What is Big O Notation?

Is there a computer science topic more terrifying than Big O notation? Don’t let the name scare you, Big O notation is not a big deal. It’s very easy to understand and you don’t need to be a math whiz to do so. In this tutorial, you'll learn the fundamentals of Big O notation, beginning with constant and linear time complexity with examples in JavaScript.

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This is the first in a series on Big O notation. If you want to stay in the loop, sign up for my weekly newsletter, The Solution.

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What Problem(s) Does Big O Notation Solve?

• Big O notation helps us answer the question, “Will it scale?”

• Big O notation equips us with a shared language for discussing performance with other developers (and mathematicians!).

What is Big O Notation?

Big O is a notation for measuring the performance of an algorithm. Big O notation mathematically describes the complexity of an algorithm in terms of time and space. We don’t measure the speed of an algorithm in seconds (or minutes!). We measure the rate of growth of an algorithm in the number of operations it takes to complete.

The O is short for “Order of magnitude”. So, if we’re discussing an algorithm with O(n), we say its order of magnitude, or rate of growth, is n, or linear complexity.

You will probably read or hear Big O referred to as asymptotic runtime, or asymptotic computational complexity. This is a fancy way of describing the limits of a function. There is a branch of mathematics, order theory, devoted to this topic. For our intents and purposes, order:

… provides a formal framework for describing statements such as "this is less than that" or "this precedes that".

We use order to evaluate the complexity of our algorithms.

Math O’Clock 🧮 🕐

You don’t need to be a math whiz to grok Big O, but there are a few basic concepts we need to cover to set you up for success.

If you recall from algebra, you worked with functions such as f(x) and g(x), and even did things like f(g(x)), where f() and g() were equations and x was a numerical value (or another equation!) passed to the functions.

When we’re programming, we give our “equations” descriptive names (at least I hope you are), such as isAuthenticated and calcuateMedian, but we could also name them f and g (please don’t).

Let’s say f(x) is equal to 3x² + 12x - 6.

We could say that the order of magnitude, or rate of growth, of f(x) is O(n²⁾. We’ll see why later.

It’s more common to simply say “f(x) is order of n^2”, or “f(x) is Big O of n^2”.

Math time over.

For now. 😀

How Does Big O Notation Work?

Big O notation measures the worst-case runtime.

Why?

Because we don’t know what we don’t know.

If we’re writing a search algorithm, we won’t always know the query ahead of time. If we’re writing a sorting algorithm, we won’t always know the dataset ahead of time. What if the query is the very last element or what if the dataset is a real mess. We want to know just how poorly our algorithm will perform.

The worst-case scenario is also known as the “upper bound”. Limits again!

You’re going to encounter a lot of tables like this:

O	Run time
O(1)	constant	fast
O(log n)	logarithmic
O(n)	linear
O(n * log n)	log linear
O(n2)	quadratic
O(n3)	cubic
O(2n)	exponential
O(n!)	factorial	slow

This lists common runtimes from fastest to slowest.

We’ll refer to this a lot as we proceed.

Before we get into any code, let’s get hands-on to get a feel (pun intended) for Big O. We’ll use an example from Grokking Algorithms.

Let's say I give you a square piece of paper and ask you to divide it into sixteen squares. How would you approach this problem?

You could take the brute force approach and draw sixteen individual squares. If you take this approach, how many steps, or computations, will you perform?

Sixteen.

Is there an approach that requires fewer steps? Of course!

Fold the paper in half. Then in half again. Four squares!

Now fold it in half two more times.

When you unfold it, the paper will be divided into sixteen squares.

How many steps, or computations, were required?

Four.

In Big O notation, our first approach, brute force, is O(n), or linear time. Creating sixteen squares requires sixteen operations. But our second, refactored and optimized, approach is O(log n), or logarithmic time (the inverse of exponentiation). Creating sixteen squares requires only four steps.

We’ll look at O(log n) later. Let’s begin with O(1), which will help us understand O(n).

