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# 14.3 &nbsp; 动态规划解题思路

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。

1. 如何判断一个问题是不是动态规划问题？
2. 求解动态规划问题该从何处入手，完整步骤是什么？

## 14.3.1 &nbsp; 问题判断

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，那么它通常适合用动态规划求解。然而，我们很难从问题描述中直接提取出这些特性。因此我们通常会放宽条件，**先观察问题是否适合使用回溯（穷举）解决**。

**适合用回溯解决的问题通常满足“决策树模型”**，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。

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。

在此基础上，动态规划问题还有一些判断的“加分项”。

- 问题包含最大（小）或最多（少）等最优化描述。
- 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递推关系。

相应地，也存在一些“减分项”。

- 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。
- 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。

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，并在求解过程中验证它。

## 14.3.2 &nbsp; 问题求解步骤

动态规划的解题流程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 $dp$ 表，推导状态转移方程，确定边界条件等。

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。

!!! question

    给定一个 $n \times m$ 的二维网格 `grid` ，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。

图 14-10 展示了一个例子，给定网格的最小路径和为 $13$ 。

![最小路径和示例数据](dp_solution_pipeline.assets/min_path_sum_example.png){ class="animation-figure" }

<p align="center"> 图 14-10 &nbsp; 最小路径和示例数据 </p>

**第一步：思考每轮的决策，定义状态，从而得到 $dp$ 表**

本题的每一轮的决策就是从当前格子向下或向右走一步。设当前格子的行列索引为 $[i, j]$ ，则向下或向右走一步后，索引变为 $[i+1, j]$ 或 $[i, j+1]$ 。因此，状态应包含行索引和列索引两个变量，记为 $[i, j]$ 。

状态 $[i, j]$ 对应的子问题为：从起始点 $[0, 0]$ 走到 $[i, j]$ 的最小路径和，解记为 $dp[i, j]$ 。

至此，我们就得到了图 14-11 所示的二维 $dp$ 矩阵，其尺寸与输入网格 $grid$ 相同。

![状态定义与 dp 表](dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png){ class="animation-figure" }

<p align="center"> 图 14-11 &nbsp; 状态定义与 dp 表 </p>

!!! note

    动态规划和回溯过程可以描述为一个决策序列，而状态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。
    
    每个状态都对应一个子问题，我们会定义一个 $dp$ 表来存储所有子问题的解，状态的每个独立变量都是 $dp$ 表的一个维度。从本质上看，$dp$ 表是状态和子问题的解之间的映射。

**第二步：找出最优子结构，进而推导出状态转移方程**

对于状态 $[i, j]$ ，它只能从上边格子 $[i-1, j]$ 和左边格子 $[i, j-1]$ 转移而来。因此最优子结构为：到达 $[i, j]$ 的最小路径和由 $[i, j-1]$ 的最小路径和与 $[i-1, j]$ 的最小路径和中较小的那一个决定。

根据以上分析，可推出图 14-12 所示的状态转移方程：

$$
dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
$$

![最优子结构与状态转移方程](dp_solution_pipeline.assets/min_path_sum_solution_state_transition.png){ class="animation-figure" }

<p align="center"> 图 14-12 &nbsp; 最优子结构与状态转移方程 </p>

!!! note

    根据定义好的 $dp$ 表，思考原问题和子问题的关系，找出通过子问题的最优解来构造原问题的最优解的方法，即最优子结构。

    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。

**第三步：确定边界条件和状态转移顺序**

在本题中，处在首行的状态只能从其左边的状态得来，处在首列的状态只能从其上边的状态得来，因此首行 $i = 0$ 和首列 $j = 0$ 是边界条件。

如图 14-13 所示，由于每个格子是由其左方格子和上方格子转移而来，因此我们使用循环来遍历矩阵，外循环遍历各行，内循环遍历各列。

![边界条件与状态转移顺序](dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png){ class="animation-figure" }

<p align="center"> 图 14-13 &nbsp; 边界条件与状态转移顺序 </p>

!!! note

    边界条件在动态规划中用于初始化 $dp$ 表，在搜索中用于剪枝。
    
    状态转移顺序的核心是要保证在计算当前问题的解时，所有它依赖的更小子问题的解都已经被正确地计算出来。

根据以上分析，我们已经可以直接写出动态规划代码。然而子问题分解是一种从顶至底的思想，因此按照“暴力搜索 $\rightarrow$ 记忆化搜索 $\rightarrow$ 动态规划”的顺序实现更加符合思维习惯。

### 1. &nbsp; 方法一：暴力搜索

从状态 $[i, j]$ 开始搜索，不断分解为更小的状态 $[i-1, j]$ 和 $[i, j-1]$ ，递归函数包括以下要素。

- **递归参数**：状态 $[i, j]$ 。
- **返回值**：从 $[0, 0]$ 到 $[i, j]$ 的最小路径和 $dp[i, j]$ 。
- **终止条件**：当 $i = 0$ 且 $j = 0$ 时，返回代价 $grid[0, 0]$ 。
- **剪枝**：当 $i < 0$ 时或 $j < 0$ 时索引越界，此时返回代价 $+\infty$ ，代表不可行。

实现代码如下：

=== "Python"

    ```python title="min_path_sum.py"
    def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -> int:
        """最小路径和：暴力搜索"""
        # 若为左上角单元格，则终止搜索
        if i == 0 and j == 0:
            return grid[0][0]
        # 若行列索引越界，则返回 +∞ 代价
        if i < 0 or j < 0:
            return inf
        # 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        up = min_path_sum_dfs(grid, i - 1, j)
        left = min_path_sum_dfs(grid, i, j - 1)
        # 返回从左上角到 (i, j) 的最小路径代价
        return min(left, up) + grid[i][j]
    ```

