Chapter 2, Problem 1E

a) f(n) is O(1)

Program Plan Intro

Asymptotic Notations:

For asymptotic analysis of algorithms, some mathematical tools are used to represent time complexity of algorithm, such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are described below:

Big-O:

A function  “g(n)” is $O\left(f\left(n\right)\right)$ if there exists a constant $c>0$ such that “cf(n)” grows faster than $"g\left(n\right)"$ for all $n>=n_0$ , that is, “g(n)” 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, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

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

Big Theta:

A function  “g(n)” is $\Theta (f(n))$ iff two positive constants “c₁” and “c₂” and a positive integer $n_0$ exists such that $c_{1}f\left(n\right)\le g\left(n\right)\le c_{2}f\left(n\right)$ for all $n>=n_0$, that is, “f(n)” denotes the exact bound for “g(n)”, which denotes that the growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

• “n”  is the data set size.
• “f(n)” denotes a function that is been calculated taking n as the parameter.
• $\Theta (f(n))$ means that curve denoted by “f(n)”  is an exact bound for the resource needs of a function.

Explanation of Solution

Explanation:

The function $"f\left(n\right)"isO\left(1\right)$ means that constant “c” denotes the upper bound for the function $"f\left(n\right)"$.

Hence, function $"f\left(n\right)"$ would never exceed the value of a certain constant “c”.

b) f(n) is $\Theta (1)$

Explanation of Solution

Explanation:

The function $"f\left(n\right)"is$ $\Theta (1)$ means that the constant “c” denotes the exact bound for the function $"f\left(n\right)"$.

Hence, function $"f\left(n\right)"$ would be a constant function.

c) f(n) is $n^{O(1)}$

Explanation of Solution

Explanation:

The function $"f\left(n\right)"is$$n^{O(1)}$ means that function $"f\left(n\right)"$ would never exceed the value of power function $n^c$ for any constant “c”.