---
category: Algorithms & Data Structures
contributors:
    - ["Jake Prather", "http://github.com/JakeHP"]
---

# Asymptotic Notations

## What are they?

Asymptotic Notations are languages that allows us to analyze an algorithm's running time by
identifying its behavior as the input size for the algorithm increases. This is also known as
an algorithm's growth rate. Does the algorithm suddenly become incredibly slow when the input
size grows? Does it mostly maintain it's quick run time as the input size increases?
Asymptotic Notation gives us the ability to answer these questions.

## Are there alternatives to answering these questions?

One way would be to count the number of primitive operations at different input sizes.
Though this is a valid solution, the amount of work this takes for even simple algorithms
does not justify its use.

Another way is to physically measure the amount of time an algorithm takes to complete
given different input sizes. However, the accuracy and relativity (times obtained would
only be relative to the machine they were computed on) of this method is bound to
environmental variables such as computer hardware specifications, processing power, etc.

## Types of Asymptotic Notation

In the first section of this doc we described how an Asymptotic Notation identifies the
behavior of an algorithm as the input size changes. Let us imagine an algorithm as a function
f, n as the input size, and f(n) being the running time. So for a given algorithm f, with input
size n you get some resultant run time f(n). This results in a graph where the Y axis is the
runtime, X axis is the input size, and plot points are the resultants of the amount of time
for a given input size.

You can label a function, or algorithm, with an Asymptotic Notation in many different ways.
Some examples are, you can describe an algorithm by it's best case, worse case, or equivalent case.
The most common is to analyze an algorithm by it's worst case. This is because if you determine an
algorithm's run time or time complexity, by it's best case, what if it's best case is only obtained
given at a low, unrealistic, input size? It is equivalent to having a 5 meter sprinting race.
That isn't the best measurement.

### Types of functions, limits, and simplification
```
Logarithmic Function - log n
Linear Function - an + b
Quadratic Function - an^2 + bn + c
Polynomial Function - an^z + . . . + an^2 + a*n^1 + a*n^0, where z is some constant
Exponential Function - a^n, where a is some constant
```

These are some basic function growth classifications used in various notations. The list starts at the least
fast growing function (logarithmic) and goes on to the fastest growing (exponential). Notice that as 'n', or the input,
increases in each of those functions, the result clearly increases much quicker in quadratic, polynomial, and
exponential, compared to logarithmic and linear.

One extremely important note is that for the notations about to be discussed you should do your best to use simplest terms.
This means to disregard constants, and lower order terms, because as the input size (or n in our f(n)
example) increases to infinity (mathematical limits), the lower order terms and constants are of little
to no importance. That being said, if you have constants that are 2^9001, or some other ridiculous
unimaginable amount, realize that simplifying will skew your notation accuracy.

Since we want simplest form, lets modify our table a bit...
```
Logarithmic - log n
Linear - n
Quadratic - n^2
Polynomial - n^z, where z is some constant
Exponential - a^n, where a is some constant
```

### Big-Oh
Big-Oh, commonly written as O, is an Asymptotic Notation for the worst case, or ceiling of growth
for a given function. Say f(n) is your algorithm runtime, and g(n) is an arbitrary time complexity
you are trying to relate to your algorithm. f(n) is O(g(n)), if for any real constant c (c>0),
f(n) <= c g(n) for every input size n (n>0).

*Example 1*  
```
f(n) = 3log n + 100  
g(n) = log n
```

is f(n) O(g(n))?  
is 3 log n + 100 O(log n)?  
Let's look to the definition of Big-Oh.  
3log n + 100 <= c * log n  
Is there some constant c that satisfies this for all n?  
3log n + 100 <= 150 * log n, n > 2 (undefined at n = 1)  
Yes! The definition of Big-Oh has been met therefore f(n) is O(g(n)).

*Example 2*  
```
f(n) = 3*n^2  
g(n) = n
```

is f(n) O(g(n))?  
is 3*n^2 O(n)?  
Let's look at the definition of Big-Oh.  
3*n^2 <= c * n  
Is there some constant c that satisfies this for all n?  
No there isn't, f(n) is NOT O(g(n)).

### Big-Omega
Big-Omega, commonly written as Ω, is an Asymptotic Notation for the best case, or a floor growth rate
for a given function.

f(n) is Ω(g(n)), if for any real constant c (c>0), f(n) is >= c g(n) for every input size n (n>0).

Feel free to head over to additional resources for examples on this. Big-Oh is the primary notation used
for general algorithm time complexity.

### Ending Notes
It's hard to keep this kind of topic short, and you should definitely go through the books and online
resources listed. They go into much greater depth with definitions and examples.
More where x='Algorithms & Data Structures' is on it's way; we'll have a doc up on analyzing actual
code examples soon.

## Books

* [Algorithms](http://www.amazon.com/Algorithms-4th-Robert-Sedgewick/dp/032157351X)
* [Algorithm Design](http://www.amazon.com/Algorithm-Design-Foundations-Analysis-Internet/dp/0471383651)

## Online Resources

* [MIT](http://web.mit.edu/16.070/www/lecture/big_o.pdf)
* [KhanAcademy](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/asymptotic-notation)