Unique Paths

Problem

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?

Example

Thoughts

typical DP

Solution

class Solution:
    """
    @param n and m: positive integer(1 <= n , m <= 100)
    @return an integer
    """ 
    def uniquePaths(self, m, n):
        #init states
        f = [[0 for i in range(n)] for j in range(m)]
        for i in range(m):
            f[i][0] = 1
        for j in range(n):
            f[0][j] = 1

        for i in range(1, m):
            for j in range(1, n):
                f[i][j] = f[i-1][j] + f[i][j-1]

        return f[m-1][n-1]