Unique Path

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 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

Analyse: Dynamic Planning. Transfer function is grid[i][j] = grid[i - 1][j] + grid[i][j - 1].

Runtime: 0ms.

class Solution {
public:
    int uniquePaths(int m, int n) {
        vector<vector<int> > grid(m, vector<int> (n, 0));
        
        for(int i = 0; i < m; i++) 
            grid[i][0] = 1; //the first colume has only one path
        for(int j = 0; j < n; j++) 
            grid[0][j] = 1; // the first row has only one path
        for(int i = 1; i < m; i++)
            for(int j = 1; j < n; j++)
                grid[i][j] = grid[i - 1][j] + grid[i][j - 1];
        
        return grid[m - 1][n - 1];
    }
};