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Unformatted text preview: CS170  Spring 2010 HW4  Solutions (DPV 3.2) The figure below shows pre and post numbers for the vertices in parentheses. Different edges are marked as indicated. A B C D E F G H (1,16) (2,11) (4,5) (7,8) (3,10) (13,16) (12,15) (6,9) Tree Edge Cross Edge Back Edge Forward Edge A B C D E F G H (1,16) (2,15) (3,14) (4,13) (5,12) (6,11) (7,10) (8,9) (a) (b) (DPV 3.3) (a) The figure below shows the pre and post times in parentheses. (1,14) (15,16) (2,13) (3,10) (11,12) (4,9) (5,6) (7,8) A B C D E F G H (b) The vertices A,B are sources and G,H are sinks. (c) Since the algorithm outputs vertices in decreasing order of post numbers, the ordering given is B,A,C,E,D,F,H,G . (d) Any ordering of the graph must be of the form { A,B } ,C, { D,E } ,F, { G,H } , where { A,B } indi cates A and B may be in any order within these two places. Hence the total number of orderings is 2 3 = 8. (DPV 3.4) To solve this question we use the algorithm provided in page 94 of the text book. (1) G R is built and a DFS is run to generate post times for all nodes. (2) we run a DFS on G picking the nodes in which to start each run from by the decreasing order of the post scores from step (1). Each time the DFS run finishes a “tree” and is ready to pick a new node at which to start at  the nodes traversed in that run are “packed” into an SCC. (i) The strongly connected components are found in the order { B } , { E } , { A } { H,G,I } , { C,D,F,J } . { B } and { E } are sources and { C,D,F,J } is a sink. The “metagraph” is shown in the figure below....
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This note was uploaded on 04/30/2010 for the course CS 170 taught by Professor Henzinger during the Spring '02 term at Berkeley.
 Spring '02
 HENZINGER
 Algorithms

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