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 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

Example 2:

Input: m = 7, n = 3
Output: 28

AC code:

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

time: O(n^2)　　space: O(m*n)

Runtime: 0 ms, faster than 100.00% of C++ online submissions for Unique Paths.

Optimization:

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

time: O(n^2)　　space:O(m)

Runtime: 0 ms, faster than 100.00% of C++ online submissions for Unique Paths.

 continue optination:

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