Asymptotic Notation Demystified: Guide for Beginners

Asymptotic notation is a widely accepted term for algorithm efficiency analysis. As a function’s input approaches infinity, asymptotic notation, a type of mathematical notation, is used to explain the limiting behaviour of the function. The performance of algorithms and data structures, particularly in terms of their time and space complexity, is frequently examined in computer science.

Three types of asymptotic notation are frequently used:

asymptotic notation
Types of asymptotic notation

Big O notation (O): Big O notation expresses the maximum growth rate of a function. It is used to define the maximum amount of time or available space that an algorithm or data structure can consume and represents the worst-case scenario for the performance of an algorithm or data structure.

Big Omega notation (Ω): The bottom bound of a function’s growth rate is shown in the notation omega. It is used to define the minimum amount of time or storage that an algorithm or data structure needs to finish a task. It depicts the performance of an algorithm or data structure at its best.

Big Theta notation (Θ): The average-case scenario for a function’s growth rate is shown in theta notation. It describes the typical amount of time or space needed by an algorithm or data structure and represents the precise growth rate of a function.

Programmers and computer scientists can rapidly and correctly compare the performance of various algorithms or data structures and select the most suitable one for a given task or problem by utilising asymptotic notation.

Suggested Reading: Approaches for Efficiency Analysis of Algorithm

How to find asymptotic notation?

You must ascertain a function’s growth rate as the input size gets closer to infinity in order to calculate its asymptotic notation. The general procedures are as follows:

Find the dominating term: Throughout the function, look for the term that expands the quickest as the input size increases. The function’s total growth rate is determined by this dominant term.

Removing any constant influences is necessary since asymptotic notation only takes the function’s development rate into account, not the particular constants at play. Hence, any constant components of the dominant term can be removed.

Choose the right notation: After determining the dominating term and removing any constants, select the right asymptotic notation. When expressing the upper bound, the lower bound, and the actual growth rate, use the Big Oh, Big Omega and Big Theta notation, respectively.

Consider the function f(n) = 8n2 + 15n + 10, for instance. As n gets closer to infinity, the main term in this function, 8n2, rises more quickly than 15n and 10 do. The constant factor of three can be removed, leaving n2. As a result, this function’s asymptotic notation is O(n2) or (n2).

It’s crucial to keep in mind that figuring out asymptotic notation takes some mathematical analysis and knowledge of growth rates. In rare circumstances, it could be required to calculate a function’s growth rate using mathematical methods like limits, derivatives, and integrals.

Suggested Reading: How to Analyze Algorithm – Big Oh, Omega, and Theta Notations

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