[Algorithm] 4. Ugly Number II

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

 1     /**
 2      * @param n: An integer
 3      * @return: the nth prime number as description.
 4      */
 5     public int nthUglyNumber(int n) {
 6         // find out the nth ugly number.
 7         int checker,count=0;
 8         int num=1;
 9         
10         do{
11             checker = num;
12             
13             while( checker%2 == 0 ) checker /= 2;
14             while( checker%3 == 0 ) checker /= 3;
15             while( checker%5 == 0 ) checker /= 5;
16             if( checker==1 )
17             {
18                 count++;
19             }
20             if( count == n )
21             {
22                 return num;
23             }
24             
25             num++;
26         }while(true);
27     }

Better Solution

 1     /**
 2      * @param n: An integer
 3      * @return: the nth prime number as description.
 4      */
 5     int nthUglyNumber(int n) {
 6         // find out the nth ugly number.
 7         int* uglyNum = new int[n];
 8         int factor2=2, factor3=3, factor5=5;
 9         int index2=0, index3=0, index5=0;
10         
11         uglyNum[0]=1;
12         
13         for(int i=1; i<n; i++)
14         {
15             factor2 = uglyNum[index2]*2; factor3 = uglyNum[index3]*3; factor5 = uglyNum[index5]*5;
16             int minNum = min(factor2, min(factor3, factor5) );
17             uglyNum[i] = minNum;
18             if ( minNum == factor2 )    index2++;
19             if ( minNum == factor3 )    index3++;
20             if ( minNum == factor5 )    index5++;
21         }
22         
23         return uglyNum[n-1];        
24     }

Hints

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.
An ugly number must be multiplied by either 2, 3, or 5 from a smaller ugly number.
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.
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...

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