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

// Dynamic Programming
int uniquePaths(int m, int n) { // time: O(m * n); space: O(m * n)
    vector<vector<int> > path(m, vector<int>(n, 0));
    for (int i = 0; i < m; ++i) {
        for (int j = 0; j < n; ++j) {
            if (i == 0 && j == 0) {
                path[i][j] = 1;
                continue;
            }
            path[i][j] = (i >= 1 ? path[i - 1][j] : 0) + (j >= 1 ? path[i][j - 1] : 0);
        }
    }
    return path.back().back();
}

// Space Optimized Dynamic Programming
int uniquePaths(int m, int n) { // time: O(m * n); space: O(n)
    vector<int> path(n, 1);
    for (int i = 1; i < m; ++i) {
        for (int j = 1; j < n; ++j) {
            path[j] += path[j - 1];
        }
    }
    return path.back();
}