_62.java

package com.fishercoder.solutions;

/**
 * 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?
 */
public class _62 {

  public static class Solution1 {
    /**
     * Another typical DP question, use a 2d array: the first row and the first column need to be
     * initialized to be 1 since there's only one way to reach every position in the first row and
     * the first column: either from left or top.
     */
    public int uniquePaths(int m, int n) {
      int[][] dp = new int[m][n];
      for (int i = 0; i < m; i++) {
        for (int j = 0; j < n; j++) {
          if (i == 0 || j == 0) {
            dp[i][j] = 1;
          } else {
            dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
          }
        }
      }
      return dp[m - 1][n - 1];
    }
  }
}