19.1.1 A first attempt: DataIndexedIntegerSet

Let us begin by considering the following approach. This approach was introduced in Hashing Video 1.

For now, we're only going to try to improve complexity from $\Theta(logN)$ to $\Theta(1)$. We're going to not worry about comparability. In fact, we're going to only consider storing and searching for ints.

Here's an idea: let's create an ArrayList of type boolean and size 2 billion. Let everything be false by default.

• The add(int x) method simply sets the x position in our ArrayList to true. This takes $\Theta(1)$ time.

• The contains(int x) method simply returns whether the x position in our ArrayList is true or false. This also takes $\Theta(1)$ time!

public class DataIndexedIntegerSet {
    private boolean[] present;

    public DataIndexedIntegerSet() {
        present = new boolean[2000000000];
    }

    public void add(int x) {
        present[i] = true;
    }

    public boolean contains(int x) {
        return present[i];
    }
}

There we have it. That's all folks.

Well, not really. What are some potential issues with this approach?

• Extremely wasteful. If we assume that a boolean takes 1 byte to store, the above needs 2GB of space per new DataIndexedIntegerSet(). Moreover, the user may only insert a handful of items...

• What do we do if someone wants to insert a String? Or other data types?

  • Let's look at this next. Of course, we may want to insert other things, like Dogs. That'll come soon!