[LeetCode] 62. Unique Paths (DP问题)

题：https://leetcode.com/problems/minimum-path-sum/description/

题目

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.

Example 1:

[[1,3,1],
 [1,5,1],
 [4,2,1]]

思路

动态规划问题

This is a typical DP problem. Suppose the minimum path sum of arriving at point (i, j) is S[i][j], then the state equation is S[i][j] = min(S[i - 1][j], S[i][j - 1]) + grid[i][j].

Well, some boundary conditions need to be handled. The boundary conditions happen on the topmost row (S[i - 1][j] does not exist) and the leftmost column (S[i][j - 1] does not exist). Suppose grid is like [1, 1, 1, 1], then the minimum sum to arrive at each point is simply an accumulation of previous points and the result is [1, 2, 3, 4].

Code

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