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Asymptotic Notations are languages 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 it mostly maintain it's quick run time as the input size increases?
Asymptotic Notation gives us the ability to answer these questions. Asymptotic Notation gives us the ability to answer these questions.
## Are there alternatives to answering these questions? ## Are there alternatives to answering these questions?
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Though this is a valid solution, the amount of work this takes for even simple algorithms Though this is a valid solution, the amount of work this takes for even simple algorithms
does not justify its use. does not justify its use.
Another way is to physically measure the amount of time the algorithm takes to complete 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 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 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. environmental variables such as computer hardware specifications, processing power, etc.
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## Types of Asymptotic Notation ## Types of Asymptotic Notation
In the first section of this doc we described how an Asymptotic Notation identifies the 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 our algorithm as a function 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 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 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 runtime, X axis is the input size, and plot points are the resultants of the amount of time
for a given input size. for a given input size.
You can label a function, or algorithm, with an Asymptotic Notation in many different ways. You can label a function, or algorithm, with an Asymptotic Notation in many different ways.
Some examples are, you can describe your algorithm by it's best case, worse case, or equivalent case. Some examples are, you can describe an algorithm by it's best case, worse case, or equivalent case.
The most common is to analyze your algorithm by it's worst case. This is because if you determine an 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 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. given at a low, unrealistic, input size? It is equivalent to having a 5 meter sprinting race.
That isn't the best measurement. That isn't the best measurement.