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final-f2008 - 6.006 Fall 2008 Final Examination...

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Introduction to Algorithms December 15, 2008 Massachusetts Institute of Technology 6.006 Fall 2008 Professors Ronald L. Rivest and Sivan Toledo Final Examination Final Examination Do not open this quiz booklet until directed to do so. Read all the instructions on this page. When the quiz begins, write your name on every page of this quiz booklet. You have 180 minutes to earn 200 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 quiz booklet contains 15 pages, including this one. Two extra sheets of scratch paper are attached. Please detach them before turning in your quiz at the end of the exam period. This quiz 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. 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. 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. 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 6 18 6 1 25 2 6 18 7 2 30 3 4 24 8 3 30 4 1 15 9 1 20 5 1 20 Total 200 Name:
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6.006 Final Examination Name 2 Problem 1. Miscellaneous True/False [18 points] (6 parts) For each of the following questions, circle either T (True) or F (False). Explain your choice. (No credit if no explanation given.) (a) T F If the load factor of a hash table is less than 1, then there are no collisions. Explain: (b) T F If SAT P A , then A is NP-hard. Explain: (c) T F The longest common subsequence problem can be solved using an algorithm for finding the longest path in a weighted DAG. Explain:
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6.006 Final Examination Name 3 (d) T F Applying a Givens rotation to a matrix changes at most one row of the matrix. Explain: (e) T F The problem of finding the shortest path from s to t in a directed, weighted graph exhibits optimal substructure. Explain: (f) T F A single rotation is sufficient to restore the AVL invariant after an insertion into an AVL tree. Explain:
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6.006 Final Examination Name 4 Problem 2. More True/False [18 points] (6 parts) For each of the following questions, circle either T (True) or F (False). Explain your choice. (No credit if no explanation given.) (a) T F Using hashing, we can create a sorting algorithm similar to COUNTING - SORT that sorts a set of n (unrestricted) integers in linear time. The algorithm works
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