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Unformatted text preview: Solutions to Midterm 1 Peng Du University of California, San Diego 1 (10 points) 1.1 Part (a) The DFS forest is showed in Fig. 1 where pre and post numbers are written in the form [pre,post] beside each node. a b c f d e h g i [1,2] [3,6] [9,18] [7,8] [4,5] [14,15] [12,13] [11,16] [10,17] Fig.1. The DFS forest. 1.2 Part (b) The shortest path tree is showed in Fig. 2 where the distances are written next to each node. 2 (10 points) 2.1 Part (a) Polynomial time. We only need to output all nodes with zero indegrees, which takes O (  V  +  E  ) time. 2.2 Part (b) Exponential time. We have proved in homework 24 that graphs can have expo nentially many cycles. It takes the computer at least exponential time to output all of them. 2 a b c e d f h g i 6 7 8 5 8 1 3 7 Fig.2. The shortest path tree. 2.3 Part (c) Exponential time. There are n ! many topological orderings for a graph with n nodes and no edges. Since n ! > 2 n when n is large enough, we need at least exponential time to output all of them.exponential time to output all of them....
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This note was uploaded on 01/21/2010 for the course CSE 661930 taught by Professor Freund,yoav during the Fall '09 term at UCSD.
 Fall '09
 Freund,Yoav

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