Prove that if f (n) = O (g (n)) and g (n) = Ohm (h (n)),...

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1. Prove that if f(n) = 0(g(n)) and g(n) = Ω(h(n), it is not necessarily true that f(n) = 0(h(n)). You may assume that low degree (i.e., low- exponent) polynomials do not dominate higher degree polynomials, while higher degree polynomials dominate lower ones. For example, n3メ0(n2), but n2 = 0(n3). 2. Prove that if f(n) = 0(g(n)) and g(n) = Ω(h(n), it is not necessarily true that f(n) = Ω(h(n)). 3. Use the properties of Big-Oh, Big-Omega, and Big-Theta discussed in class to prove that if f(n) n Ign-10n + 42 and g(n) = Θ(Vf(n)), then g(n)2-29(n) + 1-Θ(n Ign). You may assume that 1 = 0(n lg n), lg(n) = Ω(1), and g(n) = Ω(1).

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Answer 1

Answer #1

Asymptotic Notation is the languages used to analyze an algorithm’s running time by knowing its behavior as the input size for the algorithm increases. This is also known as an algorithm’s growth rate. Asymptotic basically measure the efficiency of algorithms. Types of asymptotic notation :

Big-O
Big-Omega
Small- o
Small - Omega
Theta

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