Use of Recurrence Relation | How to calculate the time complexity of Algorithm

When we are preparing for the job as a software developer in MNC's like Google, Mata, IBM, etc., we usually focus mostly on the data structure and algorithms, and sometimes we fall behind in the analysis of the algorithms.

So it is also a very demanding skill for tech companies to analyze the performance of the algorithms in terms of time and space complexity.
For analyzing the algorithms, we first create the recurrence relation.

What is the recurrence relation in the context of algorithms?

A recurrence relation is a mathematical relation or expression that is used to describe the time complexity of a problem in terms of its solution to smaller instances of the same problem.

A typical recurrence relation is in the form:

T(n) = f(n) + T(h(n))

where T(n) is the time complexity of the problem of size n, f(n) is the time complexity of the current problem, and h(n) is the size of the subproblem being solved.

The recurrence relation is used to calculate the time complexity of various algorithms, like divide-and-conquer or recursive algorithms.
It helps to define the time complexity in general and in a more concise form.

Recurrence Relation

A brief explanation of how the recurrence relation works:

1. Divide and conquer algorithms: In divide and conquer algorithms, we divide the problems into subproblems, and after that, we solve and produce the results by combining the results of the subproblems.

2. Algorithms Analysis: The recurrence relation is the mathematical framework used to calculate the time complexity of the problem. By using that, we can get the overall time complexity of the algorithms.

3. Algorithms Design: Recurrence relations work as a guide during the design of any algorithms; they provide the analysis of the optimization of the algorithms.

4. Time complexity analysis: It helps to find the time complexity of the algorithms by using mathematical formulas and techniques and provides closed-form or asymptotic bounds on the time complexity.

So the conclusion for the above is that the recurrence relation is the mathematical relation that helps us design and analyze the algorithms. It provides a concise and general form for analyzing the time complexity of algorithms.
It provides a view of how complexity grows with the size of the input data in the analysis and design of the algorithms.