Unique Paths问题及解法

问题描述：

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.

问题分析：

本题是一个动态规划问题。经过观察我们发现：

start到finish的路径总数 = start到finish的上边一个格的路径总数 + start到finish的左边一个格的路径总数。依次依次从start开始，计算到每个位置(i,j)时的路径总数。最终可确定到位置(m,n)的路径总数。

过程详见代码：

class Solution {public:    int uniquePaths(int m, int n) {vector<vector<int>> mat(m + 1, vector<int>(n + 1, 0));mat[1][1] = 1;for (int i = 1; i <= m; i++){for (int j = 1; j <= n; j++){if (mat[i][j] == 0) mat[i][j] = mat[i][j - 1] + mat[i - 1][j];}}return mat[m][n];}};