O(1): Constant Time Complexity

Say you’re working with an API that returns a users full name in an array, like so:

[“Jared”, “Nielsen”];

Your task is to get the users first name. Easy, in JavaScript:

const getFirstName = data => {
    return data[0];
}

No matter how many times you run your ‘algorithm’, it only needs to perform one operation to return the desired value. That’s O(1), or constant time.

Here’s another JavaScript example:

const isEven = num => num % 2 === 0;

Our algorithm checks whether or not a number is even or odd and will return true or false accordingly. It only needs to perform one operation. Again, O(1).

What is Big O Notation?

Big O notation is not a big deal. It’s very easy to understand and you don’t need to be a math whiz to do so. In this tutorial, you learned the fundamentals of Big O notation, as well as constant and linear time complexity with examples in JavaScript.

Stay tuned for part two of this series on Big O notation where we'll look at O(n), or linear time complexity. If you want to stay in the loop, sign up for my weekly newsletter, The Solution.

Comments

[Jared Nielsen]

Thanks for catching my error, Aleksandr. That was a holdover from an earlier draft. The perils of being your own editor. But it generated a great discussion! Update forthcoming.

[Paceaux]

I've read at least a dozen articles on Big-O throughout the years. This by far is the best explanation I've seen. Thanks for writing it and I look forward to reading your future posts.

[Paceaux]

I've probably read a dozen article on Big-O notation and this is the first one I was able to get through by reading it exactly once.

Jared's examples are intentionally trivial because his intended audience is beginners. Content of your comment makes it pretty clear that you are not the target audience.

Coming into an introductory article and saying, "the examples ... are trivial", as though it were a bad thing, is not helpful; in fact it's hurtful. Why should beginners be given complex examples first? What possible gain is there?

[aRtoo]

gotcha. much love sir. thanks.

[Rob Conery]

Hi Jared - nice article. I wanted to echo what others have said that your example re alertColor() is always O(1) so you may want to tweak this:

What if state is not equal to danger? Then we perform multiple operations and the order of alertColor() is n.

The reason it's O(1) is that this algorithm will always perform the same tasks no matter the input (as there is no input).

I would encourage you to update the post so people don't get confused :).

A really helpful way to think about Big-O (in my opinion) is to ponder how many loops you have that loop over your input argument. One loop will always be O(n), nested loops are O(n²⁾ and so on. No loops at all (as in your example) is always O(1).

I wrote about Big-O here: rob.conery.io/2019/03/25/wtf-is-bi... and I also wrote a big fat book about core programming concepts which is linked on my blog... didn't come here to be cheesy I promise!

[Jared Nielsen]

Hi Rob! The Impostor's Handbook is fantastic! I'm entirely self-taught and it helped me fill in gaps I didn't know I had.

Thank you for kindly flagging my error. Editorial oversight. Update in the works.

Cheers!

[Rob Conery]

Cheers mate - no prob at all. This is tough stuff and my goodness did I get it wrong way too many times!

[aRtoo]

yea i got you now. on your example since arrayOfLengthN is just being compared then run time would always be constant unless n is being run by or evaluated by other method such as forEach. In other words compare instruction is O(1) regardless of how many are being chained.

[aRtoo]

In the example with all the if statements, if we define n to be > the number of if statements, then the algorithm is in fact O(n). > But n is just a constant, so it's really O(1).

Im guessing the author is talking about n as number of ifs thats how I was understanding it. Thats why he had to create a hash for a constant look up.

[aRtoo]

ohh. I really thought its O(n) because it will try to evaluate every ifs.

Unless for switch statement, compiler will try its best to optimized and it will be much faster lookup. Im talking about c++ btw. thanks

[aRtoo]

question though. What if there's a nested ifs then It would be an O(n) isn't it? since it will try to evaluate the nested ifs?

[Jessica Garson]

Awesome!

[AxelleDRouge]

Woah that was so well explained, while still being entertaining!
thanks, I look forward to the next part of this series