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Chapter 1: Problem 15

Show directly that \(f(n)=n^{2}+3 n^{3} \in \Theta\left(n^{3}\right)\). That is, use the definitions of \(O\) and \(\Omega\) to show that \(f(n)\) is in both \(O\left(n^{3)} \text { and } \Omega\left(n^{3)}\right.\right.\)

Short Answer

Expert verified
The function \(f(n) = n^{2}+3n^{3}\) belongs to both \(O(n^{3})\) and \(\Omega(n^{3})\). Therefore, it is in \(\Theta(n^{3})\).

Step by step solution

01

Proving \(f(n) \in O(n^{3})\)

Define \(f(n)=n^{2} + 3n^{3}\). To show \(f(n) \in O(n^{3})\), we need to find positive constants \(c\) and \(n_{0}\) such that \(0 \leq f(n) \leq c*n^{3}\) for all \(n \geq n_{0}\). Let's select \(c=4\) and \(n_{0}=1\). For \(n \geq n_{0}\), we have \(0 \leq n^{2} + 3n^{3} \leq n^{3} + 3n^{3} = 4n^{3}\). Therefore, \(f(n) = n^{2}+3n^{3} \in O(n^{3})\).
02

Proving \(f(n) \in \Omega(n^{3})\)

To show \(f(n) \in \Omega(n^{3})\), we need to find positive constants \(c\) and \(n_{0}\) such that \(0 \leq c*n^{3} \leq f(n)\) for all \(n \geq n_{0}\). Let's select \(c=1\) and \(n_{0}=1\). For \(n \geq n_{0}\), we have \(0 \leq n^{3} \leq n^{2} + 3n^{3} = f(n)\). Therefore, \(f(n) = n^{2}+3n^{3} \in \Omega(n^{3})\).
03

Conclusion

Since we have shown that \(f(n) = n^{2}+3n^{3}\) is a member of both \(O(n^{3})\) and \(\Omega(n^{3})\), we can conclude that \(f(n) = n^{2}+3n^{3} \in \Theta(n^{3})\).

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Most popular questions from this chapter

Using the Properties of Order in Section \(1.4 .2,\) show that $$5 n^{5}+4 n^{4}+6 n^{3}+2 n^{2}+n+7 \in \Theta\left(n^{5}\right)$$

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

Write an algorithm that prints out all the subsets of three elements of a set of \(n\) elements. The elements of this set are stored in a list that is the input to the algorithm.

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. \(T(n)=10^{n}\)

Show the correctness of the following statements. a. \(\lg n \in O(n)\) b. \(n \in O(n \lg n)\) c. \(n \lg n \in O\left(n^{2}\right)\) d. \(2^{n} \in \Omega\left(5^{\ln n}\right.\) e. \(\lg ^{3} n \in o\left(n^{0.5}\right)\)
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