左边格子 < span class = "arithmatex" > \([i, j-1]\)< / span > 转移而来。因此最优子结构为：到达 < span class = "arithmatex" > \([i, j]\)< / span > 的最小路径和由 < span class = "arithmatex" > \([i, j-1]\)< / span > 的最小路径和与 < span class = "arithmatex" > \([i-1, j]\)< / span > 的最小路径和，这两者较小的那一个决定。< / p >
< p > 根据以上分析，可推出以下状态转移方程：< / p >
< div class = "arithmatex" > \[
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
\]< / div >
< p > < img alt = "最优子结构与状态转移方程" src = "../dp_solution_pipeline.assets/min_path_sum_solution_step2.png" / > < / p >
< p align = "center" > Fig. 最优子结构与状态转移方程 < / p >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 基于定义好的 < span class = "arithmatex" > \(dp\)< / span > 表，我们思考原问题和子问题的关系，找出如何通过子问题的解来构造原问题的解。< / p >
< p > 最优子结构揭示了原问题和子问题的递推关系，一旦我们找到了最优子结构，就可以使用它来构建出状态转移方程。< / p >
< / div >
< p > < strong > 第三步：确定边界条件和状态转移顺序< / strong > < / p >
< p > 在本题中，当 < span class = "arithmatex" > \(i=0\)< / span > 或 < span class = "arithmatex" > \(j=0\)< / span > 时只有一种可能的路径，即只能向右移动或只能向下移动，因此首行和首列是边界条件。< / p >
< p > 每个格子是由其左方格子和上方格子转移而来，因此我们使用两层循环来遍历矩阵即可，即外循环正序遍历各行、内循环正序遍历各列。< / p >
< p > < img alt = "边界条件与状态转移顺序" src = "../dp_solution_pipeline.assets/min_path_sum_solution_step3.png" / > < / p >
< p align = "center" > Fig. 边界条件与状态转移顺序 < / p >
< div class = "admonition note" >
< p class = "admonition-title" > Note< / p >
< p > 边界条件即初始状态，在搜索中用于剪枝，在动态规划中用于初始化 < span class = "arithmatex" > \(dp\)< / span > 表。状态转移顺序的核心是要保证在计算当前问题时，所有它依赖的更小子问题都已经被正确地计算出来。< / p >
< / div >
< p > 最后，我们基于以上结果实现解法即可。熟练度较高同学可以直接写出动态规划解法，初学者可以按照“暴力搜索 < span class = "arithmatex" > \(\rightarrow\)< / span > 记忆化搜索 < span class = "arithmatex" > \(\rightarrow\)< / span > 动态规划” 的顺序实现。< / p >
< h2 id = "1333" > 13.3.3. 方法一：暴力搜索< a class = "headerlink" href = "#1333" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 从状态 < span class = "arithmatex" > \([i, j]\)< / span > 开始搜索，不断分解为更小的状态 < span class = "arithmatex" > \([i-1, j]\)< / span > 和 < span class = "arithmatex" > \([i, j-1]\)< / span > ，包括以下递归要素：< / p >
< ul >
< li > < strong > 递归参数< / strong > ：状态 < span class = "arithmatex" > \([i, j]\)< / span > ； < strong > 返回值< / strong > ：从 < span class = "arithmatex" > \([0, 0]\)< / span > 到 < span class = "arithmatex" > \([i, j]\)< / span > 的最小路径和 < span class = "arithmatex" > \(dp[i, j]\)< / span > ； < / li >
< li > < strong > 终止条件< / strong > ：当 < span class = "arithmatex" > \(i = 0\)< / span > 且 < span class = "arithmatex" > \(j = 0\)< / span > 时，返回代价 < span class = "arithmatex" > \(grid[0][0]\)< / span > ； < / li >
< li > < strong > 剪枝< / strong > ：当 < span class = "arithmatex" > \(i < 0\)< / span > 时或 < span class = "arithmatex" > \(j < 0\)< / span > 时索引越界，此时返回代价 < span class = "arithmatex" > \(+\infty\)< / span > ，代表不可行；< / li >
< / ul >
< div class = "tabbed-set tabbed-alternate" data-tabs = "1:11" > < 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" / > < div class = "tabbed-labels" > < label for = "__tabbed_1_1" > Java< / label > < label for = "__tabbed_1_2" > C++< / label > < label for = "__tabbed_1_3" > Python< / label > < label for = "__tabbed_1_4" > Go< / label > < label for = "__tabbed_1_5" > JavaScript< / label > < label for = "__tabbed_1_6" > TypeScript< / label > < label for = "__tabbed_1_7" > C< / label > < label for = "__tabbed_1_8" > C#< / label > < label for = "__tabbed_1_9" > Swift< / label > < label for = "__tabbed_1_10" > Zig< / label > < label for = "__tabbed_1_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-0-1" name = "__codelineno-0-1" href = "#__codelineno-0-1" > < / a > < span class = "cm" > /* 最小路径和：暴力搜索 */< / span >
< a id = "__codelineno-0-2" name = "__codelineno-0-2" href = "#__codelineno-0-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "o" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-3" name = "__codelineno-0-3" href = "#__codelineno-0-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-0-4" name = "__codelineno-0-4" href = "#__codelineno-0-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-5" name = "__codelineno-0-5" href = "#__codelineno-0-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "o" > [< / span > < span class = "mi" > 0< / span > < span class = "o" > ][< / span > < span class = "mi" > 0< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-6" name = "__codelineno-0-6" href = "#__codelineno-0-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-7" name = "__codelineno-0-7" href = "#__codelineno-0-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-0-8" name = "__codelineno-0-8" href = "#__codelineno-0-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-0-9" name = "__codelineno-0-9" href = "#__codelineno-0-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > Integer< / span > < span class = "p" > .< / span > < span class = "na" > MAX_VALUE< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-10" name = "__codelineno-0-10" href = "#__codelineno-0-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-0-11" name = "__codelineno-0-11" href = "#__codelineno-0-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价< / span >
< a id = "__codelineno-0-12" name = "__codelineno-0-12" href = "#__codelineno-0-12" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-0-13" name = "__codelineno-0-13" href = "#__codelineno-0-13" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-0-14" name = "__codelineno-0-14" href = "#__codelineno-0-14" > < / a > < span class = "w" > < / span > < span class = "c1" > // 返回从左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-0-15" name = "__codelineno-0-15" href = "#__codelineno-0-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "na" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-0-16" name = "__codelineno-0-16" href = "#__codelineno-0-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-1-1" name = "__codelineno-1-1" href = "#__codelineno-1-1" > < / a > < span class = "cm" > /* 最小路径和：暴力搜索 */< / span >
< a id = "__codelineno-1-2" name = "__codelineno-1-2" href = "#__codelineno-1-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-3" name = "__codelineno-1-3" href = "#__codelineno-1-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-1-4" name = "__codelineno-1-4" href = "#__codelineno-1-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-5" name = "__codelineno-1-5" href = "#__codelineno-1-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-1-6" name = "__codelineno-1-6" href = "#__codelineno-1-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-7" name = "__codelineno-1-7" href = "#__codelineno-1-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-1-8" name = "__codelineno-1-8" href = "#__codelineno-1-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-1-9" name = "__codelineno-1-9" href = "#__codelineno-1-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-1-10" name = "__codelineno-1-10" href = "#__codelineno-1-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-1-11" name = "__codelineno-1-11" href = "#__codelineno-1-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价< / span >
< a id = "__codelineno-1-12" name = "__codelineno-1-12" href = "#__codelineno-1-12" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-1-13" name = "__codelineno-1-13" href = "#__codelineno-1-13" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-1-14" name = "__codelineno-1-14" href = "#__codelineno-1-14" > < / a > < span class = "w" > < / span > < span class = "c1" > // 返回从左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-1-15" name = "__codelineno-1-15" href = "#__codelineno-1-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > !=< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-1-16" name = "__codelineno-1-16" href = "#__codelineno-1-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-2-1" name = "__codelineno-2-1" href = "#__codelineno-2-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_path_sum_dfs< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ]],< / span > < span class = "n" > i< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-2" name = "__codelineno-2-2" href = "#__codelineno-2-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 最小路径和：暴力搜索" " " < / span >
