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Practical Session 2 – “Constants Don’t Matter!!!” Algorithm Analysis There exist c > 0 and n0 > 0 such that: f(n) = O(g(n)) 0 ≤ f(n) ≤ cg(n) for each n ≥ n0 There exist c > 0 and n0 > 0 such that: f(n) = Ω(g(n)) 0 ≤ cg(n) ≤ f(n) for each n ≥ n0 There exist c1, c2 > 0 and n0 > 0 such that: f(n) = Θ(g(n)) 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for each n ≥ n0 Some Basics Prove: 100 * n + 12 = Θ(n). Solution: Prove: . Solution: Question 1 Prove : (log n)log n = Ω(n3/2). Question 2 Prove that log(n!) = Θ(n logn). Question 3 Prove : (log n)! = Ω (log(n!)). Question 4 Give an example for two functions g(n) and f(n) such that neither g(n) = O(f(n)) nor g(n) = Ω(f(n)). Question 5 Prove: For any two functions f(n) and g(n), f(n) +g(n) = Θ(max(f(n),g(n))). Question 6 Prove or disprove: 1. For any functions f(n) and g(n), f(n)=O(g(n))→g(n)= Ω (f(n)). 2. For any function f(n), f(n)=(f(n/2)) . Question 7 Given a sorted array A of n distinct integers. a) Describe an O(1) time algorithm that determines whether or not there exists an element x, such that A[1] < x < A[n] and x is not in A. b) Describe an O(logn) time algorithm that finds such an x. c) Describe an efficient algorithm that determines whether or not there exists an element i in A, such that A[i]=i. Analyze the running time of the algorithm. Question 8 You have two jars made from a special secret material. You are in a tall building with N floors, and you must determine what is the top floor M from which you can drop a jar to the ground so that it doesn't break on impact. You can throw a jar out a window as many times as you want as long as it doesn't break. Describe a strategy for finding the highest safe rung that requires you to drop a jar at most f(n) times, for some function f(n) that grows slower than linearly. (In other words, it should be the case that .

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