[LeetCode] Unique Paths

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.

解题思路：

数学题。共有路径数C(m-1 + n - 1, n-1)。计算的时候会用到阶层运算，因此要用longlong类型防止溢出，而且需要尽量先做除法运算。

class Solution {public:    int uniquePaths(int m, int n) {        long long result = 1;        if(m<=1 || n<=1){            return result;        }        int j=n-1;        for(int i=m+n-2; i>=m; i--){            while(j>1 && result%j==0){                result = result/j;                j--;            }            result = result * i;        }        while(j>1 && result%j==0){            result = result/j;            j--;        }        return (int)result;    }};

update in 2015-07-28

方法2：回溯法。

d[i][j]表示从0,0到i,j的走法数，则有

d[i][j]=d[i-1][j]+d[i][j-1]

代码如下：

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