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.

Version 1. Initialize first row and first column separately

public int uniquePaths(int m, int n) { if (m <= 0 || n <= 0) { return 0; } int[][] f = new int[m][n]; //initialize the first column for (int i = 0; i < m; i++) { f[i][0] = 1; } //initialize the first row for (int j = 0; j < n; j++) { f[0][j] = 1; } for (int i = 1; i < m; i++) { for (int j = 1; j < n; j++) { f[i][j] = f[i - 1][j] + f[i][j - 1]; } } return f[m - 1][n - 1]; }

Version 2. Put initialization in the 2 for loop.

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