< a id = "__codelineno-2-3" name = "__codelineno-2-3" href = "#__codelineno-2-3" > < / a > < span class = "c1" > # 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-2-4" name = "__codelineno-2-4" href = "#__codelineno-2-4" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "ow" > and< / span > < span class = "n" > j< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-5" name = "__codelineno-2-5" href = "#__codelineno-2-5" > < / a > < span class = "k" > return< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-2-6" name = "__codelineno-2-6" href = "#__codelineno-2-6" > < / a > < span class = "c1" > # 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-2-7" name = "__codelineno-2-7" href = "#__codelineno-2-7" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > < < / span > < span class = "mi" > 0< / span > < span class = "ow" > or< / span > < span class = "n" > j< / span > < span class = "o" > < < / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-2-8" name = "__codelineno-2-8" href = "#__codelineno-2-8" > < / a > < span class = "k" > return< / span > < span class = "n" > inf< / span >
< a id = "__codelineno-2-9" name = "__codelineno-2-9" href = "#__codelineno-2-9" > < / a > < span class = "c1" > # 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价< / span >
< a id = "__codelineno-2-10" name = "__codelineno-2-10" href = "#__codelineno-2-10" > < / a > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > min_path_sum_dfs< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-11" name = "__codelineno-2-11" href = "#__codelineno-2-11" > < / a > < span class = "n" > up< / span > < span class = "o" > =< / span > < span class = "n" > min_path_sum_dfs< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-2-12" name = "__codelineno-2-12" href = "#__codelineno-2-12" > < / a > < span class = "c1" > # 返回从左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-2-13" name = "__codelineno-2-13" href = "#__codelineno-2-13" > < / a > < span class = "k" > return< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-3-1" name = "__codelineno-3-1" href = "#__codelineno-3-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-4-1" name = "__codelineno-4-1" href = "#__codelineno-4-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-5-1" name = "__codelineno-5-1" href = "#__codelineno-5-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-6-1" name = "__codelineno-6-1" href = "#__codelineno-6-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-7-1" name = "__codelineno-7-1" href = "#__codelineno-7-1" > < / a > < span class = "cm" > /* 最小路径和：暴力搜索 */< / span >
< a id = "__codelineno-7-2" name = "__codelineno-7-2" href = "#__codelineno-7-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-3" name = "__codelineno-7-3" href = "#__codelineno-7-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-7-4" name = "__codelineno-7-4" href = "#__codelineno-7-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ){< / span >
< a id = "__codelineno-7-5" name = "__codelineno-7-5" href = "#__codelineno-7-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-6" name = "__codelineno-7-6" href = "#__codelineno-7-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-7" name = "__codelineno-7-7" href = "#__codelineno-7-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-7-8" name = "__codelineno-7-8" href = "#__codelineno-7-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-7-9" name = "__codelineno-7-9" href = "#__codelineno-7-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > .< / span > < span class = "n" > MaxValue< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-7-10" name = "__codelineno-7-10" href = "#__codelineno-7-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-7-11" name = "__codelineno-7-11" href = "#__codelineno-7-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价< / span >
< a id = "__codelineno-7-12" name = "__codelineno-7-12" href = "#__codelineno-7-12" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-7-13" name = "__codelineno-7-13" href = "#__codelineno-7-13" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-7-14" name = "__codelineno-7-14" href = "#__codelineno-7-14" > < / a > < span class = "w" > < / span > < span class = "c1" > // 返回从左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-7-15" name = "__codelineno-7-15" href = "#__codelineno-7-15" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-7-16" name = "__codelineno-7-16" href = "#__codelineno-7-16" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-8-1" name = "__codelineno-8-1" href = "#__codelineno-8-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-9-1" name = "__codelineno-9-1" href = "#__codelineno-9-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-10-1" name = "__codelineno-10-1" href = "#__codelineno-10-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFS< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > 我们尝试画出以 < span class = "arithmatex" > \(dp[2, 1]\)< / span > 为根节点的递归树。观察下图，递归树包含一些重叠子问题，其数量会随着网格 < code > grid< / code > 的尺寸变大而急剧增多。< / p >
< p > 直观上看，< strong > 存在多条路径可以从左上角到达同一单元格< / strong > ，这便是该问题存在重叠子问题的内在原因。< / p >
< p > < img alt = "暴力搜索递归树" src = "../dp_solution_pipeline.assets/min_path_sum_dfs.png" / > < / p >
< p align = "center" > Fig. 暴力搜索递归树 < / p >
< p > 每个状态都有向下和向右两种选择，从左上角走到右下角总共需要 < span class = "arithmatex" > \(m + n - 2\)< / span > 步，所以最差时间复杂度为 < span class = "arithmatex" > \(O(2^{m + n})\)< / span > 。请注意，这种计算方式未考虑临近网格边界的情况，当到达网络边界时只剩下一种选择。因此实际的路径数量会少一些。< / p >
< h2 id = "1334" > 13.3.4. 方法二：记忆化搜索< a class = "headerlink" href = "#1334" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 为了避免重复计算重叠子问题，我们引入一个和网格 < code > grid< / code > 相同尺寸的记忆列表 < code > mem< / code > ，用于记录各个子问题的解，提升搜索效率。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "2:11" > < input checked = "checked" id = "__tabbed_2_1" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_2" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_3" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_4" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_5" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_6" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_7" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_8" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_9" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_10" name = "__tabbed_2" type = "radio" / > < input id = "__tabbed_2_11" name = "__tabbed_2" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_2_1" > Java< / label > < label for = "__tabbed_2_2" > C++< / label > < label for = "__tabbed_2_3" > Python< / label > < label for = "__tabbed_2_4" > Go< / label > < label for = "__tabbed_2_5" > JavaScript< / label > < label for = "__tabbed_2_6" > TypeScript< / label > < label for = "__tabbed_2_7" > C< / label > < label for = "__tabbed_2_8" > C#< / label > < label for = "__tabbed_2_9" > Swift< / label > < label for = "__tabbed_2_10" > Zig< / label > < label for = "__tabbed_2_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-11-1" name = "__codelineno-11-1" href = "#__codelineno-11-1" > < / a > < span class = "cm" > /* 最小路径和：记忆化搜索 */< / span >
