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practice_quiz1

# practice_quiz1 - Introduction to Algorithms Massachusetts...

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Introduction to Algorithms October 4, 2005 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik D. Demaine and Charles E. Leiserson Handout 10 6.046 fall 2005 Quiz Review—modified from 6.046 Spring 2005 1. Recurrences Solve the following recurrences by giving tight Θ -notation bounds. You do not need to justify your answers, but any justification that you provide will help when assigning partial credit. (a) T ( n ) = T ( n/ 3) + T ( n/ 6) + Θ( n log n ) (b) T ( n ) = T ( n/ 2) + T ( n ) + n (c) T ( n ) = 3 T ( n/ 5) + lg 2 n (d) T ( n ) = 2 T ( n/ 3) + n lg n (e) T ( n ) = T ( n/ 5) + lg 2 n 3 (f) T ( n ) = 8 T ( n/ 2) + n 3 (g) T ( n ) = 7 T ( n/ 2) + n (h) T ( n ) = T ( n 2) + lg n 2. True or False 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. No points will be given even for a correct solution if no justification is presented. T F For all asymptotically positive f ( n ) , f ( n ) + o ( f ( n )) = Θ( f ( n )) . T F The worst-case running time and expected running time are equal to within constant factors for any randomized algorithm. T F The collection H = { h 1 , h 2 , h 3 } of hash functions is universal, where the three hash functions map the universe { A, B, C, D } of keys into the range { 0 , 1 , 2 } according

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