[leetcode] Unique Paths 解题报告

题目链接：https://leetcode.com/problems/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.

很基本的一道动态规划题目。可以将问题划分为子问题看待，到终点有多少种走法是由到终点左边方格有多少种走法 + 到终点上边有多少种走法，因此很容易得出状态转移方程为：dp[i][j] = dp[i][j-1] + dp[i-1][j]。其中初始化第一列和第一行都为1，因为只有一种走法。时间复杂度为O(M*N), 空间复杂度为O(M*N)。具体代码如下：

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