62 Unique Paths – Medium

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

Note: m and n will be at most 100.

Thoughts:

This is a very basic and easy dynamic programming problem.

Each time the robot needs to make a choice is either go down or go right. Do the optimized solution is the better one from these two choices.

d[i][j] is the number of unique paths to reach point(i,j)

d[0][0] = 1

d[i][j] = d[i-1][j] +d[i][j-1]

Solutions:

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