62. Unique Paths

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.

代码如下：

class Solution {
public:
    int uniquePaths(int m, int n) {
      vector<vector<int>> path(m, vector<int> (n, 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];
    }
};

解题思路：

• 典型的动态规划，对于每一步，相对于上一步都有两种走法，向下或者向右；
• 用 path 来记录到达当前点的最大走法，即为到达上一步的两种走法的最大走法之和，且 path[0][1] 和 path[1][0] 必为1，因为第一步分别只有一种走法（要想走到这一步，要么只能向下，要么只能向右）。