unique_paths.cpp

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.





// Recursive (Memoized)

class Solution {
public:
    vector <vector<int>> dp;
    
    int solve(int i, int j, int m, int n) {
        if (i == m && j == n) return 1;
        if (i > m || j > n) return 0;
        
        if (dp[i][j] != -1) return dp[i][j];
        
        int down = 0, right = 0;
        
        if (i < m) down = solve(i + 1, j, m, n);
        if (j < n) right = solve(i, j + 1, m, n);
        
        return dp[i][j] = down + right;
    }
    
    int uniquePaths(int m, int n) {
        dp.resize(m + 1, vector <int> (n + 1, -1));
        
        return solve(0, 0, m - 1, n - 1);   // 0, 0 -> m - 1, n - 1
    }
};





// Dp (Accepted)

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