=== "C++"

    ```cpp title="min_path_sum.cpp"
    /* 最小路径和：暴力搜索 */
    int minPathSumDFS(vector<vector<int>> &grid, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return INT_MAX;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        int up = minPathSumDFS(grid, i - 1, j);
        int left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
    }
    ```

=== "Java"

    ```java title="min_path_sum.java"
    /* 最小路径和：暴力搜索 */
    int minPathSumDFS(int[][] grid, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Integer.MAX_VALUE;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        int up = minPathSumDFS(grid, i - 1, j);
        int left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return Math.min(left, up) + grid[i][j];
    }
    ```

=== "C#"

    ```csharp title="min_path_sum.cs"
    /* 最小路径和：暴力搜索 */
    int MinPathSumDFS(int[][] grid, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return int.MaxValue;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        int up = MinPathSumDFS(grid, i - 1, j);
        int left = MinPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return Math.Min(left, up) + grid[i][j];
    }
    ```

=== "Go"

    ```go title="min_path_sum.go"
    /* 最小路径和：暴力搜索 */
    func minPathSumDFS(grid [][]int, i, j int) int {
        // 若为左上角单元格，则终止搜索
        if i == 0 && j == 0 {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return math.MaxInt
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        up := minPathSumDFS(grid, i-1, j)
        left := minPathSumDFS(grid, i, j-1)
        // 返回从左上角到 (i, j) 的最小路径代价
        return int(math.Min(float64(left), float64(up))) + grid[i][j]
    }
    ```

=== "Swift"

    ```swift title="min_path_sum.swift"
    /* 最小路径和：暴力搜索 */
    func minPathSumDFS(grid: [[Int]], i: Int, j: Int) -> Int {
        // 若为左上角单元格，则终止搜索
        if i == 0, j == 0 {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return .max
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        let up = minPathSumDFS(grid: grid, i: i - 1, j: j)
        let left = minPathSumDFS(grid: grid, i: i, j: j - 1)
        // 返回从左上角到 (i, j) 的最小路径代价
        return min(left, up) + grid[i][j]
    }
    ```

=== "JS"

    ```javascript title="min_path_sum.js"
    /* 最小路径和：暴力搜索 */
    function minPathSumDFS(grid, i, j) {
        // 若为左上角单元格，则终止搜索
        if (i === 0 && j === 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Infinity;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        const up = minPathSumDFS(grid, i - 1, j);
        const left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return Math.min(left, up) + grid[i][j];
    }
    ```

=== "TS"

    ```typescript title="min_path_sum.ts"
    /* 最小路径和：暴力搜索 */
    function minPathSumDFS(
        grid: Array<Array<number>>,
        i: number,
        j: number
    ): number {
        // 若为左上角单元格，则终止搜索
        if (i === 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Infinity;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        const up = minPathSumDFS(grid, i - 1, j);
        const left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return Math.min(left, up) + grid[i][j];
    }
    ```

=== "Dart"

    ```dart title="min_path_sum.dart"
    /* 最小路径和：暴力搜索 */
    int minPathSumDFS(List<List<int>> grid, int i, int j) {
      // 若为左上角单元格，则终止搜索
      if (i == 0 && j == 0) {
        return grid[0][0];
      }
      // 若行列索引越界，则返回 +∞ 代价
      if (i < 0 || j < 0) {
        // 在 Dart 中，int 类型是固定范围的整数，不存在表示“无穷大”的值
        return BigInt.from(2).pow(31).toInt();
      }
      // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
      int up = minPathSumDFS(grid, i - 1, j);
      int left = minPathSumDFS(grid, i, j - 1);
      // 返回从左上角到 (i, j) 的最小路径代价
      return min(left, up) + grid[i][j];
    }
    ```

=== "Rust"

    ```rust title="min_path_sum.rs"
    /* 最小路径和：暴力搜索 */
    fn min_path_sum_dfs(grid: &Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
        // 若为左上角单元格，则终止搜索
        if i == 0 && j == 0 {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return i32::MAX;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        let up = min_path_sum_dfs(grid, i - 1, j);
        let left = min_path_sum_dfs(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        std::cmp::min(left, up) + grid[i as usize][j as usize]
    }
    ```

=== "C"

    ```c title="min_path_sum.c"
    /* 最小路径和：暴力搜索 */
    int minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return INT_MAX;
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        int up = minPathSumDFS(grid, i - 1, j);
        int left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;
    }
    ```

=== "Kotlin"

    ```kotlin title="min_path_sum.kt"
    /* 最小路径和：暴力搜索 */
    fun minPathSumDFS(grid: Array<IntArray>, i: Int, j: Int): Int {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Int.MAX_VALUE
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        val up = minPathSumDFS(grid, i - 1, j)
        val left = minPathSumDFS(grid, i, j - 1)
        // 返回从左上角到 (i, j) 的最小路径代价
        return min(left, up) + grid[i][j]
    }
    ```

=== "Ruby"

    ```ruby title="min_path_sum.rb"
    ### 最小路径和：暴力搜索 ###
    def min_path_sum_dfs(grid, i, j)
      # 若为左上角单元格，则终止搜索
      return grid[i][j] if i == 0 && j == 0
      # 若行列索引越界，则返回 +∞ 代价
      return Float::INFINITY if i < 0 || j < 0
      # 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
      up = min_path_sum_dfs(grid, i - 1, j)
      left = min_path_sum_dfs(grid, i, j - 1)
      # 返回从左上角到 (i, j) 的最小路径代价
      [left, up].min + grid[i][j]
    end
    ```

