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/*
space complexity: O(n)

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

dp[n]代表ith row的各种最短路径
*/

class Solution {
    public int minPathSum(int[][] grid) {
        int m = grid.length;
        int n = grid[0].length;
        int[] dp = new int[n];
        dp[0] = grid[0][0];
        for(int i = 1; i < n; i++)
            dp[i] = dp[i - 1] + grid[0][i];
        
        for(int i = 1; i < m; i++) {
            for(int j = 0; j < n; j++) {
                if(j == 0) {
                    dp[j] = dp[j] + grid[i][j];
                } else {
                    dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];
                }
            }
        }       
        
        return dp[n - 1];
    }
}

class Solution {
    public int minPathSum(int[][] grid) {
        int m = grid.length;
        int n = grid[0].length;
        int[][] dp = new int[m][n];
        dp[0][0] = grid[0][0];
        
        for(int i = 1; i < m; i++) {
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        for(int i = 1; i < n; i++) {
            dp[0][i] = dp[0][i - 1] + grid[0][i];
        }       
        
        for(int i = 1; i < m; i++) {
            for(int j = 1; j < n; j++) {
                dp[i][j] = Math.min(dp[i - 1][j], dp[i][j - 1]) + grid[i][j];
            }
        }
        
        return dp[m - 1][n - 1];
    }
}