62 Unique Paths

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?

three ways credit to youtube edward shi

• [x] Basic Dynamic Programming, time O(m * n), space O(m * n)
• [x] Dynamic Programming, space O(n)
• [x] combination (m,n) m = m + n - 2, n = m - 1

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

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

class Solution {
    public int uniquePaths(int m, int n) {
        if (m <= 0 || n <= 0) {
            return 0;
        }
        double res = 1;
        int count = m + n - 2;
        int k = m - 1;
        for (int i = 1; i <= k; i++) {
            res = res * (count - k + i) / i;
        }
        return (int)res;
    }
}