< a id = "__codelineno-11-2" name = "__codelineno-11-2" href = "#__codelineno-11-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "o" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "o" > [][]< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-3" name = "__codelineno-11-3" href = "#__codelineno-11-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-11-4" name = "__codelineno-11-4" href = "#__codelineno-11-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-5" name = "__codelineno-11-5" href = "#__codelineno-11-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "o" > [< / span > < span class = "mi" > 0< / span > < span class = "o" > ][< / span > < span class = "mi" > 0< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-6" name = "__codelineno-11-6" href = "#__codelineno-11-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-11-7" name = "__codelineno-11-7" href = "#__codelineno-11-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-11-8" name = "__codelineno-11-8" href = "#__codelineno-11-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-9" name = "__codelineno-11-9" href = "#__codelineno-11-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > Integer< / span > < span class = "p" > .< / span > < span class = "na" > MAX_VALUE< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-10" name = "__codelineno-11-10" href = "#__codelineno-11-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-11-11" name = "__codelineno-11-11" href = "#__codelineno-11-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若已有记录，则直接返回< / span >
< a id = "__codelineno-11-12" name = "__codelineno-11-12" href = "#__codelineno-11-12" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > mem< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > !=< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-11-13" name = "__codelineno-11-13" href = "#__codelineno-11-13" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-14" name = "__codelineno-11-14" href = "#__codelineno-11-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-11-15" name = "__codelineno-11-15" href = "#__codelineno-11-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 左边和上边单元格的最小路径代价< / span >
< a id = "__codelineno-11-16" name = "__codelineno-11-16" href = "#__codelineno-11-16" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-17" name = "__codelineno-11-17" href = "#__codelineno-11-17" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-11-18" name = "__codelineno-11-18" href = "#__codelineno-11-18" > < / a > < span class = "w" > < / span > < span class = "c1" > // 记录并返回左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-11-19" name = "__codelineno-11-19" href = "#__codelineno-11-19" > < / a > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "na" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-20" name = "__codelineno-11-20" href = "#__codelineno-11-20" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "o" > [< / span > < span class = "n" > i< / span > < span class = "o" > ][< / span > < span class = "n" > j< / span > < span class = "o" > ]< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-11-21" name = "__codelineno-11-21" href = "#__codelineno-11-21" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-12-1" name = "__codelineno-12-1" href = "#__codelineno-12-1" > < / a > < span class = "cm" > /* 最小路径和：记忆化搜索 */< / span >
< a id = "__codelineno-12-2" name = "__codelineno-12-2" href = "#__codelineno-12-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-3" name = "__codelineno-12-3" href = "#__codelineno-12-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-12-4" name = "__codelineno-12-4" href = "#__codelineno-12-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-5" name = "__codelineno-12-5" href = "#__codelineno-12-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-12-6" name = "__codelineno-12-6" href = "#__codelineno-12-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-12-7" name = "__codelineno-12-7" href = "#__codelineno-12-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-12-8" name = "__codelineno-12-8" href = "#__codelineno-12-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "mi" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-9" name = "__codelineno-12-9" href = "#__codelineno-12-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-12-10" name = "__codelineno-12-10" href = "#__codelineno-12-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-12-11" name = "__codelineno-12-11" href = "#__codelineno-12-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若已有记录，则直接返回< / span >
< a id = "__codelineno-12-12" name = "__codelineno-12-12" href = "#__codelineno-12-12" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > !=< / span > < span class = "w" > < / span > < span class = "mi" > -1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-12-13" name = "__codelineno-12-13" href = "#__codelineno-12-13" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-12-14" name = "__codelineno-12-14" href = "#__codelineno-12-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-12-15" name = "__codelineno-12-15" href = "#__codelineno-12-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 左边和上边单元格的最小路径代价< / span >
< a id = "__codelineno-12-16" name = "__codelineno-12-16" href = "#__codelineno-12-16" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-12-17" name = "__codelineno-12-17" href = "#__codelineno-12-17" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-12-18" name = "__codelineno-12-18" href = "#__codelineno-12-18" > < / a > < span class = "w" > < / span > < span class = "c1" > // 记录并返回左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-12-19" name = "__codelineno-12-19" href = "#__codelineno-12-19" > < / a > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > !=< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "w" > < / span > < span class = "o" > ?< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > :< / span > < span class = "w" > < / span > < span class = "n" > INT_MAX< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-12-20" name = "__codelineno-12-20" href = "#__codelineno-12-20" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-12-21" name = "__codelineno-12-21" href = "#__codelineno-12-21" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-13-1" name = "__codelineno-13-1" href = "#__codelineno-13-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_path_sum_dfs_mem< / span > < span class = "p" > (< / span >
< a id = "__codelineno-13-2" name = "__codelineno-13-2" href = "#__codelineno-13-2" > < / a > < span class = "n" > grid< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ]],< / span > < span class = "n" > mem< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ]],< / span > < span class = "n" > i< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "p" > :< / span > < span class = "nb" > int< / span >
< a id = "__codelineno-13-3" name = "__codelineno-13-3" href = "#__codelineno-13-3" > < / a > < span class = "p" > )< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-4" name = "__codelineno-13-4" href = "#__codelineno-13-4" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 最小路径和：记忆化搜索" " " < / span >
< a id = "__codelineno-13-5" name = "__codelineno-13-5" href = "#__codelineno-13-5" > < / a > < span class = "c1" > # 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-13-6" name = "__codelineno-13-6" href = "#__codelineno-13-6" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "ow" > and< / span > < span class = "n" > j< / span > < span class = "o" > ==< / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-7" name = "__codelineno-13-7" href = "#__codelineno-13-7" > < / a > < span class = "k" > return< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-13-8" name = "__codelineno-13-8" href = "#__codelineno-13-8" > < / a > < span class = "c1" > # 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-13-9" name = "__codelineno-13-9" href = "#__codelineno-13-9" > < / a > < span class = "k" > if< / span > < span class = "n" > i< / span > < span class = "o" > < < / span > < span class = "mi" > 0< / span > < span class = "ow" > or< / span > < span class = "n" > j< / span > < span class = "o" > < < / span > < span class = "mi" > 0< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-10" name = "__codelineno-13-10" href = "#__codelineno-13-10" > < / a > < span class = "k" > return< / span > < span class = "n" > inf< / span >
