5 lines, Java, Θ(mn) time, Θ(1) space


  • 0
    G
            for(int i = 1 ; i < grid.length ; ++i) grid[i][0] += grid[i-1][0];
            for(int i = 1 ; i < grid[0].length ; ++i) grid[0][i] += grid[0][i-1];
            for(int i = 1 ; i < grid.length ; ++i)
                for(int j = 1 ; j < grid[0].length ; ++j)
                    grid[i][j] += Math.min(grid[i-1][j], grid[i][j-1]);
            return grid[grid.length-1][grid[0].length-1];
    

    At any point, you will take the path that minimizes the total sum at current position.


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