EularProject 15: 方格迷宫的路径数

Lattice paths

Problem 15

Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.



How many such routes are there through a 20×20 grid?

Answer: 137846528820 Completed on Tue, 27 Jan 2015, 03:57

dict={}
def func(i,j):
    rest=dict.get(str(i)+'_'+str(j))
    if rest!=None:
        return rest
    else:
        if i==0 or j==0:
            return 1
        temp=func(i-1,j)+func(i,j-1)
        dict[str(i)+'_'+str(j)]=temp
        return temp

result=func(20,20)
print(result)

上面的代码用到了带记忆的自上向下的动态规划算法。鉴于题目的特殊性，可以发现用组合数公式可以直接求解。

import operator
from functools import reduce

def c(n,k):
    return  reduce(operator.mul,range(n - k + 1, n + 1)) //reduce(operator.mul, range(1, k +1))

def fac(n):
    return reduce(operator.mul, range(1,n+1))

a,b=20,20
print(c(a+b,a))

time : <1s

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