[LeetCode]: 62: Unique Paths

题目：

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.

思路1：递归

终点的路径数= 所有能达到终点的点的路径数之和，即为：路径[m][n] = 路径[m-1][n] + 路径[m][n-1]

依次递归，直到start点（0，0）

代码：

    public static int uniquePaths(int m, int n) {
        if(n == 1 && m==1){
            //初始的位置
            return 1;
        }
        else if(n == 1 && m>1){
            return uniquePaths(1,m-1);
        }
        else if(n >1 && m==1){
            return uniquePaths(n-1,1);
        }
        else{
            return uniquePaths(n-1,m)+uniquePaths(n,m-1);
        }
    }

结果在m=17 n=23的时候超时了。所以意识到题目的限制条件是需要速度！

思路2：动态规划

某一个点的路径数= 所有能达到该点的点的路径数之和，即为：路径[m][n] = 路径[m-1][n] + 路径[m][n-1]

与递归不同的是，递归是从大往小算，动态规划是从小往大算

代码如下：

    public static int uniquePaths(int m, int n) {
        int[][] arrResult  = new int[m][n];
        
        for(int i = 0;i<m;i++){
            for(int j = 0;j<n;j++){
                if(i == 0 && j== 0){  
                    arrResult[0][0] = 1;
                }
                else if(i == 1 && j== 0){
                    arrResult[1][0] = 1;
                }else if(i == 0 && j== 1){
                    arrResult[0][1] = 1;
                }
                //以上是填充基础值
                else{
                    int row = 0;
                    int column  = 0;
                    
                    if(i> 0 ){
                        row = arrResult[i-1][j];
                    }
                    
                    if(j> 0 ){
                        column = arrResult[i][j-1];
                    }

                    arrResult[i][j] = row + column;    
                }
            }
        }
        
        
        return arrResult[m-1][n-1];
    }