Big O Notation: Understanding  Time Complexity using Flowcharts 
Tags: beginners, javascript
Published: 2020-08-30 21:46:42

I highly recommend Edison's post on Big-O complexity in JavaScript. It's the friendliest article I've seen on the topic. {% link https://dev.to/edisonpebojots/data-structure-algorithm-chapter-01-2538 %}

I'll be taking points from Edison here as I visualize Big-O time complexity with flowcharts.

##O log(n)
**Logarithmic Time**
![log time](https://dev-to-uploads.s3.amazonaws.com/i/vgkhh6w9wp1oh9mxcjnv.png)

The way I visually understand time complexity is by looking at the iterator, `i*2` for example , and looking at how many loops the function has. 
##O(n)
**Linear Time**
![linear time](https://dev-to-uploads.s3.amazonaws.com/i/lxd2z7wwm33ai75guffs.png)
Linear time and logarithmic time look similar but the output is different because of the conditions of the loop. `exampleLogarithmic(100)` will return `1, 2, 4, 8, 16, 32, 64`, whereas `exampleLinear(100)` simply loops through all positive integers under 100. 
##O(n^2)
**Quadratic Time**
![quadratic time](https://dev-to-uploads.s3.amazonaws.com/i/3ahf3mv89rd3it5pri17.png)
The number of loops coincides with the exponent which `n` is raised to. You can literally see the function grow bigger as time complexity increases.
##O(n^3)
**Cubic Time**
![cubic time](https://dev-to-uploads.s3.amazonaws.com/i/gt3d7cfim46z63w8qpp8.png)
This isn't the only way to understand time complexity, but it is really helpful to literally see the function grow longer as time complexity increases. Sometimes code written in black and white `<pre>` blocks doesn't get the point across to visual learners.

Now let's have a quiz. What is the time complexity of this function?

Make your guess...
![Alt Text](https://dev-to-uploads.s3.amazonaws.com/i/2brt9wkn8lov5fazu8q8.png)
It's linear! I can tell because there's one loop and the iterator doesn't cause the loop to skip over any integers.

What is the time complexity of this function?
![Alt Text](https://dev-to-uploads.s3.amazonaws.com/i/baqtxsbi1orwipiooi63.png)
Don't doubt yourself. Although this is a bit different from the first examples, it has linear time complexity.

What is the time complexity of this function?
![linear](https://dev-to-uploads.s3.amazonaws.com/i/0q0yf4jkwxey80xs1zic.png)
You may see a pattern here. It's linear!

Now, if you've been following my train of logic, this may be a trick question:
![Alt Text](https://dev-to-uploads.s3.amazonaws.com/i/coxkx17qi1o533gmfgcs.png)

I said that the number of loops denoted the exponent n is raised to. So why is does this have linear time complexity and not quadratic?

This would have quadratic time complexity if it showed a `for` loop inside of another `for` loop. However, one `for` loop that runs *after* another `for` loop does not have quadratic but rather linear time complexity.

Okay, so what is the time complexity of this function?
![linear](https://dev-to-uploads.s3.amazonaws.com/i/36ygfhjfl64geb7h6cra.png)
There's nothing tricky here. This has quadratic time complexity.

Now, for your last question - a question that questions all the other questions - what is this function's time complexity?
![Alt Text](https://dev-to-uploads.s3.amazonaws.com/i/fkyg8nteqakcthhpa37v.png)
I hope you're looking at the conditions of the for loop as well as the sheer number of loops. This has quadratic time complexity because of the loop condition `i<n*n`.


I generated the images in this post with my [app](https://jsvisual.netlify.app), whose development process I described in another post: 
{% link https://dev.to/ender_minyard/how-i-got-100-on-lighthouse-2icd %}