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midterm_702_07_sol - Solutions for CAS 702 Midterm 2007...

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Solutions for CAS 702 Midterm 2007 Problem 1a Let k j and i j be the values of k and i respectively after statement i := i + i is executed j times. Then we observe that i 0 = 2 , k 0 = 1 and i j = 2 j +1 , k j = i 2 j - 1 = 2 2 j . The loop ends when k j = 2 2 j n , or j lg n 2 . So j = lg n 2 = Θ(lg n ). Problem 1b The conjecture is false. A counter example is f ( n ) = 2 n and g ( n ) = n . Problem 1c Both are false. The curve f ( n ) keeps fluctuating and touching the curves of a Θ( n 2 ) and a Θ(1) function. Problem 1d We have 1 + α = α 2 since α is a root of x 2 - x - 1 = 0. We show that α i - 2 F i α i - 1 for i 2 by induction. It hold for i = 2 (base case). Assume the inequalities hold for any k smaller than i . Then, F i = F i - 1 + F i - 2 α i - 2 + α i - 3 = α i - 3 ( α + 1) = α i - 3 α 2 = α i - 1 . Similarly we can prove that α i - 2 F i . It follows immediately that F i = Θ( α i ). Problem 2a The best and worst running time of Heap sort are both Θ( n lg n ) (refer to Exercise 6.4-5). The best running time of Quick sort is n lg n (balanced partitioning),
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