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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring 10 : Mathematics for Computer Science March 1 Prof. Albert R. Meyer revised March 1, 2010, 826 minutes Solutions to In-Class Problems Week 5, Mon. Problem 1. If a and b are distinct nodes of a digraph, then a is said to cover b if there is an edge from a to b and every path from a to b traverses this edge. If a covers b , the edge from a to b is called a covering edge . (a) What are the covering edges in the following DAG? 12 6 1 8 2 4 10 5 7 11 9 3 Solution. TBA (b) Let covering ( D ) be the subgraph of D consisting of only the covering edges. Suppose D is a finite DAG. Explain why covering ( D ) has the same positive path relation as D . Hint: Consider longest paths between a pair of vertices. Solution. What we need to show is that if there is a path in D between vertices a = b , then there is a path consisting only of covering edges from a to b . But since D is a finite DAG, there must be a longest path from a to b . Now every edge on this path must be a covering edge or it could be replaced by a path of length 2 or more, yielding a longer path from a to b . (c) Show that if two DAG s have the same positive path relation, then they have the same set of covering edges. Creative Commons 2010, Prof. Albert R. Meyer . 2 Solutions to In-Class Problems Week 5, Mon. Solution. Proof. Suppose C and D are DAGs with the same positive path relation and that a b is a covering edge of C . We want to show that a b must also be a covering edge of D . Since a b itself defines a (length one) positive length path in C , there must be a positive length path in D from a to b . If this positive length path in ....
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