=== "Zig"

    ```zig title="min_path_sum.zig"
    // 最小路径和：暴力搜索
    fn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {
        // 若为左上角单元格，则终止搜索
        if (i == 0 and j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 or j < 0) {
            return std.math.maxInt(i32);
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        var up = minPathSumDFS(grid, i - 1, j);
        var left = minPathSumDFS(grid, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
    }
    ```

??? pythontutor "可视化运行"

    <div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1,%20j%29%20%E5%92%8C%20%28i,%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
    <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%AE%A1%E7%AE%97%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i-1,%20j%29%20%E5%92%8C%20%28i,%20j-1%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%BF%94%E5%9B%9E%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E6%9A%B4%E5%8A%9B%E6%90%9C%E7%B4%A2%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>

图 14-14 给出了以 $dp[2, 1]$ 为根节点的递归树，其中包含一些重叠子问题，其数量会随着网格 `grid` 的尺寸变大而急剧增多。

从本质上看，造成重叠子问题的原因为：**存在多条路径可以从左上角到达某一单元格**。

![暴力搜索递归树](dp_solution_pipeline.assets/min_path_sum_dfs.png){ class="animation-figure" }

<p align="center"> 图 14-14 &nbsp; 暴力搜索递归树 </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 $m + n - 2$ 步，所以最差时间复杂度为 $O(2^{m + n})$ ，其中 $n$ 和 $m$ 分别为网格的行数和列数。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。

### 2. &nbsp; 方法二：记忆化搜索

我们引入一个和网格 `grid` 相同尺寸的记忆列表 `mem` ，用于记录各个子问题的解，并将重叠子问题进行剪枝：

=== "Python"

    ```python title="min_path_sum.py"
    def min_path_sum_dfs_mem(
        grid: list[list[int]], mem: list[list[int]], i: int, j: int
    ) -> int:
        """最小路径和：记忆化搜索"""
        # 若为左上角单元格，则终止搜索
        if i == 0 and j == 0:
            return grid[0][0]
        # 若行列索引越界，则返回 +∞ 代价
        if i < 0 or j < 0:
            return inf
        # 若已有记录，则直接返回
        if mem[i][j] != -1:
            return mem[i][j]
        # 左边和上边单元格的最小路径代价
        up = min_path_sum_dfs_mem(grid, mem, i - 1, j)
        left = min_path_sum_dfs_mem(grid, mem, i, j - 1)
        # 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = min(left, up) + grid[i][j]
        return mem[i][j]
    ```

=== "C++"

    ```cpp title="min_path_sum.cpp"
    /* 最小路径和：记忆化搜索 */
    int minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return INT_MAX;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        int up = minPathSumDFSMem(grid, mem, i - 1, j);
        int left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
        return mem[i][j];
    }
    ```

=== "Java"

    ```java title="min_path_sum.java"
    /* 最小路径和：记忆化搜索 */
    int minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Integer.MAX_VALUE;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        int up = minPathSumDFSMem(grid, mem, i - 1, j);
        int left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = Math.min(left, up) + grid[i][j];
        return mem[i][j];
    }
    ```

=== "C#"

    ```csharp title="min_path_sum.cs"
    /* 最小路径和：记忆化搜索 */
    int MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return int.MaxValue;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        int up = MinPathSumDFSMem(grid, mem, i - 1, j);
        int left = MinPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = Math.Min(left, up) + grid[i][j];
        return mem[i][j];
    }
    ```

=== "Go"

    ```go title="min_path_sum.go"
    /* 最小路径和：记忆化搜索 */
    func minPathSumDFSMem(grid, mem [][]int, i, j int) int {
        // 若为左上角单元格，则终止搜索
        if i == 0 && j == 0 {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return math.MaxInt
        }
        // 若已有记录，则直接返回
        if mem[i][j] != -1 {
            return mem[i][j]
        }
        // 左边和上边单元格的最小路径代价
        up := minPathSumDFSMem(grid, mem, i-1, j)
        left := minPathSumDFSMem(grid, mem, i, j-1)
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]
        return mem[i][j]
    }
    ```

=== "Swift"

    ```swift title="min_path_sum.swift"
    /* 最小路径和：记忆化搜索 */
    func minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -> Int {
        // 若为左上角单元格，则终止搜索
        if i == 0, j == 0 {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return .max
        }
        // 若已有记录，则直接返回
        if mem[i][j] != -1 {
            return mem[i][j]
        }
        // 左边和上边单元格的最小路径代价
        let up = minPathSumDFSMem(grid: grid, mem: &mem, i: i - 1, j: j)
        let left = minPathSumDFSMem(grid: grid, mem: &mem, i: i, j: j - 1)
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = min(left, up) + grid[i][j]
        return mem[i][j]
    }
    ```

=== "JS"

    ```javascript title="min_path_sum.js"
    /* 最小路径和：记忆化搜索 */
    function minPathSumDFSMem(grid, mem, i, j) {
        // 若为左上角单元格，则终止搜索
        if (i === 0 && j === 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Infinity;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] !== -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        const up = minPathSumDFSMem(grid, mem, i - 1, j);
        const left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = Math.min(left, up) + grid[i][j];
        return mem[i][j];
    }
    ```

=== "TS"

