What algorithm can calculate the power set of a given set?

Question

I would like to efficiently generate a unique list of combinations of numbers based on a starting list of numbers.

example start list = [1,2,3,4,5] but the algorithm should work for [1,2,3...n]

result =

[1],[2],[3],[4],[5] [1,2],[1,3],[1,4],[1,5] [1,2,3],[1,2,4],[1,2,5] [1,3,4],[1,3,5],[1,4,5] [2,3],[2,4],[2,5] [2,3,4],[2,3,5] [3,4],[3,5] [3,4,5] [4,5]

Note. I don't want duplicate combinations, although I could live with them, eg in the above example I don't really need the combination [1,3,2] because it already present as [1,2,3]

Answer 1

Break the problem down.

How would get all the different one-item lists? Two-item lists?

Can you use your answer to the one-item case in answering the two-item case?

Answer 2

There is a name for what you're asking. It's called the power set.

Googling for "power set algorithm" led me to this recursive solution.

Ruby algorithm

For the haters:

def powerset(set)
  return [set] if set.empty?

  p = set.pop
  subset = powerset(set)  
  subset | subset.map { |x| x | [p] }
end

Answer 3

Just count 0 to 2^n - 1 and print the numbers according to the binary representation of your count. a 1 means you print that number and a 0 means you don't. Example:

set is {1, 2, 3, 4, 5}
count from 0 to 31:
count = 00000 => print {}
count = 00001 => print {1} (or 5, the order in which you do it really shouldn't matter)
count = 00010 => print {2}
        00011 => print {1, 2}
        00100 => print {3}
        00101 => print {1, 3}
        00110 => print {2, 3}
        00111 => print {1, 2, 3}
        ...
        11111 => print {1, 2, 3, 4, 5}