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Unformatted text preview: 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 = 2 , k = 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 = d lg n 2 e = 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 )....
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This note was uploaded on 10/26/2009 for the course CAS 702 taught by Professor Zera during the Fall '09 term at McMaster University.
- Fall '09