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.

 Notice

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

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Example
Tags 
Dynamic Programming
Related Problems 
Easy Triangle 
 
26 %
Medium Binary Tree Maximum Path Sum

矩陣的動歸, 要初始化邊界

public int minPathSum(int[][] grid) {
        // write your code here
        if (grid == null || grid.length == 0) {
            return -1;
        }
        
        int n = grid.length;
        int m = grid[0].length;
        
        //state
        int[][] sum = new int[n][m];
        
        //initialize
        sum[0][0] = grid[0][0];
        for (int i = 1; i < m; i++) {
            sum[0][i] = sum[0][i - 1] + grid[0][i];
        }
        for (int i = 1; i < n; i++) {
            sum[i][0] = sum[i - 1][0] + grid[i][0];
        }
        
        //top down
        for (int i = 1; i < n; i++) {
            for (int j = 1; j < m; j++) {
                sum[i][j] = Math.min(sum[i - 1][j], sum[i][j - 1]) + grid[i][j];
            }
        }
        
        //ans
        return sum[n - 1][m - 1];
    }

Minimum Path Sum