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Unformatted text preview: Introduction to Algorithms May 21, 2008 Massachusetts Institute of Technology 6.006 Spring 2008 Professors Srini Devadas and Erik Demaine Final Exam Final Exam Do not open this exam booklet until you are directed to do so. Read all the instructions on this page. When the exam begins, write your name on every page of this exam booklet. This exam contains 12 problems, some with multiple parts. You have 180 minutes to earn 180 points. This exam booklet contains 24 pages, including this one. Two extra sheets of scratch paper are attached. Please detach them before turning in your exam at the end of the exam period. This exam is closed book. You may use three 8 1 2 00 11 00 or A4 crib sheets (both sides). No calculators or programmable devices are permitted. Write your solutions in the space provided. If you need more space, write on the back of the sheet containing the problem. Do not put part of the answer to one problem on the back of the sheet for another problem, since the pages may be separated for grading. Do not waste time and paper rederiving facts that we have studied. It is sufficient to cite known results. Do not spend too much time on any one problem. Read them all through first, and attack them in the order that allows you to make the most progress. Show your work, as partial credit will be given. You will be graded not only on the correct ness of your answer, but also on the clarity with which you express it. Be neat. Good luck! Problem Parts Points Grade Grader Problem Parts Points Grade Grader 1 1 10 7 2 10 2 10 40 8 2 10 3 1 10 9 1 10 4 2 10 10 3 15 5 1 10 11 4 20 6 2 15 12 4 20 Total 180 Name: Circle your recitation time: Hueihan Jhuang: (10AM) (11AM) Victor Costan (2PM) (3PM) 6.006 Final Exam Name 2 Problem 1. Asymptotics [10 points] For each pair of functions f ( n ) and g ( n ) in the table below, write O , , or in the appropriate space, depending on whether f ( n ) = O ( g ( n )) , f ( n ) = ( g ( n )) , or f ( n ) = ( g ( n )) . If there is more than one relation between f ( n ) and g ( n ) , write only the strongest one. The first line is a demo solution. We use lg to denote the base2 logarithm. n n lg n n 2 n lg 2 n O 2 lg 2 n lg( n !) n lg 3 6.006 Final Exam Name 3 Problem 2. True or False [40 points] (10 parts) Decide whether these statements are True or False . You must briefly justify all your answers to receive full credit. (a) An algorithm whose running time satisfies the recurrence P ( n ) = 1024 P ( n/ 2) + O ( n 100 ) is asymptotically faster than an algorithm whose running time satisfies the recurrence E ( n ) = 2 E ( n 1024) + O (1) . True False Explain: (b) An algorithm whose running time satisfies the recurrence A ( n ) = 4 A ( n/ 2) + O (1) is asymptotically faster than an algorithm whose running time satisfies the recurrence B ( n ) = 2 B ( n/ 4) + O (1) ....
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 Fall '08
 ErikDemaine
 Algorithms

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