final-s2008-sol-1

final-s2008-sol-1 - Introduction to Algorithms May 21, 2008...

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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 Solutions Final Exam Solutions Problem 1. Asymptotics [10 points] For each pair of functions f ( n ) and g ( n ) in the table below, write O , Ω , or Θ in the appropriate space, depending on whether f ( n ) = O ( g ( n )) , f ( n ) = Ω( g ( n )) , or f ( n ) = Θ( 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 base-2 logarithm. Solution: n n lg n n 2 n lg 2 n Ω Ω O 2 lg 2 n Ω Ω Ω lg( n !) Ω Θ O n lg 3 Ω Ω O 6.006 Final Exam Solutions Name 2 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: Solution: True. The first recurrence leads to a result that is polynomial in n , while the second recurrence produces a result that is exponential in n . (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) . True False Explain: Solution: False. Considering the recursion trees for A ( n ) and B ( n ) , it is easy to see that the tree for A has both a smaller height ( log 4 ( n ) vs. log 2 ( n ) ), and a smaller branching factor. 6.006 Final Exam Solutions Name 3 (c) Radix sort works in linear time only if the elements to sort are integers in the range { , 1 ,...,cn } for some c = O (1) . True False Explain: Solution: False. Radix sort also works in linear time if the elements to sort are integers in the range { 1 ,...,n d } for any constant d . (d) Given an undirected graph, it can be tested to determine whether or not it is a tree in O ( V + E ) time. A tree is a connected graph without any cycles. True False Explain: Solution: True. Using either DFS or BFS yields a running time of O ( V + E ) . 6.006 Final Exam Solutions Name 4 (e) The Bellman-Ford algorithm applies to instances of the single-source shortest path problem which do not have a negative-weight directed cycle, but it does not detect the existence of a negative-weight directed cycle if there is one. True False Explain: Solution: False. Bellman-Ford detects negative-weight directed cycles in its input graph. (f) The topological sort of an arbitrary directed acyclic graph G = ( V,E ) can be com- puted in linear time....
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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-s2008-sol-1 - Introduction to Algorithms May 21, 2008...

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