Issues with what we've seen so far

So far, we've looked at a few data structures for efficiently searching for the existence of items within the data structure. We looked at Binary Search Trees, then made them balanced using 2-3 Trees.

However, there are some limitations that these structures impose (yes, even 2-3 trees!)

  1. They require that items be comparable. How do you decide where a new item goes in a BST? You have to answer the question "are you smaller than or bigger than the root"? For some objects, this question may make no sense.
  2. They give a complexity of Θ(logN)\Theta(\log N). Is this good? Absolutely. But maybe we can do better.

A first attempt: DataIndexedIntegerSet

Let us begin by considering the following approach.

For now, we're only going to try to improve issue #2 above (improve complexity from Θ(logN)\Theta(\log N) to Θ(1)\Theta(1). We're going to not worry about issue #1 (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 Θ(1)\Theta(1) time.
  • The contains(int x) method simply returns whether the x position in our ArrayList is true or false. This also takes Θ(1)\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?
    • Let's look at this next. Of course, we may want to insert other things, like Dogs. That'll come soon!

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