What is a PowerSet?

Why is it called a PowerSet?
Is it good or bad for us?
Will I need it on my next Angular/Machine learning project?

If we look at Wikipedia for the definition of powerset we see the below:

In mathematics, the power set (or powerset) of any set S is the set of all subsets of S,...

Cool, so that means if we have a set of a few item's {1, 2} then the power set would be the set of all sets, in the {1, 2} case it would be:

{ {1}, {2}, {1, 2}}

Continuing to read the Wikipedia definition we see the guys there say:

including the empty set and S itself

You know what, those guys say we need to add the empty set let's just add it and move on:

{ {}, {1}, {2}, {1, 2}}

So we added the empty set, the definition is now happy. :)

All right I think now we are aligned with the definition, nice.

Computer makes a power set

Now the question is, how do we program the computer, to give us the PowerSet? The computer did not read the Wikipedia definition, so it doesn't have a clue about how to code that, let's help him.

Recursion comes immediately to mind. With recursion, if we manage the solve the problem in terms of a smaller problem, we could merge the smaller problem solution with the larger one, so we take our original set:

{1, 2}, a smaller set or a smaller problem here would be: {1, 2}.tail => {2}. If we magically (or in other words - recursively) new the PowerSet of it:

{ {}, {2}} . All that would be left to us then would be to merge it with the PowerSet of the head - {1, 2}.head => {1} which is: { {}, {1}} and then add the set itself which is: { {1, 2} }.

I don't know about you but, as much as recursion is romantic, I always feel like it's cheating, so how would we do it without cheating, that is, without recursion?

If we scan the set only once and do a single calculation on each item, we wouldn't really have a chance to do all the combinations of the subsets, so it appears as if we need to have nested loop. Let's start writing that, but wait before that, let's warm up our java syntax engines a bit.

Power Set with Java

Warming up java set skills:

Define a new set in java:

Set<Integer> set = new HashSet<>(); // define a set of integers in java.
Set<Set<Integer>> setOfSets = new HashSet<>(); // PowerSet is a set of sets..
Set<Set<Integer>> copiedSet = new HashSet(setOfSets); // Copy a set into a new variable.

We begin with the signature, we get a set and we return a set, both are a set of integers:

public Set<Set<Integer>> powerSet(Set<Integer> set) // we get a set and return a set of sets, a power set is a set of sets.

Now that we got a set let's start iterating the set items:

public Set<Set<Integer>> powerSet(Set<Integer> set) {
    Set<Set<Integer> powerSet = new HashSet<>();

    for (Integer item: set) {
        powerSet.add(item);
    }

    // omg omgI'm stuck here...
    // but now what after we added all items how do we combine them all to the power set?
}

It looks like this is not enough, we cannot just scan the items and build a set of sets. Let's ponder about this a little.

hmm...

Well we said a set of sets, let's repeat this a set of sets, so it's like multiple sets but not only multiple sets it's like each set is including other sets.

Each set that we manage to create, we are going to need this set with each other item in the set, so see, for each set we find, no matter which subset it is, for each subset, another valid subset would be this subset with each other item on the original set.

Found { 1, 2 } subset? and you have { 4 } in the original set: here is another subset for you { 1, 2, 4 }.

For that reason what we are going to do is store all the temporal subsets we manage to find and for each such subset add each item from the original set meaning it's going to follow:

1. Find the first subset - for example, the empty set.
2. Add it to the subsets already found.
3. Add each item from original set to.

Alright, all set? let's move on to the full PowerSet Implementation!:

     public static Set<Set<Integer>> powerSet(Set<Integer> set) {
       Set<Set<Integer>> powerSets = new HashSet<>();  // we define powerset.

       Set<Integer> emptySet = new HashSet<>();
       powerSets.add(emptySet); // start with the empty set.


       for (Integer item: set) { // We want to add each item to all previous subsets.
         Set<Set<Integer>> foundSets = new HashSet(powerSets);
         for (Set<Integer> foundSet: foundSets) {
           Set<Integer> newSet = new HashSet<Integer>(foundSet);
           newSet.add(item); // Add each item to all previously found subsets.
           powerSets.add(newSet); // Add new subset to all subsets found.

         }
       }

       return powerSets;

     }

And if we run it with:

         Set<Integer> set = new HashSet<Integer>();
         set.add(1);
         set.add(2);
         set.add(3);
         System.out.println(powerSet(set));

We get:

// [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] Looks like a power set to me! What do you say? ;)

Back to recursion

Now that we have the imperative method behind us, let's see the scala :: functional :: declarative :: concise method :))

  def powerSet(set: Set[Int]): Set[Set[Int]] = {

    set.toList match {
      case Nil => Set(Set.empty[Int])
      case x :: xs => powerSet(xs.toSet).map(xsi => xsi + x) ++ powerSet(xs.toSet)
    }
  }

  println(powerSet(Set(1,2,3))) // Set(Set(), Set(3, 1), Set(2), Set(2, 1), Set(3, 2), Set(3), Set(3, 2, 1), Set(1))

So cool! we just spelled the magic words of the powerSet definition, that each powerset is the powerset of the head added to each of the powerset found for the tail! and we have it done and ready! so hot!

Summary

To sum up we started by adding the empty set. Then we scan each item from 1..n and add this item to the empty set, we end up having each time only once in the powerSet.

Next, we take each such item and add it to the currently existing sets, when we add { 1 } to the set { 1 } it would not be { 1, 1 } with just { 1 } (psst it's a set remember?). And we continue and scan each item in the set and add each of them to the current sets which would mean we get to have all subsets - power sets.

How will that help us with AngularJs or machine learning? it's simple at least in my mind, by mastering syntax and the language, by mastering the basic data structures, be it sets, lists, maps, from them we can build queues, heaps, trees, and from them even more complex data structures all these are important, and as software engineers we deal with them every day, either directly or indirectly, so it's best to master those very simple basic terms.

The for loop, the set, the list, are basic, but they are powerful with these basic constructs we have built the powerSet.