LeetCode Minimum Path Sum

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.

思路:用动态规划的一种思维，申请一片辅助空间dp[m][n]，dp[i][j]表示以该位置为终点的最短路径

dp[i][j]的值的决策方案：

　　因为题目要求只能向左或者向右，所以dp[i][j]的取值只能来自dp[i][j-1]+arr[i][j]或者dp[i-1][j]+dp[i][j]

参考代码：

public class Solution {
    /**
     * @param grid: a list of lists of integers.
     * @return: An integer, minimizes the sum of all numbers along its path
     */
    public int minPathSum(int[][] grid) {
        if (grid == null)
            return 0;
        // write your code here
        int[] dp = new int[grid[0].length];
        dp[0] = grid[0][0];//进行了空间压缩
        for (int i = 1; i < grid[0].length; i++) {
            dp[i] = dp[i - 1] + grid[0][i];
        }

        for (int i = 1; i < grid.length; i++) {
            for (int j = 0; j < grid[0].length; 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[dp.length - 1];
    }
}

进阶：该题的空间复杂度还可以进行压缩