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)