# Big O vs. Big Theta vs. Big Omega Notation Differences Explained

Big O, Big Theta and Big Omega notations express an algorithm’s time and space complexity. Discover what each one is and what the differences between them are.

Written by Dr. Derek Austin
Published on Aug. 08, 2023
Image: Shutterstock / Built In

Big O, Big Theta and Big Omega notations express an algorithm’s time and space complexity. If you’ve ever wondered how to quantify the efficiency of your code, these notations help you do it.

## Big O vs. Big Theta vs. Big Omega Explained

• Big O: This represents the worst-case performance for an algorithm, setting an upper bound on how slow your code can be. It’s noted as `O(n²)`.
• Big Theta (Θ): This represents the average, typical case performance for an algorithm. It’s noted as `Θ(n×p)`.
• Big Omega (Ω): This represents the best case performance for an algorithm, setting a lower bound on how fast the code can perform. It’s noted as `Ω(n)`.

While it can be a dry subject, understanding how Big O and the other notations work is important for technical interviews. So, to make it more fun, we’ll unravel the mysteries using the universal language of emojis. Buckle up for a roller coaster of fun, knowledge and a few emojis along the way!

## What Is Big O Notation?

Big O notation (O) is like the overprotective parent of algorithmic complexity. It expresses the worst-case performance of an algorithm, setting an upper bound on how slow your code can be.

Imagine you’re catering a number (p) of parties (🎉), and the more guests (n) arrive, the longer it takes for you to deliver your delicious pizzas (🍕).

Let’s look at a fun emoji representation of Big O complexity, `O(n²)`, reflecting the exponential growth of the worst case.

Here, every guest is at a different party, so you have to make n deliveries, which means p is the same as n.

As the number of guests (👩‍💻) and parties (🎉) increases, so does the time it takes to deliver the pizza, since these are parties of one.

For this simple example, we’ll assume the worst case is that each guest (n) is at a separate party (p), so Big O is `O(n×p)===O(n×n)===O(n²)` exponential.

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## What Is Big Theta Notation?

Big Theta (Θ) notation is the cool cousin of Big O notation, representing the average-case performance of an algorithm.

Big Theta isn’t just fun to say, it’s the sweet spot between the best and worst-case scenarios.

When you have an algorithm with matching upper and lower bounds, you can use Big Theta notation to describe its complexity.

Let’s stick with our pizza party example (🍕) and assume we know the average time it takes to deliver pizza based on the number of guests.

Here’s an emoji representation of a linear Big Theta complexity, Θ(n×p), where if p is constant, then where we scale linearly with n:

As the number of guests (n) increases, the average time to deliver pizza increases linearly. Here, p is constant value, resulting in `Θ(n×2)` or `Θ(n×2)`.

The difference from before is that we might imagine that in the worst case, you have to deliver n pizzas to p different parties, one pizza per party.

That’s basically the traveling salesman problem, but the bottom line is that more trips will mean more travel time than fewer parties: Big `O(n²)`.

On average, each party will have more than one guest, so we’ll scale slower than that upper bound but faster than the lower bound of `p===1`.

To recap: Big Theta is the “typical” case. It’s not the optimal, best case for one single party, but it’s also not the worst case of n separate parties.

## What Is Big Omega Notation?

What if all the guests are at just one party? In that situation, we will need to use Big Omega notation, for the best case scenario: `Ω(n)`.

In the best case, Big Omega (Ω), one pizza gets delivered to just one party, so we just scale linearly with n, the number of guests.

Here’s an emoji representation of linear Big Omega complexity, `Ω(n)`, where if p is one then where we scale linearly with n:

In the best case, you still scale with n, since you have to bake more pizza to feed each additional guest.

But if you can multithread, by baking more than one pizza simultaneously, then you could even be faster than that.

For example, if you have two pizza ovens, your Big Omega best case would become `Ω(1/2n)`, assuming you have an even number of pizzas.

For simplicity, we often exclude the constants from algorithmic complexity notation, so you might still say `Ω(n)` instead.

The important takeaway from Big Omega is that if we only have one party, we don’t have travel time between multiple parties, so we only scale to n.

## When to Use Big O vs. Big Theta

So, how do you choose between Big O and Big Theta (Θ) notations? It depends on the context.

Big O notation is crucial when you need to determine the worst-case scenario, like ensuring your app doesn’t crash during peak hours.

For Big O, think of the Diablo 4 beta weekend in which millions of users simultaneously tried to play at the same time.

On the other hand, Big Theta notation is ideal when you want to analyze the average performance of an algorithm.

For example, if you’re thinking about how quickly your AI search engine can find results for your users, would you care more about O or Θ? Well, both the worst case (Big O / P99, for 99th percentile) and average case (Big Theta / P50, for 50th percentile) are probably important.

But, of the two, the average performance is probably most important, since it’s the one that the typical user will see. And that’s Big Theta.

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## Big O vs. Big Theta vs. Big Omega Tips

Here’s a fun emoji summary to help you remember the three measures of algorithmic complexity:

You can also remember the difference between Big O, Big Omega and Big Theta with this mnemonic:

• Big O is the worst case, representing a one-sided bound. Big O is “Woah, woah, worst case, and “O” has one side.”
• Big Theta (Θ) is the average case, representing a two-sided bound. Theta starts with “T” as in “The typical, two-sided case.”
• Big Omega (Ω) is the best case, representing a one-sided bound. Omega is the ultimate, best case: “one-sided utopia.”

And there you have it. We’ve delved into the world of algorithmic complexity and explored the differences between Big O, Big Theta and Big Omega notations.

These notations allow us to quantify the efficiency of our code and make informed decisions about optimizing it. Whether you’re developing a new app, building a website, or creating a game, understanding algorithmic complexity is a valuable skill.

So, next time you’re at a party, remember the pizza delivery (🍕) and how algorithmic complexity can help you ace your next technical interview.