    ```typescript title="min_path_sum.ts"
    /* 最小路径和：记忆化搜索 */
    function minPathSumDFSMem(
        grid: Array<Array<number>>,
        mem: Array<Array<number>>,
        i: number,
        j: number
    ): number {
        // 若为左上角单元格，则终止搜索
        if (i === 0 && j === 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Infinity;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        const up = minPathSumDFSMem(grid, mem, i - 1, j);
        const left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = Math.min(left, up) + grid[i][j];
        return mem[i][j];
    }
    ```

=== "Dart"

    ```dart title="min_path_sum.dart"
    /* 最小路径和：记忆化搜索 */
    int minPathSumDFSMem(List<List<int>> grid, List<List<int>> mem, int i, int j) {
      // 若为左上角单元格，则终止搜索
      if (i == 0 && j == 0) {
        return grid[0][0];
      }
      // 若行列索引越界，则返回 +∞ 代价
      if (i < 0 || j < 0) {
        // 在 Dart 中，int 类型是固定范围的整数，不存在表示“无穷大”的值
        return BigInt.from(2).pow(31).toInt();
      }
      // 若已有记录，则直接返回
      if (mem[i][j] != -1) {
        return mem[i][j];
      }
      // 左边和上边单元格的最小路径代价
      int up = minPathSumDFSMem(grid, mem, i - 1, j);
      int left = minPathSumDFSMem(grid, mem, i, j - 1);
      // 记录并返回左上角到 (i, j) 的最小路径代价
      mem[i][j] = min(left, up) + grid[i][j];
      return mem[i][j];
    }
    ```

=== "Rust"

    ```rust title="min_path_sum.rs"
    /* 最小路径和：记忆化搜索 */
    fn min_path_sum_dfs_mem(grid: &Vec<Vec<i32>>, mem: &mut Vec<Vec<i32>>, i: i32, j: i32) -> i32 {
        // 若为左上角单元格，则终止搜索
        if i == 0 && j == 0 {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if i < 0 || j < 0 {
            return i32::MAX;
        }
        // 若已有记录，则直接返回
        if mem[i as usize][j as usize] != -1 {
            return mem[i as usize][j as usize];
        }
        // 左边和上边单元格的最小路径代价
        let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);
        let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];
        mem[i as usize][j as usize]
    }
    ```

=== "C"

    ```c title="min_path_sum.c"
    /* 最小路径和：记忆化搜索 */
    int minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return INT_MAX;
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j];
        }
        // 左边和上边单元格的最小路径代价
        int up = minPathSumDFSMem(grid, mem, i - 1, j);
        int left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;
        return mem[i][j];
    }
    ```

=== "Kotlin"

    ```kotlin title="min_path_sum.kt"
    /* 最小路径和：记忆化搜索 */
    fun minPathSumDFSMem(
        grid: Array<IntArray>,
        mem: Array<IntArray>,
        i: Int,
        j: Int
    ): Int {
        // 若为左上角单元格，则终止搜索
        if (i == 0 && j == 0) {
            return grid[0][0]
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 || j < 0) {
            return Int.MAX_VALUE
        }
        // 若已有记录，则直接返回
        if (mem[i][j] != -1) {
            return mem[i][j]
        }
        // 左边和上边单元格的最小路径代价
        val up = minPathSumDFSMem(grid, mem, i - 1, j)
        val left = minPathSumDFSMem(grid, mem, i, j - 1)
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[i][j] = min(left, up) + grid[i][j]
        return mem[i][j]
    }
    ```

=== "Ruby"

    ```ruby title="min_path_sum.rb"
    ### 最小路径和：记忆化搜索 ###
    def min_path_sum_dfs_mem(grid, mem, i, j)
      # 若为左上角单元格，则终止搜索
      return grid[0][0] if i == 0 && j == 0
      # 若行列索引越界，则返回 +∞ 代价
      return Float::INFINITY if i < 0 || j < 0
      # 若已有记录，则直接返回
      return mem[i][j] if mem[i][j] != -1
      # 左边和上边单元格的最小路径代价
      up = min_path_sum_dfs_mem(grid, mem, i - 1, j)
      left = min_path_sum_dfs_mem(grid, mem, i, j - 1)
      # 记录并返回左上角到 (i, j) 的最小路径代价
      mem[i][j] = [left, up].min + grid[i][j]
    end
    ```

=== "Zig"

    ```zig title="min_path_sum.zig"
    // 最小路径和：记忆化搜索
    fn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {
        // 若为左上角单元格，则终止搜索
        if (i == 0 and j == 0) {
            return grid[0][0];
        }
        // 若行列索引越界，则返回 +∞ 代价
        if (i < 0 or j < 0) {
            return std.math.maxInt(i32);
        }
        // 若已有记录，则直接返回
        if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {
            return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
        }
        // 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
        var up = minPathSumDFSMem(grid, mem, i - 1, j);
        var left = minPathSumDFSMem(grid, mem, i, j - 1);
        // 返回从左上角到 (i, j) 的最小路径代价
        // 记录并返回左上角到 (i, j) 的最小路径代价
        mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];
    }
    ```

??? pythontutor "可视化运行"

