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?

思路很容易想，但是要尽量做到简化。

public class Solution {
    public int uniquePaths(int m, int n) {
       int[][] map =new int[m][n];
       map[0][0]=1;
       return find(m-1,n-1,map);      
    } 
    
    int find(int m,int n,int[][] map){
        if(map[m][n]!=0){
            return map[m][n];
        }else{
            if(m==0){map[m][n]= 1;}
            else if(n==0){map[m][n]=1;}
            else{
            map[m][n]= find(m-1,n,map)+find(m,n-1,map);
            }
            return map[m][n];
        }     
    }           
}

上面的方法是第一次写的，其实没有用到dp， 用的是 记忆化搜索（memory search），下面这种方法是第二次写用的dp：

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