[leetcode]Unique Paths

问题叙述性说明：

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.

基本思想：

本题能够这么考虑。设f[i][j]表示从左上到i行j列的格子处的全部路径数。考虑我们能够怎样到f[i][j]这个格子呢？有两种方法：

从（i-1,j）向右走一步
从（i，j-1）向左走一步

所以我们能够列出恒等式 f[i][j] = f[i-1][j]+f[i][j-1];初始化（i，0）和（0,j）为1;如此便能够迭代的求出到最后右下时的方法数。

代码：

int uniquePaths(int m, int n) {  //c++
        if(m == 1 || n == 1)
            return 1;
        int array[m][n];
        
        //init
        for(int i = 0; i < n; i++)
            array[0][i] = 1;
        for(int i = 0; i < m; i++)
            array[i][0] = 1;
            
        for(int i = 1; i < m; i++)
            for(int j = 1; j < n; j++){
                array[i][j] = array[i-1][j]+array[i][j-1];
            }
        return array[m-1][n-1];
    }