62. Unique Paths

题目描述：

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.

题目翻译

机器人位于m x n网格的左上角（标有'Start'）。

在任何时间，机器人只能向下或向右移动。机器人试图到达网格的右下角（标有'Finish'）。

有多少种可能的不同路径？

注意：

m和n最大为​​100。

解题方案

标签： Dynamic Programming

思路：

• 使用二维数组来实现。规律为除了第一行和第一列全为1外，其他格的路径数为其上一格和左一格的和。

• 算法复杂度O(m*n)。

Python代码：

class Solution:  
    def uniquePaths(self, m, n):  
        paths = [[1] for i in range(m)]  # initialize the 1st column to be 1   

        for i in range(n - 1):           # initialize the 1st row to be 1   
            paths[0].append(1)   

        for i in range(m - 1):  
            for j in range(n - 1):  
                paths[i + 1].append(paths[i][j + 1] + paths[i + 1][j])  
        return paths[m - 1][n - 1]

Java代码：

public class Solution {  
    public int uniquePaths(int m, int n) {  
        int[][] A = new int[m][n];  

        for (int i = 0; i < m; ++i) {  
            A[i][0] = 1;  
        }  

        for (int i = 1; i < n; ++i) {  
            A[0][i] = 1;  
        } 

        for (int i = 1; i < m; ++i)  
            for (int j = 1; j < n; ++j) {  
                A[i][j] = A[i][j - 1] + A[i - 1][j];  
            }  

        return A[m - 1][n - 1];  
    }  
}

参考资料

• http://blog.csdn.net/lilong_dream/article/details/19771225
• 作者：lilong_dream