    <div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20%E5%B7%A6%E8%BE%B9%E5%92%8C%E4%B8%8A%E8%BE%B9%E5%8D%95%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
    <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%22%22%22%0A%20%20%20%20%23%20%E8%8B%A5%E4%B8%BA%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%8D%95%E5%85%83%E6%A0%BC%EF%BC%8C%E5%88%99%E7%BB%88%E6%AD%A2%E6%90%9C%E7%B4%A2%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E8%8B%A5%E8%A1%8C%E5%88%97%E7%B4%A2%E5%BC%95%E8%B6%8A%E7%95%8C%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20%2B%E2%88%9E%20%E4%BB%A3%E4%BB%B7%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20%E8%8B%A5%E5%B7%B2%E6%9C%89%E8%AE%B0%E5%BD%95%EF%BC%8C%E5%88%99%E7%9B%B4%E6%8E%A5%E8%BF%94%E5%9B%9E%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20!%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20%E5%B7%A6%E8%BE%B9%E5%92%8C%E4%B8%8A%E8%BE%B9%E5%8D%95%E5%85%83%E6%A0%BC%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E5%B9%B6%E8%BF%94%E5%9B%9E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%20%28i,%20j%29%20%E7%9A%84%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E4%BB%A3%E4%BB%B7%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20%E8%AE%B0%E5%BF%86%E5%8C%96%E6%90%9C%E7%B4%A2%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>

如图 14-15 所示，在引入记忆化后，所有子问题的解只需计算一次，因此时间复杂度取决于状态总数，即网格尺寸 $O(nm)$ 。

![记忆化搜索递归树](dp_solution_pipeline.assets/min_path_sum_dfs_mem.png){ class="animation-figure" }

<p align="center"> 图 14-15 &nbsp; 记忆化搜索递归树 </p>

### 3. &nbsp; 方法三：动态规划

基于迭代实现动态规划解法，代码如下所示：

=== "Python"

    ```python title="min_path_sum.py"
    def min_path_sum_dp(grid: list[list[int]]) -> int:
        """最小路径和：动态规划"""
        n, m = len(grid), len(grid[0])
        # 初始化 dp 表
        dp = [[0] * m for _ in range(n)]
        dp[0][0] = grid[0][0]
        # 状态转移：首行
        for j in range(1, m):
            dp[0][j] = dp[0][j - 1] + grid[0][j]
        # 状态转移：首列
        for i in range(1, n):
            dp[i][0] = dp[i - 1][0] + grid[i][0]
        # 状态转移：其余行和列
        for i in range(1, n):
            for j in range(1, m):
                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
        return dp[n - 1][m - 1]
    ```

=== "C++"

    ```cpp title="min_path_sum.cpp"
    /* 最小路径和：动态规划 */
    int minPathSumDP(vector<vector<int>> &grid) {
        int n = grid.size(), m = grid[0].size();
        // 初始化 dp 表
        vector<vector<int>> dp(n, vector<int>(m));
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (int j = 1; j < m; j++) {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (int i = 1; i < n; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        return dp[n - 1][m - 1];
    }
    ```

=== "Java"

    ```java title="min_path_sum.java"
    /* 最小路径和：动态规划 */
    int minPathSumDP(int[][] grid) {
        int n = grid.length, m = grid[0].length;
        // 初始化 dp 表
        int[][] dp = new int[n][m];
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (int j = 1; j < m; j++) {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (int i = 1; i < n; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        return dp[n - 1][m - 1];
    }
    ```

=== "C#"

    ```csharp title="min_path_sum.cs"
    /* 最小路径和：动态规划 */
    int MinPathSumDP(int[][] grid) {
        int n = grid.Length, m = grid[0].Length;
        // 初始化 dp 表
        int[,] dp = new int[n, m];
        dp[0, 0] = grid[0][0];
        // 状态转移：首行
        for (int j = 1; j < m; j++) {
            dp[0, j] = dp[0, j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (int i = 1; i < n; i++) {
            dp[i, 0] = dp[i - 1, 0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];
            }
        }
        return dp[n - 1, m - 1];
    }
    ```

=== "Go"

    ```go title="min_path_sum.go"
    /* 最小路径和：动态规划 */
    func minPathSumDP(grid [][]int) int {
        n, m := len(grid), len(grid[0])
        // 初始化 dp 表
        dp := make([][]int, n)
        for i := 0; i < n; i++ {
            dp[i] = make([]int, m)
        }
        dp[0][0] = grid[0][0]
        // 状态转移：首行
        for j := 1; j < m; j++ {
            dp[0][j] = dp[0][j-1] + grid[0][j]
        }
        // 状态转移：首列
        for i := 1; i < n; i++ {
            dp[i][0] = dp[i-1][0] + grid[i][0]
        }
        // 状态转移：其余行和列
        for i := 1; i < n; i++ {
            for j := 1; j < m; j++ {
                dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]
            }
        }
        return dp[n-1][m-1]
    }
    ```

=== "Swift"

    ```swift title="min_path_sum.swift"
    /* 最小路径和：动态规划 */
    func minPathSumDP(grid: [[Int]]) -> Int {
        let n = grid.count
        let m = grid[0].count
        // 初始化 dp 表
        var dp = Array(repeating: Array(repeating: 0, count: m), count: n)
        dp[0][0] = grid[0][0]
        // 状态转移：首行
        for j in 1 ..< m {
            dp[0][j] = dp[0][j - 1] + grid[0][j]
        }
        // 状态转移：首列
        for i in 1 ..< n {
            dp[i][0] = dp[i - 1][0] + grid[i][0]
        }
        // 状态转移：其余行和列
        for i in 1 ..< n {
            for j in 1 ..< m {
                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
            }
        }
        return dp[n - 1][m - 1]
    }
    ```

=== "JS"

