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?

Above is a 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

动态规划，很经典的问题。

转移方程dp[i][j] = dp[i – 1][j] + dp[i][j – 1];

说明几点：

1. 构造动态规划矩阵map时，长宽分别取m+1, n+1。并把第一行和第一列置0；map[1][1]=1.这样可以避免进行边界检查。

2. 遍历矩阵从1，1开始，但跳过1，1.

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