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Chapter 1: Problem 21

Discuss the reflexive, symmetric, and transitive properties for asymptotic comparisons \((O, \Omega, \Theta, o)\)

Short Answer

Expert verified
Big-O, \(\Omega\), and \(\Theta\) follow the reflexive and transitive properties. \(\Theta\) also follows the symmetric property. Little-o does not follow the reflexive property but does follow the transitive.

Step by step solution

01

Understanding the properties

Before discussing whether these properties apply to the asymptotic notations or not, let's understand what these properties mean. Reflexive property means that for any set 'a', \(aRa\) holds true. Symmetric property means that for any set 'a' and 'b', if \(aRb\) holds true, then \(bRa\) should also be true. Transitive property means that if \(aRb\) and \(bRc\) are true, then \(aRc\) should be true.
02

Discussing the Reflexive Property

In terms of Big-O notation, for any function 'f', we can say \(f = O(f)\). Similarly \(\Omega\) and \(\Theta\) notations satisfy the reflexive property, as \(f = \Omega(f), f = \Theta(f)\). However, little-o notation does not satisfy, because for \(f = o(f)\) to be true, \(f\) should be strictly less than 'f', which clearly is not.
03

Discussing the Symmetric Property

In terms of Big-O notation, if \(f = O(g)\), then it doesn't necessarily mean that \(g = O(f)\). So, Big-O doesn't satisfy the symmetric property. Same goes for \(\Omega\) and little-o notation. However, for \(\Theta\) notation, if \(f = \Theta(g)\), then it means that \(g = \Theta(f)\). Hence, \(\Theta\) follows the symmetric property.
04

Discussing the Transitive Property

In terms of Big-O notation, if \(f = O(g)\) and \(g = O(h)\), then \(f = O(h)\). Therefore, Big-O follows the transitive property. Same goes for \(\Omega\) and \(\Theta\) notation and little-o notation. All follow the transitive property.

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Most popular questions from this chapter

Give a \(\Theta(n \text { lg } n)\) algorithm that computes the reminder when \(x^{n}\) is divided by \(p .\) For simplicity, you may assume that \(n\) is a power of 2 That is, \(n=2^{k}\) for some positive integer \(k\)

Write an algorithm that finds both the smallest and largest numbers in a list of \(n\) numbers. Try to find a method that does at most \(1.5 n\) comparisons of array items.

Justify the correctness of the following statements assuming that \(f(n)\) and \(g(n)\) are asymptotically positive functions. a. \(f(n)+g(n) \in O(f(n)), g(n))\) b. \(f^{2}(n) \in \Omega(f(n))\) c. \(f(n)+o(f(n)) \in \Theta(f(n), \text { where of } o(f(n))\) means any function \\[g(n) \in o(f(n))\\]

There are two algorithms called Algl and Alg2 for a problem of size n. Algl runs in \(n^{2}\) microseconds and Alg2 runs in 100 n log \(n\) microseconds. Algl can be implemented using 4 hours of programmer time and needs 2 minutes of CPU time. On the other hand, Alg2 require 15 hours of programmer time and 6 minutes of CPU time. If programmers are paid 20 dollars per hour and CPU time costs 50 dollars per minute, how many times must a problem instance of size 500 be solved using Alg2 in order to justify its development cost?

Suppose you have a computer that requires 1 minute to solve problem instances of size \(n=1,000 .\) Suppose you buy a new computer that runs 1,000 times faster than the old one. What instance sizes can be run in 1 minute, assuming the following time complexities \(T(n)\) for our algorithm? a. \(T(n)=n\) b. \(T(n)=n^{3}\) c. \(T(n)=10^{n}\)
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