[Leetcode] 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.

见代码。

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

第二遍做的时候果然有新收获，感觉Leetcode刷再多遍都不嫌多，每一次都会有新的收获。下面为第二次刷的时候的代码，我们完全用不着m*n的空间，其实只要m跟n相对较短的那个长度的空间就好了。下面是用了O（n）空间的代码。

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