Minimum Path Sum

一个矩阵，从左下角开始走，走到右上角经过节点和最小的路径

例如:

1, 2, 3, 1

1, 2, 1, 4

1, 1, 1, 4

1, 2, 3, 4

结果是9，up -> right -> right -> up -> right

 

递归法

int grid[4][4]这样的二维数组代表每个节点的值，从grid[0][0]走到grid[3][3]，如果走到grid[3][3]可以看做是走到grid[2][3]的最小路径+1，或者是走到grid[3][2]的最小路径+1，分别对应最后一步是向右走还是向上走。

public int minPathSum(int[][] grid, int p, int q) {
    if (p == 0 && q == 0) {
        return grid[p][q];
    }

    int up = Integer.MAX_VALUE,right = Integer.MAX_VALUE;
    if (p-1 >= 0) {
        up = minPathSum(grid, p - 1, q) + grid[p][q];
    }
    if (q-1 >= 0) {
        right = minPathSum(grid, p, q - 1) + grid[p][q];
    }
    return Math.min(up, right);
}

@Test
public void sumTest() {
    int[][] grid = new int[4][];
    // result 9 -> up -> right -> right -> up -> right
    grid[3] = new int[]{1, 2, 3, 1};
    grid[2] = new int[]{1, 2, 1, 4};
    grid[1] = new int[]{1, 1, 1, 4};
    grid[0] = new int[]{1, 2, 3, 4};
    System.out.println(minPathSum(grid, 3, 3));
}

需要注意两点：

1. 递归终止条件是走到grid[0][0]

2. 如果已经走到边界，就不递归了

以上的代码会重复求解子问题，可以通过用一个数组保存子问题的接，也就是minPathSum(p, q)的解

 

动态规划

public int minPathSum(int[][] grid, int row, int col) {
    int[][] res = new int[row][row];
    for (int i=0; i<row; i++) {
        for (int j=0; j<col; j++) {
            if (i==0 && j==0) {
                res[i][j] = grid[i][j];
            } else if (i-1 < 0) {
                res[i][j] = grid[i][j] + res[i][j - 1];
            } else if (j-1 < 0) {
                res[i][j] = grid[i][j] + res[i - 1][j];
            } else {
                res[i][j] = grid[i][j] + Math.min(res[i - 1][j], res[i][j - 1]);
            }
        }
    }
    return res[row - 1][row - 1];
}

@Test
public void sumTest() {
    int[][] grid = new int[4][];
    // result 9 -> up -> right -> right -> up -> right
    grid[3] = new int[]{1, 2, 3, 1};
    grid[2] = new int[]{1, 2, 1, 4};
    grid[1] = new int[]{1, 1, 1, 4};
    grid[0] = new int[]{1, 2, 3, 4};
    System.out.println(minPathSum(grid, 4, 4));
}