【LeetCOde】62. Unique Paths

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?

Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

【思路】

设s[i][j] 为从起点到（i,j）位置处的路径数。

通过分析得到：第一行，第一列都为1

到其他位置处（i,j）：到达位置（i,j）只能从上面或者左面过来，因此决定到位置（i,j）的路径数由到达上面位置（i-1,j）的路径数和到达左面位置（i,j-1）的路径所决定的。

状态转移方程：

s[i][j] = s[i-1][j] + s[i][j-1]

时间复杂度：O(n^2)  空间复杂度：O(n^2)

class Solution {
public:
    int uniquePaths(int m, int n) {
        int s[m][n] = {0};
        for(int i=0; i < m; i++)
        {
            for(int j =0; j < n; j++)
            {
                if(i==0 || j==0)
                    s[i][j] =1;
                else
                s[i][j]=s[i][j-1]+s[i-1][j];
            }
        }
        return s[m-1][n-1];
    }
};