< a id = "__codelineno-13-11" name = "__codelineno-13-11" href = "#__codelineno-13-11" > < / a > < span class = "c1" > # 若已有记录，则直接返回< / span >
< a id = "__codelineno-13-12" name = "__codelineno-13-12" href = "#__codelineno-13-12" > < / a > < span class = "k" > if< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > !=< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > :< / span >
< a id = "__codelineno-13-13" name = "__codelineno-13-13" href = "#__codelineno-13-13" > < / a > < span class = "k" > return< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-13-14" name = "__codelineno-13-14" href = "#__codelineno-13-14" > < / a > < span class = "c1" > # 左边和上边单元格的最小路径代价< / span >
< a id = "__codelineno-13-15" name = "__codelineno-13-15" href = "#__codelineno-13-15" > < / a > < span class = "n" > left< / span > < span class = "o" > =< / span > < span class = "n" > min_path_sum_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-16" name = "__codelineno-13-16" href = "#__codelineno-13-16" > < / a > < span class = "n" > up< / span > < span class = "o" > =< / span > < span class = "n" > min_path_sum_dfs_mem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span >
< a id = "__codelineno-13-17" name = "__codelineno-13-17" href = "#__codelineno-13-17" > < / a > < span class = "c1" > # 记录并返回左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-13-18" name = "__codelineno-13-18" href = "#__codelineno-13-18" > < / a > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-13-19" name = "__codelineno-13-19" href = "#__codelineno-13-19" > < / a > < span class = "k" > return< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-14-1" name = "__codelineno-14-1" href = "#__codelineno-14-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-15-1" name = "__codelineno-15-1" href = "#__codelineno-15-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
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< div class = "highlight" > < span class = "filename" > min_path_sum.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-16-1" name = "__codelineno-16-1" href = "#__codelineno-16-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
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< div class = "highlight" > < span class = "filename" > min_path_sum.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-17-1" name = "__codelineno-17-1" href = "#__codelineno-17-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-18-1" name = "__codelineno-18-1" href = "#__codelineno-18-1" > < / a > < span class = "cm" > /* 最小路径和：记忆化搜索 */< / span >
< a id = "__codelineno-18-2" name = "__codelineno-18-2" href = "#__codelineno-18-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [][]< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-3" name = "__codelineno-18-3" href = "#__codelineno-18-3" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若为左上角单元格，则终止搜索< / span >
< a id = "__codelineno-18-4" name = "__codelineno-18-4" href = "#__codelineno-18-4" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "w" > < / span > < span class = "o" > & & < / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > ==< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-5" name = "__codelineno-18-5" href = "#__codelineno-18-5" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-6" name = "__codelineno-18-6" href = "#__codelineno-18-6" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-18-7" name = "__codelineno-18-7" href = "#__codelineno-18-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若行列索引越界，则返回 +∞ 代价< / span >
< a id = "__codelineno-18-8" name = "__codelineno-18-8" href = "#__codelineno-18-8" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "w" > < / span > < span class = "o" > ||< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-9" name = "__codelineno-18-9" href = "#__codelineno-18-9" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > .< / span > < span class = "n" > MaxValue< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-18-10" name = "__codelineno-18-10" href = "#__codelineno-18-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-18-11" name = "__codelineno-18-11" href = "#__codelineno-18-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 若已有记录，则直接返回< / span >
< a id = "__codelineno-18-12" name = "__codelineno-18-12" href = "#__codelineno-18-12" > < / a > < span class = "w" > < / span > < span class = "k" > if< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > !=< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "m" > 1< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-18-13" name = "__codelineno-18-13" href = "#__codelineno-18-13" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-14" name = "__codelineno-18-14" href = "#__codelineno-18-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-18-15" name = "__codelineno-18-15" href = "#__codelineno-18-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 左边和上边单元格的最小路径代价< / span >
< a id = "__codelineno-18-16" name = "__codelineno-18-16" href = "#__codelineno-18-16" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > left< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > );< / span >
< a id = "__codelineno-18-17" name = "__codelineno-18-17" href = "#__codelineno-18-17" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > );< / span >
< a id = "__codelineno-18-18" name = "__codelineno-18-18" href = "#__codelineno-18-18" > < / a > < span class = "w" > < / span > < span class = "c1" > // 记录并返回左上角到 (i, j) 的最小路径代价< / span >
< a id = "__codelineno-18-19" name = "__codelineno-18-19" href = "#__codelineno-18-19" > < / a > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > left< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > up< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-20" name = "__codelineno-18-20" href = "#__codelineno-18-20" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > mem< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-18-21" name = "__codelineno-18-21" href = "#__codelineno-18-21" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-19-1" name = "__codelineno-19-1" href = "#__codelineno-19-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-20-1" name = "__codelineno-20-1" href = "#__codelineno-20-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-21-1" name = "__codelineno-21-1" href = "#__codelineno-21-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDFSMem< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< / div >
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< p > 如下图所示，引入记忆化可以消除所有重复计算，时间复杂度取决于状态总数，即网格尺寸 < span class = "arithmatex" > \(O(nm)\)< / span > 。< / p >
< p > < img alt = "记忆化搜索递归树" src = "../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" / > < / p >
< p align = "center" > Fig. 记忆化搜索递归树 < / p >
< h2 id = "1335" > 13.3.5. 方法三：动态规划< a class = "headerlink" href = "#1335" title = "Permanent link" > ¶ < / a > < / h2 >
