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Unformatted text preview: CAS 702 – Data Structures and Algorithms Midterm Exam- 30 points, 4 problems, 2 pages October 11, 2007 Justify your answers Name: Student number: Signature: Problem 1 a. Find a theta notation, i.e. Θ( ... ), for the number of times the statements i = i + i is executed. 2p i = 2 k = 1 while ( k < n ) k := i × i i := i + i b. Let f ( n ) and g ( n ) be asymptotically positive functions. Prove or disprove the follow- ing conjecture: f ( n ) = O ( g ( n )) implies 2 f ( n ) = O (2 g ( n ) ). 2p c. For f ( n ) = 1 + ( n sin( nπ 2 )) 2 , and g ( n ) = 3 n , determine whether f ( n ) = O ( g ( n )) and whether g ( n ) = O ( f ( n )). 2p d. Fibonacci numbers are defined by the recurrence: F 1 = F 2 = 1, and F i +2 = F i +1 + F i . Prove that F i = Θ( α i ) where α = 1+ √ 5 2 is the positive root of x 2- x- 1 = 0. (Hint: you may prove by induction that α i- 2 ≤ F i ≤ α i- 1 for i ≥ 2) 2p Problem 2 a. Give the best and worst running time for Heap sort and Quick sort using theta nota-...
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