Question

I have an algorithm to compute the powerset of a set using all of the bits between 0 and 2^n:

public static <T> void findPowerSetsBitwise(Set<T> set, Set<Set<T>> results){
        T[] arr = (T[]) set.toArray();
        int length = arr.length;

        for(int i = 0; i < 1<<length; i++){
            int k = i;
            Set<T> newSubset = new HashSet<T>();
            int index = arr.length - 1;
            while(k > 0){
                if((k & 1) == 1){
                    newSubset.add(arr[index]);
                }
                k>>=1;
                index --;
            }
            results.add(newSubset);
        }

    }

My question is: What is the running time of this algorithm. The loop is running 2^n times and in each iteration the while loop runs lg(i) times. So I think the running time is

T(n) = the sum from i=0 to i=2^n of lg(i)

But I don’t know how to simplify this further, I know this can be solved in O(2^n) time (not space) recursively, so I’m wondering if the method above is better or worse than this, timewise as it’s better in space.

Answer 1

sigma(lg(i)) where i in (1,2^n) 
= lg(1) + lg(2) + ... + lg(2^n)     
= lg(1*2*...*2^n) 
= lg((2^n)!) 
> lg(2^2^n) 
  = 2^n

thus, the suggested solution is worth in terms of time complexity then the recursive O(2^n) one.

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EDIT:

To be exact, we know that for each k – log(k!) is in Theta(klogk), thus for k=2^n we get that lg((2^n)!) is in Theta(2^nlog(2^n) = Theta(n*2^n)