## Intro

If you have followed through the Ruby on Rails tutorials on this website, then you should have a basic working application with an API to create, read, update, or delete items from your database. Following these concepts, you can build a lot of different applications, since this is essentially what almost every web application does. For example, a website that allows you to see movie ratings and reviews (such as IMDB) follows the same concept:

- Has a database of objects, in this case it stores movies.
- The movies have columns such as name, year, rating or summary.
- There is something that pulls this information from the database and renders it for the public to see.

Even things like Twitter or Airbnb build on these same concepts (they are definitely more complicated than this, but the point is that they are all just applications that interact with a database and render things to users). So if you just build upon this template application, you can make a lot of different inventions. However, when the projects get really big (maybe you end up storing a million books in the database), then the application will naturally take longer to process this. In these cases, you will want to make sure that your code is as optimal as you can make it so that the application can keep running fast and scale better with time. For example, if you are implementing a search function, you don’t want the code to go through a simple For Loop and loop through the millions of books before finding the resulting book.

The next few articles will also go through important concepts such as data structures and algorithms to help get a clearer picture of what efficient code looks like. However, it would be beneficial to first understand what Big O is to know how to evaluate the efficiency of code. This article will introduce you to some of the main points of the Big O notation so that you will have a basic understanding of what it is about.

## Big O

Big O is a mathematical representation that describes how much time and space an algorithm requires as the input grows. We are going to go over 4 different run times that are most commonly seen: O(n), O(n^2), O(log n), and O(1). If you are afraid of math and feel like this makes no sense already, don’t worry. I will go over this in plain English and will give simple examples to illustrate what each of these mean. This also means that the explanations in this article will be very basic and will definitely not go into as much detail as an academic textbook would, so if you are interested in this area then I would recommend taking a longer course on this since it is a very significant area in computer science. That being said, I do believe that even having a basic understanding of this is very valuable and will help you improve the way you code.

### O(n)

When people refer to something as having a run time of O(n), it means that it will take as long as the size of “n” (the variable). For example, the below code would be considered to have O(n) run time:

for (i = 0; i < n; i++) {

console.log("I will run n times. ");

}

If you recall from the Programming article, the for loop will run as long as the condition of “i < n” is true. This means that this function will run 5 times if the value of “n” is 5, and 5000 times if the value of “n” is 5000. This has a “linear” run time because the size of “n” is proportionate to how many times it will run at a 1:1 ratio. So in the Search function example above, if the function loops through a million Book entries in alphabetical order, users will have to wait for the function to run a million times before it reaches the end. In the best case (“maybe the book being searched starts with “A”), it might only have to loop through a handful of times before returning the result. Similarly, in the worst case scenario (maybe a Book’s name is “ZZZZZZ”), then it will run the whole million times before returning the result. Although this can be very fast if the book’s name started with “A”, Big O generally refers to the worst case scenario as its run time. So a simple for loop in this case would be O(n).

### O(n^2)

This is O(n) to the power of 2, which means it runs exponentially slower than O(n). The code below would be considered O(n^2):

for (i = 0; i < n; i++) {

for (j =0; j < n; i++) {

console.log("I will run n * n times. ");

}

}

You can see that there is a for loop nested within a for loop, and both of them depend on the size of “n”. Which means that if n is 10, then both the “j” and “i” loops have to run 10 times each. Since the “j” loop is nested within the “i” loop, the “j” loop has to run 10 times before the “i” value increments by 1. To finish, “i” will also need to run 10 times, which means 10 times 10. The point here is that both loops depend on the same variable and one of the loops is nested in the other. This makes it take exponentially longer to run as the size of n increases.

### O(log n)

This one might be a bit more complicated to understand in terms of the code. However, the concept is much easier to understand. Algorithms that have a O(log n) or logarithmic run time are often regarded as very efficient and scalable. The code below is regarded as O(log n):

function divideByTwo(n) {

if (n == 1) {

console.log(n);

return 1;

} else if (n > 1) {

var previousNumber = divideByTwo(n / 2);

console.log(n);

return n;

}

}

We haven’t went through “recursion” so the above code might look a bit confusing (recursion is essentially a function that calls itself). The code takes the input “n”, prints out the number, then divide it by 2, and print out that number, then divide by 2 and so on until the resulting number is less than 1. You can see that this is scalable because even if the input is a million, if you keep dividing it by 2, the number of times this function has to run is still less than 30. The number of times this function has to run only increases by 1 every time “n” doubles in size.

We implement these kind of functions in real life all the time. For example, when you open up a dictionary and want to find a word, you only need to flip through the dictionary a few times before you find your word even though it may have thousands of pages. If the word is “happy”, you might start by opening the dictionary from the middle. If you reach the “m” section, you automatically flip back (maybe halfway). If this next section starts with “e”, then you flip forward again and repeat until you reach the “h” section to find your word. No matter how big the dictionary is, it is always a doable task because you are able to halve your results with each flip. This is much more efficient than flipping through each page one at a time, similar to what a for loop would do.

### O(1)

This one just means constant time. It doesn’t depend on any input size, the amount of time it runs is always going to be the same. A function like below would be considered constant time:

function doBunchOfThings() {

console.log("Do something. ");

console.log("Do another thing. ");

console.log("Do even more things. ");

console.log("Keep doing things. ");

console.log("Just do one more thing. ");

}

No matter how many console.log() functions is fit in there, it is always going to be O(1) run time because there is no variable input.

There are a lot more notations out there and definitely more to analyzing run times, however, these are the essential concepts that should give you a decent understanding of how to determine the run time of a function.

For a chart of some different Big O notations and their run times, please see this link: http://bigocheatsheet.com/

#### Navigation

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