题目描述

Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Note: You can only move either down or right at any point in time.

样例

Input:
[
  [1,3,1],
  [1,5,1],
  [4,2,1]
]
Output: 7
Explanation: Because the path 1→3→1→1→1 minimizes the sum.

算法1

(矩阵型DP) $O(m*n)$

f[i][j]=min(f[i-1][j],f[i][j-1])+grid[i][j];

初始化：f[0][0]=grid[0][0];
i:1->m f[i][0]=f[i-1][0]+grid[i][0];
j:1->n f[0][j]=f[0][j-1]+grid[0][j-1];

C++ 代码

class Solution {
public:
    int minPathSum(vector<vector<int>>& grid) {
        if(grid.size()==0||grid[0].size()==0)
            return 0;
        int m=grid.size(),n=grid[0].size();
        vector<vector<int>> f(m,vector<int>(n,0));
        f[0][0]=grid[0][0];
        for(int i=1;i<m;i++){
            f[i][0]=f[i-1][0]+grid[i][0];
        }
        for(int j=1;j<n;j++){
            f[0][j]=f[0][j-1]+grid[0][j];
        }
        for(int i=1;i<m;i++){
            for(int j=1;j<n;j++){
                f[i][j]=min(f[i-1][j],f[i][j-1])+grid[i][j];
            }
        }
        return f[m-1][n-1];
    }
};