Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

• such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are
  • Big-O.
  • Big Theta

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(n)” 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 size of data set
• “f(n)” denotes a function calculated while taking n as 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 exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

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

Explanation of Solution

b) $f\left(n\right)+g\left(n\right) is\Theta (min(f(n),g(n)))$

Explanation:

• Take f(n) = n and g(n) = 1/n.
• Now, $f\left(n\right)+g\left(n\right)\&$

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

• The function 2^na is not $O\left(2^n\right)$, as there is no constant $c \ge 2^{na}/2^n=2^{n(a-}$