< p > 动态规划代码是从底至顶的，仅需循环即可实现。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "3:11" > < input checked = "checked" id = "__tabbed_3_1" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_2" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_3" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_4" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_5" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_6" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_7" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_8" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_9" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_10" name = "__tabbed_3" type = "radio" / > < input id = "__tabbed_3_11" name = "__tabbed_3" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_3_1" > Java< / label > < label for = "__tabbed_3_2" > C++< / label > < label for = "__tabbed_3_3" > Python< / label > < label for = "__tabbed_3_4" > Go< / label > < label for = "__tabbed_3_5" > JavaScript< / label > < label for = "__tabbed_3_6" > TypeScript< / label > < label for = "__tabbed_3_7" > C< / label > < label for = "__tabbed_3_8" > C#< / label > < label for = "__tabbed_3_9" > Swift< / label > < label for = "__tabbed_3_10" > Zig< / label > < label for = "__tabbed_3_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-22-1" name = "__codelineno-22-1" href = "#__codelineno-22-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > min< / span > < span class = "p" > }< / span > < span class = "o" > -[< / span > < span class = "n" > func< / span > < span class = "o" > ]< / span > < span class = "p" > {< / span > < span class = "n" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-23-1" name = "__codelineno-23-1" href = "#__codelineno-23-1" > < / a > < span class = "cm" > /* 最小路径和：动态规划 */< / span >
< a id = "__codelineno-23-2" name = "__codelineno-23-2" href = "#__codelineno-23-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDP< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > grid< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-3" name = "__codelineno-23-3" href = "#__codelineno-23-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > .< / span > < span class = "n" > size< / span > < span class = "p" > (),< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ].< / span > < span class = "n" > size< / span > < span class = "p" > ();< / span >
< a id = "__codelineno-23-4" name = "__codelineno-23-4" href = "#__codelineno-23-4" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表< / span >
< a id = "__codelineno-23-5" name = "__codelineno-23-5" href = "#__codelineno-23-5" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "p" > (< / span > < span class = "n" > m< / span > < span class = "p" > ));< / span >
< a id = "__codelineno-23-6" name = "__codelineno-23-6" href = "#__codelineno-23-6" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-7" name = "__codelineno-23-7" href = "#__codelineno-23-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首行< / span >
< a id = "__codelineno-23-8" name = "__codelineno-23-8" href = "#__codelineno-23-8" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-9" name = "__codelineno-23-9" href = "#__codelineno-23-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-10" name = "__codelineno-23-10" href = "#__codelineno-23-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-11" name = "__codelineno-23-11" href = "#__codelineno-23-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首列< / span >
< a id = "__codelineno-23-12" name = "__codelineno-23-12" href = "#__codelineno-23-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-13" name = "__codelineno-23-13" href = "#__codelineno-23-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-14" name = "__codelineno-23-14" href = "#__codelineno-23-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-15" name = "__codelineno-23-15" href = "#__codelineno-23-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余行列< / span >
< a id = "__codelineno-23-16" name = "__codelineno-23-16" href = "#__codelineno-23-16" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-17" name = "__codelineno-23-17" href = "#__codelineno-23-17" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-23-18" name = "__codelineno-23-18" href = "#__codelineno-23-18" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-19" name = "__codelineno-23-19" href = "#__codelineno-23-19" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-20" name = "__codelineno-23-20" href = "#__codelineno-23-20" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-23-21" name = "__codelineno-23-21" href = "#__codelineno-23-21" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-23-22" name = "__codelineno-23-22" href = "#__codelineno-23-22" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-24-1" name = "__codelineno-24-1" href = "#__codelineno-24-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_path_sum_dp< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ]])< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-24-2" name = "__codelineno-24-2" href = "#__codelineno-24-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 最小路径和：动态规划" " " < / span >
< a id = "__codelineno-24-3" name = "__codelineno-24-3" href = "#__codelineno-24-3" > < / a > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ),< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ])< / span >
< a id = "__codelineno-24-4" name = "__codelineno-24-4" href = "#__codelineno-24-4" > < / a > < span class = "c1" > # 初始化 dp 表< / span >
< a id = "__codelineno-24-5" name = "__codelineno-24-5" href = "#__codelineno-24-5" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [[< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "n" > m< / span > < span class = "k" > for< / span > < span class = "n" > _< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "n" > n< / span > < span class = "p" > )]< / span >
< a id = "__codelineno-24-6" name = "__codelineno-24-6" href = "#__codelineno-24-6" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-7" name = "__codelineno-24-7" href = "#__codelineno-24-7" > < / a > < span class = "c1" > # 状态转移：首行< / span >
< a id = "__codelineno-24-8" name = "__codelineno-24-8" href = "#__codelineno-24-8" > < / a > < span class = "k" > for< / span > < span class = "n" > j< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-9" name = "__codelineno-24-9" href = "#__codelineno-24-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-10" name = "__codelineno-24-10" href = "#__codelineno-24-10" > < / a > < span class = "c1" > # 状态转移：首列< / span >
< a id = "__codelineno-24-11" name = "__codelineno-24-11" href = "#__codelineno-24-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-12" name = "__codelineno-24-12" href = "#__codelineno-24-12" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-13" name = "__codelineno-24-13" href = "#__codelineno-24-13" > < / a > < span class = "c1" > # 状态转移：其余行列< / span >
< a id = "__codelineno-24-14" name = "__codelineno-24-14" href = "#__codelineno-24-14" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-15" name = "__codelineno-24-15" href = "#__codelineno-24-15" > < / a > < span class = "k" > for< / span > < span class = "n" > j< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-24-16" name = "__codelineno-24-16" href = "#__codelineno-24-16" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-24-17" name = "__codelineno-24-17" href = "#__codelineno-24-17" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ][< / span > < span class = "n" > m< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-25-1" name = "__codelineno-25-1" href = "#__codelineno-25-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-26-1" name = "__codelineno-26-1" href = "#__codelineno-26-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-27-1" name = "__codelineno-27-1" href = "#__codelineno-27-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-28-1" name = "__codelineno-28-1" href = "#__codelineno-28-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-29-1" name = "__codelineno-29-1" href = "#__codelineno-29-1" > < / a > < span class = "cm" > /* 最小路径和：动态规划 */< / span >
