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## What are they? ## What are they?
Asymptotic Notations is a language that allows us to analyze an algorithm's running time by 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 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 an algorithm's growth rate. Does the algorithm suddenly become incredibly slow when the input
size grows? Does the algorithm mostly maintain it's quick run time as the input size increases? size grows? Does the algorithm mostly maintain it's quick run time as the input size increases?
@@ -42,11 +42,13 @@ given at a low, unrealistic, input size? It is equivalent to having a 5 meter sp
That isn't the best measurement. That isn't the best measurement.
### Types of functions, limits, and simplification ### Types of functions, limits, and simplification
```
Logarithmic Function - log n Logarithmic Function - log n
Linear Function - an + b Linear Function - an + b
Quadratic Function - an^2 + bn + c Quadratic Function - an^2 + bn + c
Polynomial Function - an^z + . . . + an^2 + a*n^1 + a*n^0, where z is some constant 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 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 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, fast growing function (logarithmic) and goes on to the fastest growing (exponential). Notice that as 'n', or the input,
@@ -60,11 +62,13 @@ to no importance. That being said, if you have constants that are 2^9001, or som
unimaginable amount, realize that simplifying will skew your notation accuracy. unimaginable amount, realize that simplifying will skew your notation accuracy.
Since we want simplest form, lets modify our table a bit... Since we want simplest form, lets modify our table a bit...
```
Logarithmic - log n Logarithmic - log n
Linear - n Linear - n
Quadratic - n^2 Quadratic - n^2
Polynomial - n^z, where z is some constant Polynomial - n^z, where z is some constant
Exponential - a^n, where a is some constant Exponential - a^n, where a is some constant
```
### Big-Oh ### Big-Oh
Big-Oh, commonly written as O, is an Asymptotic Notation for the worst case, or ceiling of growth Big-Oh, commonly written as O, is an Asymptotic Notation for the worst case, or ceiling of growth
@@ -104,8 +108,8 @@ f(n) is Ω(g(n)), if for any real constant c (c>0), f(n) is >= c g(n) for every
Feel free to head over to additional resources for examples on this. Big-Oh is the primary notation used Feel free to head over to additional resources for examples on this. Big-Oh is the primary notation used
for general algorithm time complexity. for general algorithm time complexity.
### Ending Note ### Ending Notes
It's hard to keep this kind of topic short and you should definitely go through the books and online 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. 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 More where x='Algorithms & Data Structures' is on it's way; we'll have a doc up on analyzing actual
code examples soon. code examples soon.
@@ -118,4 +122,4 @@ code examples soon.
## Online Resources ## Online Resources
* [MIT](http://web.mit.edu/16.070/www/lecture/big_o.pdf) * [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) * [KhanAcademy](https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/asymptotic-notation)