[leetcode] python 62、63、64 动态规划 Grid有多少种路径

慢慢的动态规划的感觉来了，今天连做三道AC。套路也差不多，关键找到状态和状态转移方法，分享一下。

62. Unique Paths(Medium)

题目描述：A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).
The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).
How many possible unique paths are there?
一个机器人在一个m*n的网格起点，只能向下和向右运动一个点，请问到达右下角有多少种不同的路径。

image.png

思路：
动态规划令 dp[i][j] 为运动到i，j位置有多少种不同的路径，每一个网格的前一步一定是该网格正上或者正左节点，状态转移方程式也由该两网格之和决定：
dp[i][j] = dp[i-1][j] + dp[i][j-1]，代码如下所示：

    #:type m: int
    #:type n: int
    #:rtype: int
    def uniquePaths(self, m, n):
        dp = [[1] * m for _ in range(n)]
        for i in range(1, n):
            for j in range(1, m):
                dp[i][j] = dp[i-1][j] + dp[i][j-1]
        return dp[n-1][m-1]

63. Unique Paths II(Medium)

题目描述：A robot is located at the top-left corner of a m x n grid (marked 'Start' in the diagram below).
The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).
一个机器人在一个m*n的网格起点，只能向下和向右运动一个点。
Now consider if some obstacles are added to the grids. How many unique paths would there be?
现在在网格里面有一些障碍物，请问到达右下角有多少种不同的路径。

思路：该题与上题的区别在与障碍物，加个判断和初始化即可

if obstacleGrid[i][j] == 0:
    dp[i][j] = dp[i-1][j] + dp[i][j-1]

最终代码如下：

    """
    :type obstacleGrid: List[List[int]]
    :rtype: int
    """
    def uniquePathsWithObstacles(self, obstacleGrid):
        n = len(obstacleGrid)
        m = len(obstacleGrid[0])
        dp = [[0] * m for _ in range(n)]
        for i in range(n):
            if obstacleGrid[i][0] == 1:
                break
            dp[i][0] = 1
        for i in range(m):
            if obstacleGrid[0][i] == 1:
                break
            dp[0][i] = 1
        for i in range(1, n):
            for j in range(1, m):
                if obstacleGrid[i][j] == 0:
                    dp[i][j] = dp[i-1][j] + dp[i][j-1]
        return dp[n-1][m-1]

64. Minimum Path Sum(Medium)

问腿描述：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.
同样是上两题的图，给定一个m*n的网格，每个网格里包含一个非负整数，找到一条从左上角到右下的最短路径。

思路：同样DP，每一个网格的路径取最小值，

grid[i][j] += min(grid[i-1][j], grid[i][j-1])

不过这道题不需要初始化直接用给的grid即可。

    """
    :type grid: List[List[int]]
    :rtype: int
    """
    def minPathSum(self, grid):
        m = len(grid)
        n = len(grid[0])
        for i in range(1, n):
            grid[0][i] += grid[0][i-1]
        for i in range(1, m):
            grid[i][0] += grid[i-1][0]
        for i in range(1, m):
            for j in range(1, n):
                grid[i][j] += min(grid[i-1][j], grid[i][j-1])
        return grid[-1][-1]