< a id = "__codelineno-29-2" name = "__codelineno-29-2" href = "#__codelineno-29-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDP< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-3" name = "__codelineno-29-3" href = "#__codelineno-29-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ].< / span > < span class = "n" > Length< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-29-4" name = "__codelineno-29-4" href = "#__codelineno-29-4" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表< / span >
< a id = "__codelineno-29-5" name = "__codelineno-29-5" href = "#__codelineno-29-5" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [,]< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-6" name = "__codelineno-29-6" href = "#__codelineno-29-6" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-7" name = "__codelineno-29-7" href = "#__codelineno-29-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首行< / span >
< a id = "__codelineno-29-8" name = "__codelineno-29-8" href = "#__codelineno-29-8" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-9" name = "__codelineno-29-9" href = "#__codelineno-29-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-10" name = "__codelineno-29-10" href = "#__codelineno-29-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-29-11" name = "__codelineno-29-11" href = "#__codelineno-29-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首列< / span >
< a id = "__codelineno-29-12" name = "__codelineno-29-12" href = "#__codelineno-29-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-13" name = "__codelineno-29-13" href = "#__codelineno-29-13" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-14" name = "__codelineno-29-14" href = "#__codelineno-29-14" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-29-15" name = "__codelineno-29-15" href = "#__codelineno-29-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余行列< / span >
< a id = "__codelineno-29-16" name = "__codelineno-29-16" href = "#__codelineno-29-16" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-17" name = "__codelineno-29-17" href = "#__codelineno-29-17" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-29-18" name = "__codelineno-29-18" href = "#__codelineno-29-18" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-19" name = "__codelineno-29-19" href = "#__codelineno-29-19" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-29-20" name = "__codelineno-29-20" href = "#__codelineno-29-20" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-29-21" name = "__codelineno-29-21" href = "#__codelineno-29-21" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-29-22" name = "__codelineno-29-22" href = "#__codelineno-29-22" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-30-1" name = "__codelineno-30-1" href = "#__codelineno-30-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-31-1" name = "__codelineno-31-1" href = "#__codelineno-31-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-32-1" name = "__codelineno-32-1" href = "#__codelineno-32-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDP< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< / div >
< / div >
< p > 下图展示了最小路径和的状态转移过程。该过程遍历了整个网格，因此时间复杂度为 < span class = "arithmatex" > \(O(nm)\)< / span > ；数组 < code > dp< / code > 使用 < span class = "arithmatex" > \(O(nm)\)< / span > 空间。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "4:12" > < input checked = "checked" id = "__tabbed_4_1" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_2" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_3" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_4" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_5" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_6" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_7" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_8" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_9" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_10" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_11" name = "__tabbed_4" type = "radio" / > < input id = "__tabbed_4_12" name = "__tabbed_4" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_4_1" > < 1> < / label > < label for = "__tabbed_4_2" > < 2> < / label > < label for = "__tabbed_4_3" > < 3> < / label > < label for = "__tabbed_4_4" > < 4> < / label > < label for = "__tabbed_4_5" > < 5> < / label > < label for = "__tabbed_4_6" > < 6> < / label > < label for = "__tabbed_4_7" > < 7> < / label > < label for = "__tabbed_4_8" > < 8> < / label > < label for = "__tabbed_4_9" > < 9> < / label > < label for = "__tabbed_4_10" > < 10> < / label > < label for = "__tabbed_4_11" > < 11> < / label > < label for = "__tabbed_4_12" > < 12> < / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< p > < img alt = "最小路径和的动态规划过程" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step1.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step2" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step2.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step3" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step3.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step4" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step4.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step5" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step5.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step6" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step6.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step7" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step7.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step8" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step8.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step9" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step9.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step10" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step10.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step11" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step11.png" / > < / p >
< / div >
< div class = "tabbed-block" >
< p > < img alt = "min_path_sum_dp_step12" src = "../dp_solution_pipeline.assets/min_path_sum_dp_step12.png" / > < / p >
< / div >
< / div >
< / div >
< p > 如果希望进一步节省空间使用，可以考虑进行状态压缩。每个格子只与左边和上边的格子有关，因此我们可以只用一个单行数组来实现 < span class = "arithmatex" > \(dp\)< / span > 表。< / p >
< p > 由于数组 < code > dp< / code > 只能表示一行的状态，因此我们无法提前初始化首列状态，而是在遍历每行中更新它。< / p >
< div class = "tabbed-set tabbed-alternate" data-tabs = "5:11" > < input checked = "checked" id = "__tabbed_5_1" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_2" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_3" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_4" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_5" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_6" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_7" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_8" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_9" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_10" name = "__tabbed_5" type = "radio" / > < input id = "__tabbed_5_11" name = "__tabbed_5" type = "radio" / > < div class = "tabbed-labels" > < label for = "__tabbed_5_1" > Java< / label > < label for = "__tabbed_5_2" > C++< / label > < label for = "__tabbed_5_3" > Python< / label > < label for = "__tabbed_5_4" > Go< / label > < label for = "__tabbed_5_5" > JavaScript< / label > < label for = "__tabbed_5_6" > TypeScript< / label > < label for = "__tabbed_5_7" > C< / label > < label for = "__tabbed_5_8" > C#< / label > < label for = "__tabbed_5_9" > Swift< / label > < label for = "__tabbed_5_10" > Zig< / label > < label for = "__tabbed_5_11" > Dart< / label > < / div >
