Unique Paths

Problem

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?

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

Constraints:

• 1 <= m, n <= 100
• It’s guaranteed that the answer will be less than or equal to 2 * 10 ^ 9.

Optimal Solution

Complexity

• Time: O(nm)
• Space: O(nm)

public int uniquePaths(int r, int c) {
    int[][] dp = new int[r][c];
    if(r == 0 && c == 0) return 0;
    if(r == 0 || c == 0) return 1;

    for(int i = 0; i < r; i++) dp[i][0] = 1;
    for(int i = 0; i < c; i++) dp[0][i] = 1;

    for(int i = 1; i < r; i++){
        for(int j = 1; j < c; j++){
            dp[i][j] = dp[i-1][j] + dp[i][j-1];
        }
    }

    return dp[r-1][c-1];
}