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Asymptotic Notation


What is Asymptotic Notation?
Asymptotic Notations are algorithms growth rate all languages allow us to analyze algorithms running time by identifying its behavior as an input size for the algorithms increases.

 Common Rate of Growth
Comparing some efficiency of algorithms. We need some common growth rates. Let's assume n in the size of input and c some constant.
The following are the common rate of growth :
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
When analyzing function considering the higher limit part of the function i.e linear function an + b here n is a large value with constant a.
Logarithmic - log n 
Linear  - n 
Quadratic - n^2
Polynomial - n^z, where z is some constant
Exponential - a^n, where a is some constant

Classifications of algorithms
  1. Big-O Notation: f(n) = Og(n)) if there are two positive constants c and n0 such that |f(n)| < c|g(n)| for all n >= n0. Big-O commonly written as O is an Asymptotic notation for the worst case. It provides as an asymptotic upper bound for the growth rate of the runtime of an algorithm.
  2. Big-Omega: Big-omega commonly written as Ω is an asymptotic notation for the best case or for the floor growth rate of a given function.
  3. Big-Theta: Big-Theta commonly written as Θ is an asymptotic notation for the average case or tight growth rate of a given function

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