< div class = "tabbed-content" >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.java< / span > < pre > < span > < / span > < code > < a id = "__codelineno-33-1" name = "__codelineno-33-1" href = "#__codelineno-33-1" > < / a > < span class = "o" > [< / span > < span class = "kd" > class< / span > < span class = "err" > ]{< / span > < span class = "nc" > min< / span > < span class = "p" > }< / span > < span class = "o" > -[< / span > < span class = "n" > func< / span > < span class = "o" > ]< / span > < span class = "p" > {< / span > < span class = "n" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cpp< / span > < pre > < span > < / span > < code > < a id = "__codelineno-34-1" name = "__codelineno-34-1" href = "#__codelineno-34-1" > < / a > < span class = "cm" > /* 最小路径和：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-34-2" name = "__codelineno-34-2" href = "#__codelineno-34-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDPComp< / span > < span class = "p" > (< / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > > < / span > < span class = "w" > < / span > < span class = "o" > & < / span > < span class = "n" > grid< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-34-3" name = "__codelineno-34-3" href = "#__codelineno-34-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > .< / span > < span class = "n" > size< / span > < span class = "p" > (),< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ].< / span > < span class = "n" > size< / span > < span class = "p" > ();< / span >
< a id = "__codelineno-34-4" name = "__codelineno-34-4" href = "#__codelineno-34-4" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表< / span >
< a id = "__codelineno-34-5" name = "__codelineno-34-5" href = "#__codelineno-34-5" > < / a > < span class = "w" > < / span > < span class = "n" > vector< / span > < span class = "o" > < < / span > < span class = "kt" > int< / span > < span class = "o" > > < / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > (< / span > < span class = "n" > m< / span > < span class = "p" > );< / span >
< a id = "__codelineno-34-6" name = "__codelineno-34-6" href = "#__codelineno-34-6" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首行< / span >
< a id = "__codelineno-34-7" name = "__codelineno-34-7" href = "#__codelineno-34-7" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-34-8" name = "__codelineno-34-8" href = "#__codelineno-34-8" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-34-9" name = "__codelineno-34-9" href = "#__codelineno-34-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-34-10" name = "__codelineno-34-10" href = "#__codelineno-34-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-34-11" name = "__codelineno-34-11" href = "#__codelineno-34-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余行< / span >
< a id = "__codelineno-34-12" name = "__codelineno-34-12" href = "#__codelineno-34-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-34-13" name = "__codelineno-34-13" href = "#__codelineno-34-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首列< / span >
< a id = "__codelineno-34-14" name = "__codelineno-34-14" href = "#__codelineno-34-14" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-34-15" name = "__codelineno-34-15" href = "#__codelineno-34-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余列< / span >
< a id = "__codelineno-34-16" name = "__codelineno-34-16" href = "#__codelineno-34-16" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-34-17" name = "__codelineno-34-17" href = "#__codelineno-34-17" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-34-18" name = "__codelineno-34-18" href = "#__codelineno-34-18" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-34-19" name = "__codelineno-34-19" href = "#__codelineno-34-19" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-34-20" name = "__codelineno-34-20" href = "#__codelineno-34-20" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "mi" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-34-21" name = "__codelineno-34-21" href = "#__codelineno-34-21" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.py< / span > < pre > < span > < / span > < code > < a id = "__codelineno-35-1" name = "__codelineno-35-1" href = "#__codelineno-35-1" > < / a > < span class = "k" > def< / span > < span class = "nf" > min_path_sum_dp_comp< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > :< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > list< / span > < span class = "p" > [< / span > < span class = "nb" > int< / span > < span class = "p" > ]])< / span > < span class = "o" > -> < / span > < span class = "nb" > int< / span > < span class = "p" > :< / span >
< a id = "__codelineno-35-2" name = "__codelineno-35-2" href = "#__codelineno-35-2" > < / a > < span class = "w" > < / span > < span class = "sd" > " " " 最小路径和：状态压缩后的动态规划" " " < / span >
< a id = "__codelineno-35-3" name = "__codelineno-35-3" href = "#__codelineno-35-3" > < / a > < span class = "n" > n< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > ),< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ])< / span >
< a id = "__codelineno-35-4" name = "__codelineno-35-4" href = "#__codelineno-35-4" > < / a > < span class = "c1" > # 初始化 dp 表< / span >
< a id = "__codelineno-35-5" name = "__codelineno-35-5" href = "#__codelineno-35-5" > < / a > < span class = "n" > dp< / span > < span class = "o" > =< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "n" > m< / span >
< a id = "__codelineno-35-6" name = "__codelineno-35-6" href = "#__codelineno-35-6" > < / a > < span class = "c1" > # 状态转移：首行< / span >
< a id = "__codelineno-35-7" name = "__codelineno-35-7" href = "#__codelineno-35-7" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-35-8" name = "__codelineno-35-8" href = "#__codelineno-35-8" > < / a > < span class = "k" > for< / span > < span class = "n" > j< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-35-9" name = "__codelineno-35-9" href = "#__codelineno-35-9" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-35-10" name = "__codelineno-35-10" href = "#__codelineno-35-10" > < / a > < span class = "c1" > # 状态转移：其余行< / span >
< a id = "__codelineno-35-11" name = "__codelineno-35-11" href = "#__codelineno-35-11" > < / a > < span class = "k" > for< / span > < span class = "n" > i< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > n< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-35-12" name = "__codelineno-35-12" href = "#__codelineno-35-12" > < / a > < span class = "c1" > # 状态转移：首列< / span >
< a id = "__codelineno-35-13" name = "__codelineno-35-13" href = "#__codelineno-35-13" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-35-14" name = "__codelineno-35-14" href = "#__codelineno-35-14" > < / a > < span class = "c1" > # 状态转移：其余列< / span >
< a id = "__codelineno-35-15" name = "__codelineno-35-15" href = "#__codelineno-35-15" > < / a > < span class = "k" > for< / span > < span class = "n" > j< / span > < span class = "ow" > in< / span > < span class = "nb" > range< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > m< / span > < span class = "p" > ):< / span >
< a id = "__codelineno-35-16" name = "__codelineno-35-16" href = "#__codelineno-35-16" > < / a > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "nb" > min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "o" > +< / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span >
