[Algorithm] 4. Ugly Number II

Description

Ugly number is a number that only havefactors 2, 3 and 5.

Design an algorithm to find the nth ugly number. The first 10 ugly numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12...

Example

If n=9, return 10.

Challenge

O(n log n) or O(n) time.

Notice

Note that 1 is typically treated as an ugly number.

Answer

     /**
      * @param n: An integer
      * @return: the nth prime number as description.
      */
     public int nthUglyNumber(int n) {
         // find out the nth ugly number.
         int checker,count=;
         int num=;

         do{
             checker = num;

             while( checker% ==  ) checker /= ;
             while( checker% ==  ) checker /= ;
             while( checker% ==  ) checker /= ;
             if( checker== )
             {
                 count++;
             }
             if( count == n )
             {
                 return num;
             }

             num++;
         }while(true);
     }

Better Solution

     /**
      * @param n: An integer
      * @return: the nth prime number as description.
      */
     int nthUglyNumber(int n) {
         // find out the nth ugly number.
         int* uglyNum = new int[n];
         int factor2=, factor3=, factor5=;
         int index2=, index3=, index5=;

         uglyNum[]=;

         for(int i=; i<n; i++)
         {
             factor2 = uglyNum[index2]*; factor3 = uglyNum[index3]*; factor5 = uglyNum[index5]*;
             int minNum = min(factor2, min(factor3, factor5) );
             uglyNum[i] = minNum;
             if ( minNum == factor2 )    index2++;
             if ( minNum == factor3 )    index3++;
             if ( minNum == factor5 )    index5++;
         }

         return uglyNum[n-];
     }

Hints

1. The naive approach is to call isUgly for every number until you reach the n^th one. Most numbers are notugly. Try to focus your effort on generating only the ugly ones.
2. An ugly number must be multiplied by either 2, 3, or 5 from a smaller ugly number.
3. The key is how to maintain the order of the ugly numbers. Try a similar approach of merging from three sorted lists: L₁, L₂, and L₃.
4. Assume you have U_k, the k^th ugly number. Then U_k+1 must be Min(L₁ * 2, L₂ * 3, L₃ * 5).

　　(1) 1x2,  2x2, 2x2, 3x2, 3x2, 4x2, 5x2...

　　(2) 1x3,  1x3, 2x3, 2x3, 2x3, 3x3, 3x3...

　　(3) 1x5,  1x5, 1x5, 1x5, 2x5, 2x5, 2x5...