Mini #1, 15451 Fall 2010 ======================== This mini is due via *email* to your TA, by 11:59pm Tuesday, August 31. Please use the subject line "15-451 MINI #1" in your email. Make sure to include your andrew id in the email. Problem 1: Suppose that we have 3 functions: f(n), g(n) and h(n) such that f(n) = O(g(n)) and f(n) = O(h(n)). Does it follow that g(n) = O(h(n))? Explain why or give a counterexample showing why not. Problem 2: Given three functions f(n), g(n) and h(n) such that f(n) = Theta(g(n)) and g(n) = O(h(n)), (a) Is h(n) = Omega(f(n))? Provide a brief (one/two sentence) justification of your answer. (b) Given another function k(n) such that h(n) = Omega(k(n)), show that it is not possible to relate k(n) and f(n) using one of the above asymptotic notations (O, Omega, o, Theta) by providing two specific examples of functions f,h,and k such that f(n) = o(k(n)) in the first example and k(n) = o(f(n)) in the second example (remember that both examples must satisfy h(n) = Omega(k(n)) and h(n) = Omega(f(n))). Example 1: f(n) = ___, h(n) = ___, k(n) = ___ Example 2: f(n) = ___, h(n) = ___, k(n) = ___ Problem 3: For each pair of functions below, list ALL applicable from the following: f(n) = o(g(n)), g(n) = o(f(n)), f(n) = Theta(g(n)), f(n) = O(g(n)), g(n) = O(f(n)). "ln" is log-base-e, and "lg" is log-base-2. "^" is "to the power" (a) f(n)=(lg(n))^(3/2) g(n)=lg(n^(3/2)) (b) f(n)=ln(n) g(n)=lg(n) (c) f(n)=n^2 g(n)=2^((1/2)lg(n))