62. Unique Paths

problem description

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

Note: m and n will be at most 100.

Example 1:

Input: m = 3, n = 2
Output: 3
Explanation:
From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
1. Right -> Right -> Down
2. Right -> Down -> Right
3. Down -> Right -> Right

algorithm thought

典型的动态规划问题，达到一个点的所有方案，是其上一个点和左边点的方案之和。因为我到这个点可以有两种情况走过来，一种是从上面往下走，一种是从左边往右走。

code

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

algorithm analysis

二维动态规划，遍历一个二维数组。最后时间复杂度O(n²)