leetcode #64 in cpp

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.

Solution:

We use DP where dp[i][j] = minimum sum along the path from position(i+1,j+1) to the destination. 

Then dp[i][j] = min(dp[i+1][j], dp[i][j+1]) + grid[i][j]. That is, we select the direction which gives us a smaller sum, when we decide to take the next step.  

Code:

class Solution {public:    int minPathSum(vector<vector<int>>& grid) {        int m = grid.size();        int n = grid[0].size();         vector<vector<int>> dp(m, vector<int>(n,0));        dp[m-1][n-1] = grid[m-1][n-1];        for(int i = m-2; i >= 0; i --){//right border            dp[i][n-1] = grid[i][n-1] + dp[i+1][n-1];        }        for(int i = n-2; i >= 0; i--){//bottom border            dp[m-1][i] = grid[m-1][i]+dp[m-1][i+1];         }        for(int i = m-2; i >= 0; i --){            for(int j = n-2; j>=0; j --){                dp[i][j] = min(dp[i+1][j], dp[i][j+1])+grid[i][j];//compare results from turning right and turning down            }        }        return dp[0][0];    }};

One optimization is to update the sum in place in grid instead of using the dp arrays. This saves us some memory.