final-s2011

final-s2011 - May 19, 2011 6.006 Spring 2011 Final Exam...

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Introduction to Algorithms May 19, 2011 Massachusetts Institute of Technology 6.006 Spring 2011 Professors Erik Demaine, Piotr Indyk, and Manolis Kellis Final Exam Final Exam Do not open this exam booklet until 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. You have 180 minutes to earn 180 points. 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. This exam booklet contains 20 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 handwritten, 8 1 2 00 × 11 00 or A4 crib sheets (both sides). No calculators or programmable devices are permitted. No cell phones or other communications 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. 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. 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 10 30 7 10 2 8 40 8 10 3 3 15 9 15 4 10 10 10 5 10 11 20 6 10 Total 180 Name: Athena username: Recitation: Nick WF10 Nick WF11 Tianren WF12 David WF1 Joe WF2 Joe WF3a Michael WF3b
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6.006 Final Exam Name 2 Problem 1. True or false [30 points] (10 parts) For each of the following questions, circle either T (True) or F (False). Explain your choice. (Your explanation is worth more than your choice of true or false.) (a) T F For all positive f ( n ) , f ( n ) + o ( f ( n )) = Θ( f ( n )) . (b) T F For all positive f ( n ) , g ( n ) and h ( n ) , if f ( n ) = O ( g ( n )) and f ( n ) = Ω( h ( n )) , then g ( n ) + h ( n ) = Ω( f ( n )) .
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6.006 Final Exam Name 3 (c) T F Under the simple uniform hashing assumption, the probability that three specific data elements (say 1 , 2 and 3 ) hash to the same slot (i.e., h (1) = h (2) = h (3) ) is 1 /m 3 , where m is a number of buckets. (d) T F Given an array of n integers, each belonging to {- 1 , 0 , 1 } , we can sort the array in O ( n ) time in the worst case.
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6.006 Final Exam Name 4 (e) T F The following array is a max heap: [10 , 3 , 5 , 1 , 4 , 2] . (f) T F R ADIX S ORT does not work correctly (i.e., does not produce the correct output) if we sort each individual digit using I NSERTION S ORT instead of C OUNTING S ORT .
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6.006 Final Exam Name 5 (g) T F Given a directed graph G , consider forming a graph G 0 as follows. Each vertex u 0 G 0 represents a strongly connected component (SCC) of G . There is an edge
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This note was uploaded on 01/20/2012 for the course CS 6.006 taught by Professor Erikdemaine during the Fall '08 term at MIT.

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final-s2011 - May 19, 2011 6.006 Spring 2011 Final Exam...

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