62. Unique Paths

1. Description

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?

2. Example

Example 1:
Input: m = 3, n = 7
Output: 28

Example 2:
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 -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

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

Example 4:
Input: m = 3, n = 3
Output: 6

3. Constraints

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

4. Solutions

My Accepted Solution

m = rows, n = colums
Time complexity : O(mn)
Space complexity : O(mn)

class Solution 
{
public:
    // int uniquePaths(int m, int n)
    int uniquePaths(int rows, int colums) 
    {
        vector<vector<int>> possiblePaths(rows, vector<int>(colums));
        
        for(int row = 0; row < rows; row++)
        {
            for(int colum = 0; colum < colums; colum++)
            {
                if(row == 0 || colum == 0)
                    possiblePaths[row][colum] = 1;
                else
                    possiblePaths[row][colum] = possiblePaths[row][colum - 1] + possiblePaths[row - 1][colum];
            }
        }
        
        return possiblePaths.back().back();
    }
};

4.1 Math - Permutation

// TO DO

class Solution 
{
public:
    // int uniquePaths(int m, int n)
    int uniquePaths(int rows, int colums) 
    {
        
    }
};

4.2 Dynamic Programming(Less Space)

// TO DO

class Solution 
{
public:
    // int uniquePaths(int m, int n)
    int uniquePaths(int rows, int colums) 
    {
        
    }
};