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quiz1-practicesol

# quiz1-practicesol - Introduction to Algorithms...

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Introduction to Algorithms 21 February 2005 Massachusetts Institute of Technology 6.046J/18.410J Professors Charles E. Leiserson and Ronald L. Rivest Practice Quiz 1 Solutions Practice Quiz 1 Solutions Problem 1. Recurrences Solve the following recurrences by giving tight Θ -notation bounds. (a) T ( n ) = 3 T ( n/ 5) + lg 2 n Solution: By Case 1 of the Master Method, we have T ( n ) = Θ( n log 5 (3) ) . (b) T ( n ) = 2 T ( n/ 3) + n lg n Solution: By Case 3 of the Master Method, we have T ( n ) = Θ( n lg n ) . (c) T ( n ) = T ( n/ 5) + lg 2 n Solution: By Case 2 of the Master Method, we have T ( n ) = Θ(lg 3 n ) . (d) T ( n ) = 8 T ( n/ 2) + n 3 Solution: By Case 2 of the Master Method, we have T ( n ) = Θ( n 2 log n ) . (e) T ( n ) = 7 T ( n/ 2) + n 3 Solution: By Case 3 of the Master Method, we have T ( n ) = Θ( n 3 ) . (f) T ( n ) = T ( n - 2) + lg n Solution: T ( n ) = Θ( n log n ) . This is n/ 2 i =1 lg 2 i n/ 2 i =1 lg i ( n/ 4)(lg n/ 4) = Ω( n lg n ) . For the upper bound, note that T ( n ) S ( n ) , where S ( n ) = S ( n - 1)+lg n , which is clearly O ( n lg n ) .

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6.046J/18.410J Practice Quiz 1 Solutions Name 2 Problem 2. True or False, and Justify Circle T or F for each of the following statements, and briefly explain why. The better your argument, the higher your grade, but be brief. Your justification is worth more points than your true-or-false designation. (a) T F If f ( n ) does not belong to the set o ( g ( n )) , then f ( n ) = Ω( g ( n )) . Solution: False. For example f ( n ) = n sin( n ) and g ( n ) = n cos( n ) . (b) T F A set of n integers in the range { 1 , 2 , . . . , n } can be sorted by R ADIX -S ORT in O ( n ) time by running C OUNTING -S ORT on each bit of the binary representation. Solution: False. This results in Θ(log n ) iterations of counting sort, and thus an overall running time of Θ( n log n ) . (c) T F An adversary can construct an input of size n to force R ANDOMIZED -M EDIAN to run in Ω( n 2 ) time. Solution: False. The expected running time of R ANDOMIZED M EDIAN is Θ( n ) . This applies to any input. Problem 3. Short Solution Give brief , but complete, solutions to the following questions. (a) Consider any priority queue (supporting I NSERT and E XTRACT -M AX operations) in the comparison model. Explain why there must exist a sequence of n operations such that at least one operation in the sequence requires Ω(lg n ) time to execute.
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quiz1-practicesol - Introduction to Algorithms...

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