    ```javascript title="min_path_sum.js"
    /* 最小路径和：动态规划 */
    function minPathSumDP(grid) {
        const n = grid.length,
            m = grid[0].length;
        // 初始化 dp 表
        const dp = Array.from({ length: n }, () =>
            Array.from({ length: m }, () => 0)
        );
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (let j = 1; j < m; j++) {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (let i = 1; i < n; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (let i = 1; i < n; i++) {
            for (let j = 1; j < m; j++) {
                dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        return dp[n - 1][m - 1];
    }
    ```

=== "TS"

    ```typescript title="min_path_sum.ts"
    /* 最小路径和：动态规划 */
    function minPathSumDP(grid: Array<Array<number>>): number {
        const n = grid.length,
            m = grid[0].length;
        // 初始化 dp 表
        const dp = Array.from({ length: n }, () =>
            Array.from({ length: m }, () => 0)
        );
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (let j = 1; j < m; j++) {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (let i = 1; i < n; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (let i = 1; i < n; i++) {
            for (let j: number = 1; j < m; j++) {
                dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        return dp[n - 1][m - 1];
    }
    ```

=== "Dart"

    ```dart title="min_path_sum.dart"
    /* 最小路径和：动态规划 */
    int minPathSumDP(List<List<int>> grid) {
      int n = grid.length, m = grid[0].length;
      // 初始化 dp 表
      List<List<int>> dp = List.generate(n, (i) => List.filled(m, 0));
      dp[0][0] = grid[0][0];
      // 状态转移：首行
      for (int j = 1; j < m; j++) {
        dp[0][j] = dp[0][j - 1] + grid[0][j];
      }
      // 状态转移：首列
      for (int i = 1; i < n; i++) {
        dp[i][0] = dp[i - 1][0] + grid[i][0];
      }
      // 状态转移：其余行和列
      for (int i = 1; i < n; i++) {
        for (int j = 1; j < m; j++) {
          dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
        }
      }
      return dp[n - 1][m - 1];
    }
    ```

=== "Rust"

    ```rust title="min_path_sum.rs"
    /* 最小路径和：动态规划 */
    fn min_path_sum_dp(grid: &Vec<Vec<i32>>) -> i32 {
        let (n, m) = (grid.len(), grid[0].len());
        // 初始化 dp 表
        let mut dp = vec![vec![0; m]; n];
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for j in 1..m {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for i in 1..n {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for i in 1..n {
            for j in 1..m {
                dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        dp[n - 1][m - 1]
    }
    ```

=== "C"

    ```c title="min_path_sum.c"
    /* 最小路径和：动态规划 */
    int minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {
        // 初始化 dp 表
        int **dp = malloc(n * sizeof(int *));
        for (int i = 0; i < n; i++) {
            dp[i] = calloc(m, sizeof(int));
        }
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (int j = 1; j < m; j++) {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (int i = 1; i < n; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        int res = dp[n - 1][m - 1];
        // 释放内存
        for (int i = 0; i < n; i++) {
            free(dp[i]);
        }
        return res;
    }
    ```

=== "Kotlin"

    ```kotlin title="min_path_sum.kt"
    /* 最小路径和：动态规划 */
    fun minPathSumDP(grid: Array<IntArray>): Int {
        val n = grid.size
        val m = grid[0].size
        // 初始化 dp 表
        val dp = Array(n) { IntArray(m) }
        dp[0][0] = grid[0][0]
        // 状态转移：首行
        for (j in 1..<m) {
            dp[0][j] = dp[0][j - 1] + grid[0][j]
        }
        // 状态转移：首列
        for (i in 1..<n) {
            dp[i][0] = dp[i - 1][0] + grid[i][0]
        }
        // 状态转移：其余行和列
        for (i in 1..<n) {
            for (j in 1..<m) {
                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]
            }
        }
        return dp[n - 1][m - 1]
    }
    ```

=== "Ruby"

    ```ruby title="min_path_sum.rb"
    ### 最小路径和：动态规划 ###
    def min_path_sum_dp(grid)
      n, m = grid.length, grid.first.length
      # 初始化 dp 表
      dp = Array.new(n) { Array.new(m, 0) }
      dp[0][0] = grid[0][0]
      # 状态转移：首行
      (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }
      # 状态转移：首列
      (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }
      # 状态转移：其余行和列
      for i in 1...n
        for j in 1...m
          dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]
        end
      end
      dp[n -1][m -1]
    end
    ```

=== "Zig"

    ```zig title="min_path_sum.zig"
    // 最小路径和：动态规划
    fn minPathSumDP(comptime grid: anytype) i32 {
        comptime var n = grid.len;
        comptime var m = grid[0].len;
        // 初始化 dp 表
        var dp = [_][m]i32{[_]i32{0} ** m} ** n;
        dp[0][0] = grid[0][0];
        // 状态转移：首行
        for (1..m) |j| {
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        // 状态转移：首列
        for (1..n) |i| {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        // 状态转移：其余行和列
        for (1..n) |i| {
            for (1..m) |j| {
                dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
            }
        }
        return dp[n - 1][m - 1];
    }
    ```

??? pythontutor "可视化运行"

    <div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
    <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20*%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%E5%92%8C%E5%88%97%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>

图 14-16 展示了最小路径和的状态转移过程，其遍历了整个网格，**因此时间复杂度为 $O(nm)$** 。

数组 `dp` 大小为 $n \times m$ ，**因此空间复杂度为 $O(nm)$** 。

=== "<1>"
    ![最小路径和的动态规划过程](dp_solution_pipeline.assets/min_path_sum_dp_step1.png){ class="animation-figure" }

