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Unformatted text preview: UMass Lowell CS Fall, 2001 Sample Review Questions & Answers for 91.404 Material I : Function Order of Growth (20 points) 1 . (5 points) Given the following list of 3 functions: 3 (n 2 ) 25 + ( 8 n ) (n lg n)  6 Circle the one ordering below in which the 3 functions appear in increasing asymptotic growth order. That is, find the ordering f 1 , f 2 , f 3 , such that f 1 = O(f 2 ) and f 2 = O(f 3 ) . (a) 3 (n 2 ) 25 + ( 8 n ) (n lg n)  6 (b) 25 + ( 8 n ) (n lg n)  6 3 (n 2 ) (c) (n lg n)  6 3 (n 2 ) 25 + ( 8 n ) 2. (5 points) Given the following list of 3 functions: (1/6)n! (2 lg n ) + 5 9 (lg( lg n)) Circle the one ordering below in which the 3 functions appear in increasing asymptotic growth order. That is, find the ordering f 1 , f 2 , f 3 , such that f 1 = O(f 2 ) and f 2 = O(f 3 ) . (a) (1/6)n! (2 lg n ) + 5 9 (lg( lg n)) (b) (2 lg n ) + 5 9 (lg( lg n)) (1/6)n! (c) 9 (lg( lg n)) (2 lg n ) + 5 (1/6)n! (page 1 of 37) UMass Lowell CS Fall, 2001 Sample Review Questions & Answers for 91.404 Material For problems 3 and 4, assume that: f 1 = Ω (n lgn) f 2 = Ο ( n 2 ) f 3 = Θ (n) f 4 = Ω (1) For each statement: Circle TRUE if the statement is true. Circle FALSE if the statement is false. Circle only one choice. 3. (5 points) f 1 = Ω ( f 3 ) TRUE FALSE 4. (5 points) f 3 = Ο ( f 2 ) TRUE FALSE (page 2 of 37) UMass Lowell CS Fall, 2001 Sample Review Questions & Answers for 91.404 Material II: Solving a Recurrence (10 points) In each of the 3 problems below, solve the recurrence by finding a closedform function f(n) that represents a tight bound on the asymptotic running time of T(n) . That is, find f(n) such that T(n) = Θ (f(n)) . 1 . (5 points) Solve the recurrence : T(n) = T(n/4) + n/2 [You may assume that T(1) = 1 and that n is a power of 2.] Circle the one answer that gives a correct closedform solution for T(n) (a) Θ (n) (b) Θ (n 2 ) (c) Θ (n lg n) 2 . (5 points) Solve the recurrence : T(n) = n T( n ) + n [You may assume that T(2) = 1 and that n is of the form 2 2 .] Circle the one answer that gives a correct closedform solution for T(n) (a) Θ (n 2 lg n) (b) Θ (n lg 2 n) (c) Θ (n lg(lg n)) (page 3 of 37) k UMass Lowell CS Fall, 2001 Sample Review Questions & Answers for 91.404 Material III: PseudoCode Analysis (30 points) Here you’ll use the pseudocode below for two functions Mystery1 and Mystery3 . Mystery1 has three arguments: A : an array of integers; p, r : indices into A Mystery1 (A, p, r) if p is equal to r then return 0 q (p+r)/2 g1 Mystery1(A, p, q) print "Mystery1 result1= ", g1 print contents of A[p]..A[q] g2 Mystery1(A, q+1, r) print "Mystery1 result2= ", g2 print contents of A[q+1]..A[r] g3 Mystery3(A, p, q, r) print "Mystery3 result= ", g3 print contents of A[p]..A[r] return g3 Mystery3 (A, p, q, r) initialize integer array B to be empty bb p pp p qq q+1 while pp <= q and qq <= r do if A[pp] <= A[qq] then B[bb] A[pp] pp pp + 1 bb bb + 1 else B[bb] A[qq] qq qq + 1 bb bb + 1 while qq <= r do B[bb] A[qq] qq qq + 1...
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 Fall '09
 DR.KARENDANIELS
 Graph Theory, University of Massachusetts Lowell, Sample Review Questions

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