python - 导航网格

Question

我在 Project Euler 中偶然发现了一个问题，https://projecteuler.net/problem=15 .我通过组合学解决了这个问题，但我想知道是否有动态编程解决方案来解决这个问题或这些类型的问题。并说网格的一些正方形被取消了 - 可以导航吗？我正在使用 Python。我该怎么做？任何提示表示赞赏。提前致谢。

Answer 1

你可以做一个简单的回溯并探索这样一个隐式图:(评论解释了大部分)

def explore(r, c, n, memo):
    """
        explore right and down from position (r,c)
        report a rout once position (n,n) is reached
        memo is a matrix which saves how many routes exists from each position to (n,n)
    """

    if r == n and c == n:
        # one path has been found
        return 1

    elif r > n or c > n:
        # crossing the border, go back
        return 0

    if memo[r][c] is not None:
        return memo[r][c]

    a= explore(r+1, c, n, memo)    #move down
    b= explore(r, c+1, n, memo)  #move right

    # return total paths found from this (r,c) position
    memo[r][c]= a + b

    return a+b

if __name__ == '__main__':
    n= 20
    memo = [[None] * (n+1) for _ in range(n+1)]

    paths = explore(0, 0, n, memo)
    print(paths)