=== "<2>"
    ![min_path_sum_dp_step2](dp_solution_pipeline.assets/min_path_sum_dp_step2.png){ class="animation-figure" }

=== "<3>"
    ![min_path_sum_dp_step3](dp_solution_pipeline.assets/min_path_sum_dp_step3.png){ class="animation-figure" }

=== "<4>"
    ![min_path_sum_dp_step4](dp_solution_pipeline.assets/min_path_sum_dp_step4.png){ class="animation-figure" }

=== "<5>"
    ![min_path_sum_dp_step5](dp_solution_pipeline.assets/min_path_sum_dp_step5.png){ class="animation-figure" }

=== "<6>"
    ![min_path_sum_dp_step6](dp_solution_pipeline.assets/min_path_sum_dp_step6.png){ class="animation-figure" }

=== "<7>"
    ![min_path_sum_dp_step7](dp_solution_pipeline.assets/min_path_sum_dp_step7.png){ class="animation-figure" }

=== "<8>"
    ![min_path_sum_dp_step8](dp_solution_pipeline.assets/min_path_sum_dp_step8.png){ class="animation-figure" }

=== "<9>"
    ![min_path_sum_dp_step9](dp_solution_pipeline.assets/min_path_sum_dp_step9.png){ class="animation-figure" }

=== "<10>"
    ![min_path_sum_dp_step10](dp_solution_pipeline.assets/min_path_sum_dp_step10.png){ class="animation-figure" }

=== "<11>"
    ![min_path_sum_dp_step11](dp_solution_pipeline.assets/min_path_sum_dp_step11.png){ class="animation-figure" }

=== "<12>"
    ![min_path_sum_dp_step12](dp_solution_pipeline.assets/min_path_sum_dp_step12.png){ class="animation-figure" }

<p align="center"> 图 14-16 &nbsp; 最小路径和的动态规划过程 </p>

### 4. &nbsp; 空间优化

由于每个格子只与其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 $dp$ 表。

请注意，因为数组 `dp` 只能表示一行的状态，所以我们无法提前初始化首列状态，而是在遍历每行时更新它：

=== "Python"

    ```python title="min_path_sum.py"
    def min_path_sum_dp_comp(grid: list[list[int]]) -> int:
        """最小路径和：空间优化后的动态规划"""
        n, m = len(grid), len(grid[0])
        # 初始化 dp 表
        dp = [0] * m
        # 状态转移：首行
        dp[0] = grid[0][0]
        for j in range(1, m):
            dp[j] = dp[j - 1] + grid[0][j]
        # 状态转移：其余行
        for i in range(1, n):
            # 状态转移：首列
            dp[0] = dp[0] + grid[i][0]
            # 状态转移：其余列
            for j in range(1, m):
                dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
        return dp[m - 1]
    ```

=== "C++"

    ```cpp title="min_path_sum.cpp"
    /* 最小路径和：空间优化后的动态规划 */
    int minPathSumDPComp(vector<vector<int>> &grid) {
        int n = grid.size(), m = grid[0].size();
        // 初始化 dp 表
        vector<int> dp(m);
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (int j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (int i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (int j = 1; j < m; j++) {
                dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

=== "Java"

    ```java title="min_path_sum.java"
    /* 最小路径和：空间优化后的动态规划 */
    int minPathSumDPComp(int[][] grid) {
        int n = grid.length, m = grid[0].length;
        // 初始化 dp 表
        int[] dp = new int[m];
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (int j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (int i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (int j = 1; j < m; j++) {
                dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

=== "C#"

    ```csharp title="min_path_sum.cs"
    /* 最小路径和：空间优化后的动态规划 */
    int MinPathSumDPComp(int[][] grid) {
        int n = grid.Length, m = grid[0].Length;
        // 初始化 dp 表
        int[] dp = new int[m];
        dp[0] = grid[0][0];
        // 状态转移：首行
        for (int j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (int i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (int j = 1; j < m; j++) {
                dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

=== "Go"

    ```go title="min_path_sum.go"
    /* 最小路径和：空间优化后的动态规划 */
    func minPathSumDPComp(grid [][]int) int {
        n, m := len(grid), len(grid[0])
        // 初始化 dp 表
        dp := make([]int, m)
        // 状态转移：首行
        dp[0] = grid[0][0]
        for j := 1; j < m; j++ {
            dp[j] = dp[j-1] + grid[0][j]
        }
        // 状态转移：其余行和列
        for i := 1; i < n; i++ {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0]
            // 状态转移：其余列
            for j := 1; j < m; j++ {
                dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]
            }
        }
        return dp[m-1]
    }
    ```

=== "Swift"

    ```swift title="min_path_sum.swift"
    /* 最小路径和：空间优化后的动态规划 */
    func minPathSumDPComp(grid: [[Int]]) -> Int {
        let n = grid.count
        let m = grid[0].count
        // 初始化 dp 表
        var dp = Array(repeating: 0, count: m)
        // 状态转移：首行
        dp[0] = grid[0][0]
        for j in 1 ..< m {
            dp[j] = dp[j - 1] + grid[0][j]
        }
        // 状态转移：其余行
        for i in 1 ..< n {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0]
            // 状态转移：其余列
            for j in 1 ..< m {
                dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
            }
        }
        return dp[m - 1]
    }
    ```

=== "JS"

    ```javascript title="min_path_sum.js"
    /* 最小路径和：空间优化后的动态规划 */
    function minPathSumDPComp(grid) {
        const n = grid.length,
            m = grid[0].length;
        // 初始化 dp 表
        const dp = new Array(m);
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (let j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (let i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (let j = 1; j < m; j++) {
                dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

