Introduction to Asymptotic Analysis and Big (O)

In this lesson, you will learn about asymptotic notation, an important tool applied to the analysis of algorithms.

We have seen that the time complexity of an algorithm can be expressed as a polynomial. To compare two algorithms, we can compare the respective polynomials. However, the analysis performed in the previous lessons is a bit cumbersome and would become intractable for bigger algorithms that we tend to encounter in practice.

Asymptotic analysis

If the input size is really small, how bad can a poorly designed algorithm get, right? Therefore, mathematicians have a tool for this sort of analysis called the asymptotic notation. The asymptotic notation compares two functions say, f(n)f(n) and g(n)g(n) for very large values of nn. This fits in nicely with our need for comparing algorithms for very large input sizes.

Big (O) notation

One of the asymptotic notations is the Big O notation. A function f(n)f(n) is considered O(g(n))O(g(n)) (read as big oh of g(n)g(n)) if there exists some positive real constant cc and an integer n0>0n_0 > 0, such that the following inequality holds for all nn0n \geq n_0:

f(n)cg(n)f(n) \leq cg(n)

The following graph shows a plot of a function f(n)f(n) and cg(n)cg(n) that demonstrates this inequality.

Level up your interview prep. Join Educative to access 70+ hands-on prep courses.