Chapter 4, Problem 8C

Explanation of Solution

Big-O:

A function $g\left(n\right)$ is $O\left(f\left(n\right)\right)$ if there exists a constant $c>0$ such that “$\text{cf}\left(\text{n}\right)$ grows faster than $g\left(n\right)$ for all $n>=n_0$ , that is, $g\left(n\right)$ is $O\left(f\left(n\right)\right)$ iff for some constants “c” and “$n_0$”, $g(n)<=cf(n)$ for all $n>=n_0$. Here, $\text{f}\left(\text{n}\right)$ denotes upper bound for $g\left(n\right)$. The important points for above statements are as follows:

• “n” is the data set size.
• $\text{f}\left(\text{n}\right)$ denotes a function that is calculated while taking n as the parameter.
• $O\left(f\left(n\right)\right)$ means that curve denoted by $\text{f}\left(\text{n}\right)$ is an upper bound for a function’s need for resources.

Big-Omega notation:

The big Omega notation is used to describe the performance or complexity of an algorithm. In asymptotic notation for lower bound, let “f” and “g” be functions from the integers or the real numbers to the real numbers. It means that $\Omega \left(g\left(n\right)\right)= f\left(n\right)$ if there exist positive constants “$c$”$\left(c>0\right)$ in real numbers, such that $0\le cg\left(n\right)\le f\left(n\right)$ for all $n\ge n_0$.

Example of positive function f(n) is neither $O\left(n\right)$ nor $\Omega (n)$:

Assume the below mentioned function as an example to prove the given statement