=== "TS"

    ```typescript title="min_path_sum.ts"
    /* 最小路径和：空间优化后的动态规划 */
    function minPathSumDPComp(grid: Array<Array<number>>): number {
        const n = grid.length,
            m = grid[0].length;
        // 初始化 dp 表
        const dp = new Array(m);
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (let j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (let i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (let j = 1; j < m; j++) {
                dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

=== "Dart"

    ```dart title="min_path_sum.dart"
    /* 最小路径和：空间优化后的动态规划 */
    int minPathSumDPComp(List<List<int>> grid) {
      int n = grid.length, m = grid[0].length;
      // 初始化 dp 表
      List<int> dp = List.filled(m, 0);
      dp[0] = grid[0][0];
      for (int j = 1; j < m; j++) {
        dp[j] = dp[j - 1] + grid[0][j];
      }
      // 状态转移：其余行
      for (int i = 1; i < n; i++) {
        // 状态转移：首列
        dp[0] = dp[0] + grid[i][0];
        // 状态转移：其余列
        for (int j = 1; j < m; j++) {
          dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
        }
      }
      return dp[m - 1];
    }
    ```

=== "Rust"

    ```rust title="min_path_sum.rs"
    /* 最小路径和：空间优化后的动态规划 */
    fn min_path_sum_dp_comp(grid: &Vec<Vec<i32>>) -> i32 {
        let (n, m) = (grid.len(), grid[0].len());
        // 初始化 dp 表
        let mut dp = vec![0; m];
        // 状态转移：首行
        dp[0] = grid[0][0];
        for j in 1..m {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for i in 1..n {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for j in 1..m {
                dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        dp[m - 1]
    }
    ```

=== "C"

    ```c title="min_path_sum.c"
    /* 最小路径和：空间优化后的动态规划 */
    int minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {
        // 初始化 dp 表
        int *dp = calloc(m, sizeof(int));
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (int j = 1; j < m; j++) {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (int i = 1; i < n; i++) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            // 状态转移：其余列
            for (int j = 1; j < m; j++) {
                dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        int res = dp[m - 1];
        // 释放内存
        free(dp);
        return res;
    }
    ```

=== "Kotlin"

    ```kotlin title="min_path_sum.kt"
    /* 最小路径和：空间优化后的动态规划 */
    fun minPathSumDPComp(grid: Array<IntArray>): Int {
        val n = grid.size
        val m = grid[0].size
        // 初始化 dp 表
        val dp = IntArray(m)
        // 状态转移：首行
        dp[0] = grid[0][0]
        for (j in 1..<m) {
            dp[j] = dp[j - 1] + grid[0][j]
        }
        // 状态转移：其余行
        for (i in 1..<n) {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0]
            // 状态转移：其余列
            for (j in 1..<m) {
                dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]
            }
        }
        return dp[m - 1]
    }
    ```

=== "Ruby"

    ```ruby title="min_path_sum.rb"
    ### 最小路径和：空间优化后的动态规划 ###
    def min_path_sum_dp_comp(grid)
      n, m = grid.length, grid.first.length
      # 初始化 dp 表
      dp = Array.new(m, 0)
      # 状态转移：首行
      dp[0] = grid[0][0]
      (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }
      # 状态转移：其余行
      for i in 1...n
        # 状态转移：首列
        dp[0] = dp[0] + grid[i][0]
        # 状态转移：其余列
        (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }
      end
      dp[m - 1]
    end
    ```

=== "Zig"

    ```zig title="min_path_sum.zig"
    // 最小路径和：空间优化后的动态规划
    fn minPathSumDPComp(comptime grid: anytype) i32 {
        comptime var n = grid.len;
        comptime var m = grid[0].len;
        // 初始化 dp 表
        var dp = [_]i32{0} ** m;
        // 状态转移：首行
        dp[0] = grid[0][0];
        for (1..m) |j| {
            dp[j] = dp[j - 1] + grid[0][j];
        }
        // 状态转移：其余行
        for (1..n) |i| {
            // 状态转移：首列
            dp[0] = dp[0] + grid[i][0];
            for (1..m) |j| {
                dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];
            }
        }
        return dp[m - 1];
    }
    ```

??? pythontutor "可视化运行"

    <div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp_comp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E7%A9%BA%E9%97%B4%E4%BC%98%E5%8C%96%E5%90%8E%E7%9A%84%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B0%5D%20*%20m%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E5%88%97%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E7%A9%BA%E9%97%B4%E4%BC%98%E5%8C%96%E5%90%8E%E7%9A%84%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
    <div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp_comp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22%E6%9C%80%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%EF%BC%9A%E7%A9%BA%E9%97%B4%E4%BC%98%E5%8C%96%E5%90%8E%E7%9A%84%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%20dp%20%E8%A1%A8%0A%20%20%20%20dp%20%3D%20%5B0%5D%20*%20m%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E8%A1%8C%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E8%A1%8C%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E9%A6%96%E5%88%97%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%EF%BC%9A%E5%85%B6%E4%BD%99%E5%88%97%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20%E7%A9%BA%E9%97%B4%E4%BC%98%E5%8C%96%E5%90%8E%E7%9A%84%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22%E4%BB%8E%E5%B7%A6%E4%B8%8A%E8%A7%92%E5%88%B0%E5%8F%B3%E4%B8%8B%E8%A7%92%E7%9A%84%E5%81%9A%E5%B0%8F%E8%B7%AF%E5%BE%84%E5%92%8C%E4%B8%BA%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>
