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：这是一道典型的动态规划问题，使用一个二维数组dp记忆到达每一点可行的走法总数。首先将左边界点和上边界点初始化为1，因为机器人起始于（1，1），左边界点和上边界点的走法只有1种。接下来的每一点（x,y），可以由（x-1，y）向下走或是（x,y-1）向右走来到达，因此在（x,y）这一点可到达的方法有dp[x-1][y]+dp[x][y-1]种，到达终点的方法则是dp最后一个点的数据。

class Solution {
public:
    int uniquePaths(int m, int n) {
        if(m == 0 || n == 0){
            return 0;
        }
            
        vector<vector<int>> dp(m+1, vector<int>(n+1,0));
        int i, j;
        //int dp[101][101], i, j;
        for(i = 1; i <= m; i++){
            dp[i][1] = 1;
        }
            
        for(j = 1; j <= n; j++){
            dp[1][j] = 1; 
        }
           
        for(i = 2; i <= m; i++){
            for(j = 2; j <= n; j++){
                dp[i][j] = dp[i-1][j] + dp[i][j-1];
            }
        }
        return dp[m][n];
    }
};