< a id = "__codelineno-35-17" name = "__codelineno-35-17" href = "#__codelineno-35-17" > < / a > < span class = "k" > return< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > m< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.go< / span > < pre > < span > < / span > < code > < a id = "__codelineno-36-1" name = "__codelineno-36-1" href = "#__codelineno-36-1" > < / a > < span class = "p" > [< / span > < span class = "nx" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.js< / span > < pre > < span > < / span > < code > < a id = "__codelineno-37-1" name = "__codelineno-37-1" href = "#__codelineno-37-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.ts< / span > < pre > < span > < / span > < code > < a id = "__codelineno-38-1" name = "__codelineno-38-1" href = "#__codelineno-38-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "nx" > func< / span > < span class = "p" > ]{< / span > < span class = "nx" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.c< / span > < pre > < span > < / span > < code > < a id = "__codelineno-39-1" name = "__codelineno-39-1" href = "#__codelineno-39-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.cs< / span > < pre > < span > < / span > < code > < a id = "__codelineno-40-1" name = "__codelineno-40-1" href = "#__codelineno-40-1" > < / a > < span class = "cm" > /* 最小路径和：状态压缩后的动态规划 */< / span >
< a id = "__codelineno-40-2" name = "__codelineno-40-2" href = "#__codelineno-40-2" > < / a > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "nf" > minPathSumDPComp< / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "p" > [][]< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-40-3" name = "__codelineno-40-3" href = "#__codelineno-40-3" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > .< / span > < span class = "n" > Length< / span > < span class = "p" > ,< / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ].< / span > < span class = "n" > Length< / span > < span class = "p" > ;< / span >
< a id = "__codelineno-40-4" name = "__codelineno-40-4" href = "#__codelineno-40-4" > < / a > < span class = "w" > < / span > < span class = "c1" > // 初始化 dp 表< / span >
< a id = "__codelineno-40-5" name = "__codelineno-40-5" href = "#__codelineno-40-5" > < / a > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > []< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "k" > new< / span > < span class = "w" > < / span > < span class = "kt" > int< / span > < span class = "p" > [< / span > < span class = "n" > m< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-6" name = "__codelineno-40-6" href = "#__codelineno-40-6" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-7" name = "__codelineno-40-7" href = "#__codelineno-40-7" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首行< / span >
< a id = "__codelineno-40-8" name = "__codelineno-40-8" href = "#__codelineno-40-8" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-40-9" name = "__codelineno-40-9" href = "#__codelineno-40-9" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-10" name = "__codelineno-40-10" href = "#__codelineno-40-10" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-40-11" name = "__codelineno-40-11" href = "#__codelineno-40-11" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余行< / span >
< a id = "__codelineno-40-12" name = "__codelineno-40-12" href = "#__codelineno-40-12" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > n< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > i< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-40-13" name = "__codelineno-40-13" href = "#__codelineno-40-13" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：首列< / span >
< a id = "__codelineno-40-14" name = "__codelineno-40-14" href = "#__codelineno-40-14" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "m" > 0< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "m" > 0< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-15" name = "__codelineno-40-15" href = "#__codelineno-40-15" > < / a > < span class = "w" > < / span > < span class = "c1" > // 状态转移：其余列< / span >
< a id = "__codelineno-40-16" name = "__codelineno-40-16" href = "#__codelineno-40-16" > < / a > < span class = "w" > < / span > < span class = "k" > for< / span > < span class = "w" > < / span > < span class = "p" > (< / span > < span class = "kt" > int< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > < < / span > < span class = "w" > < / span > < span class = "n" > m< / span > < span class = "p" > ;< / span > < span class = "w" > < / span > < span class = "n" > j< / span > < span class = "o" > ++< / span > < span class = "p" > )< / span > < span class = "w" > < / span > < span class = "p" > {< / span >
< a id = "__codelineno-40-17" name = "__codelineno-40-17" href = "#__codelineno-40-17" > < / a > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ]< / span > < span class = "w" > < / span > < span class = "o" > =< / span > < span class = "w" > < / span > < span class = "n" > Math< / span > < span class = "p" > .< / span > < span class = "n" > Min< / span > < span class = "p" > (< / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ],< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > j< / span > < span class = "p" > ])< / span > < span class = "w" > < / span > < span class = "o" > +< / span > < span class = "w" > < / span > < span class = "n" > grid< / span > < span class = "p" > [< / span > < span class = "n" > i< / span > < span class = "p" > ][< / span > < span class = "n" > j< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-18" name = "__codelineno-40-18" href = "#__codelineno-40-18" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-40-19" name = "__codelineno-40-19" href = "#__codelineno-40-19" > < / a > < span class = "w" > < / span > < span class = "p" > }< / span >
< a id = "__codelineno-40-20" name = "__codelineno-40-20" href = "#__codelineno-40-20" > < / a > < span class = "w" > < / span > < span class = "k" > return< / span > < span class = "w" > < / span > < span class = "n" > dp< / span > < span class = "p" > [< / span > < span class = "n" > m< / span > < span class = "w" > < / span > < span class = "o" > -< / span > < span class = "w" > < / span > < span class = "m" > 1< / span > < span class = "p" > ];< / span >
< a id = "__codelineno-40-21" name = "__codelineno-40-21" href = "#__codelineno-40-21" > < / a > < span class = "p" > }< / span >
< / code > < / pre > < / div >
< / div >
< div class = "tabbed-block" >
< div class = "highlight" > < span class = "filename" > min_path_sum.swift< / span > < pre > < span > < / span > < code > < a id = "__codelineno-41-1" name = "__codelineno-41-1" href = "#__codelineno-41-1" > < / a > < span class = "p" > [< / span > < span class = "kd" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "kd" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDPComp< / span > < span class = "p" > }< / span >
< / code > < / pre > < / div >
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< div class = "highlight" > < span class = "filename" > min_path_sum.zig< / span > < pre > < span > < / span > < code > < a id = "__codelineno-42-1" name = "__codelineno-42-1" href = "#__codelineno-42-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDPComp< / span > < span class = "p" > }< / span >
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< div class = "highlight" > < span class = "filename" > min_path_sum.dart< / span > < pre > < span > < / span > < code > < a id = "__codelineno-43-1" name = "__codelineno-43-1" href = "#__codelineno-43-1" > < / a > < span class = "p" > [< / span > < span class = "n" > class< / span > < span class = "p" > ]{}< / span > < span class = "o" > -< / span > < span class = "p" > [< / span > < span class = "n" > func< / span > < span class = "p" > ]{< / span > < span class = "n" > minPathSumDPComp< / span > < span class = "p" > }< / span >
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