唯一路径问题II

Unique Paths II

Follow up for "Unique Paths":

Now consider if some obstacles are added to the grids. How many unique paths would there be?

An obstacle and empty space is marked as 1 and 0 respectively in the grid.

For example,

There is one obstacle in the middle of a 3x3 grid as illustrated below.

[

  [0,0,0],

  [0,1,0],

  [0,0,0]

]

The total number of unique paths is 2.

Note: m and n will be at most 100.

--

第一种方法(uniquePathsWithObstacles)为递归实现

　　会超时，最后一个case有16亿+条路径...递归方法会走每条路径，所以一定会超时。

第二种方法(uniquePathsWithObstaclesDP)为动态规划

　　不难发现max_ways[x,y]=max_ways[x-1,y]+max_ways[x,y-1], 即满足最优子结构性质。

　　并且max_ways[x-1,y]和max_ways[x,y-1]依赖于max_ways[m,n](0<m<x, 0<n<y)，即满足子问题重叠性质，因此使用动态规划可以获得更好的效率。

 
 
'''

Created on Nov 25, 2014

@author: ScottGu<gu.kai.66@gmail.com, 150316990@qq.com>

'''

class Solution:

    def __init__(self):

        self.ways=0

        self.max_x=0

        self.max_y=0

    # @param obstacleGrid, a list of lists of integers

    # @return an integer

    def uniquePathsWithObstacles(self, obstacleGrid):

        if(obstacleGrid==None):return 0

        if(len(obstacleGrid)==0):return 0

        if(obstacleGrid[0][0] ==1): return 0

        self.__init__()

        self.max_x=len(obstacleGrid[0])-1

        self.max_y=len(obstacleGrid)-1

        self.find_ways(0,0, obstacleGrid)

        return self.ways

    def find_ways(self, x, y, grid):

        if(x==self.max_x and y==self.max_y):

            self.ways=self.ways+1

        if(x<self.max_x and grid[y][x+1]!=1):

            self.find_ways(x+1, y, grid)

        if(y<self.max_y and grid[y+1][x]!=1):

            self.find_ways(x, y+1, grid)

    # @obstacleGrid is a grid of m*n cells

    def uniquePathsWithObstaclesDP(self, obstacleGrid):

        m = len(obstacleGrid)

        if(m ==0): return 0

        n = len(obstacleGrid[0])

        if(obstacleGrid[0][0] ==1): return 0

        max_ways={}

        for x in range(n):max_ways[x]=0

        max_ways[0]=1;

        for y in range(m):

            for x in range(n):

                if(obstacleGrid[y][x] ==1):

                    max_ways[x]=0

                else:

                    if(x >0):

                        max_ways[x] = max_ways[x-1]+max_ways[x]

        return max_ways[n-1];  

if __name__ == '__main__':

    sl=Solution()

    grid=[[0,0,0],

          [0,1,0],

          [0,0,0]]

    print sl.uniquePathsWithObstacles(grid)

    grid=[[0,0,0,0,0],

          [0,1,0,0,0],

          [0,1,0,0,0],

          [0,1,0,0,0],

          [0,0,0,0,0]]

    print sl.uniquePathsWithObstacles(grid)

    grid= [

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,1,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,1,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,1,0,0,0,0,0,0,0,0]

           ]

    print sl.uniquePathsWithObstacles(grid)

    grid= [

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0],

               [0,0,0,0,0,0,0,0,0,0] ,

               [0,0,0,0,0,0,0,0,0,0] ,

               [0,0,0,0,0,0,0,0,0,0] ,

               [0,0,0,0,0,0,0,0,0,0]

           ]

    print sl.uniquePathsWithObstacles(grid)

    grid=[

[0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0],

[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],

[0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0],

[1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,1],

[0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0],

[0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0],

[1,0,1,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0],

[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0],

[0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,1,0],

[0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0],

[0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0],

[1,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1],

[0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0],

[0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0],

[0,1,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1],

[1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],

[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0],

[0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,0,0,1,1],

[0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0,1],

[1,1,1,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0],

[0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1],

[0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0]

]

    print sl.uniquePathsWithObstaclesDP(grid)

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