Omega Notation (Ω-Notation)

Omega notation represents the lower bound of the running time of an algorithm. Thus, it provides the best case complexity of an algorithm.

The execution time serves as a lower bound on the algorithm’s time complexity.

It is defined as the condition that allows an algorithm to complete statement execution in the shortest amount of time.

Let g and f be the function from the set of natural numbers to itself. The function f is said to be Ω(g), if there is a constant c > 0 and a natural number n0 such that c*g(n) ≤ f(n) for all n ≥ n0

Mathematical Representation of Omega notation :

Ω(g(n)) = { f(n): there exist positive constants c and n0 such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 }

Let us consider the same Insertion sort example here. The time complexity of Insertion Sort can be written as Ω(n), but it is not very useful information about insertion sort, as we are generally interested in worst-case and sometimes in the average case. 

Examples :

{ (n^2+n) , (2n^2) , (n^2+log(n))} belongs to Ω( n^2)
U { (n/4) , (2n+3) , (n/100 + log(n)) } belongs to Ω(n)
U { 100 , log (2000) , 10^4 } belongs to Ω(1)

Note: Here, U represents union, we can write it in these manner because Ω provides exact or lower bounds.

We have discussed Asymptotic Analysis, and Worst, Average, and Best Cases of Algorithms. The main idea of asymptotic analysis is to have a measure of the efficiency of algorithms that don’t depend on machine-specific constants and don’t require algorithms to be implemented and time taken by programs to be compared. Asymptotic notations are mathematical tools to represent the time complexity of algorithms for asymptotic analysis.

Asymptotic Notations:

• Asymptotic Notations are mathematical tools used to analyze the performance of algorithms by understanding how their efficiency changes as the input size grows.
• These notations provide a concise way to express the behavior of an algorithm’s time or space complexity as the input size approaches infinity.
• Rather than comparing algorithms directly, asymptotic analysis focuses on understanding the relative growth rates of algorithms’ complexities.
• It enables comparisons of algorithms’ efficiency by abstracting away machine-specific constants and implementation details, focusing instead on fundamental trends.
• Asymptotic analysis allows for the comparison of algorithms’ space and time complexities by examining their performance characteristics as the input size varies.
• By using asymptotic notations, such as Big O, Big Omega, and Big Theta, we can categorize algorithms based on their worst-case, best-case, or average-case time or space complexities, providing valuable insights into their efficiency.

There are mainly three asymptotic notations:

1. Big-O Notation (O-notation)
2. Omega Notation (Ω-notation)
3. Theta Notation (Θ-notation)