254. Factor Combinations

Numbers can be regarded as product of its factors. For example,

8 = 2 x 2 x 2;
  = 2 x 4.

Write a function that takes an integer n and return all possible combinations of its factors.

Note: 

1. You may assume that n is always positive.
2. Factors should be greater than 1 and less than n.

Examples: 
input: 1
output: 

[]

input: 37
output: 

[]

input: 12
output:

[
  [2, 6],
  [2, 2, 3],
  [3, 4]
]

input: 32
output:

[
  [2, 16],
  [2, 2, 8],
  [2, 2, 2, 4],
  [2, 2, 2, 2, 2],
  [2, 4, 4],
  [4, 8]
]

这道题需要注意的是如何避免重复，需要在backtracking的inputs里加一个sentinel，在backtracking的循环过程中从sentinel开始加。

public IList<IList<int>> GetFactors(int n) {
        var res = new List<IList<int>>();
        if(n ==0) return res;
        BackTracking(n,new List<int>(),res,2);
        return res;
    }
    
    private void BackTracking(int n, List<int> cur, IList<IList<int>> res,int last)
    {
        if(n==1)
        {
            if(cur.Count()>1)
            res.Add(new List<int>(cur));
        }
        else
        {
            for(int i = last;i<= n;i++)
            {
                if(n%i ==0)
                {
                    cur.Add(i);
                    BackTracking(n/i,cur,res,i);
                    cur.RemoveAt(cur.Count()-1);
                }
            }
        }
    }

上面算法是可以优化的，因为一个整数分为两个整数乘积的时候，较小的乘数也不会超过原来数字的平方根。则可以在loop的上界限制为sqrt(n)。

public IList<IList<int>> GetFactors(int n) {
        var res = new List<IList<int>>();
        if(n ==0) return res;
        BackTracking(n,new List<int>(),res,2);
        return res;
    }
    
    private void BackTracking(int n, List<int> cur, IList<IList<int>> res,int last)
    {
        if(cur.Count()>0)
        {
            cur.Add(n);
            res.Add(new List<int>(cur));
            cur.RemoveAt(cur.Count()-1);
        }
        
            for(int i = last;i<= (int)Math.Sqrt(n);i++)
            {
                if(n%i ==0)
                {
                    cur.Add(i);
                    BackTracking(n/i,cur,res,i);
                    cur.RemoveAt(cur.Count()-1);
                }
            }
        
    }