This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 InClass 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 InClass 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 ....
View Full
Document
 Spring '11
 Prof.AlbertR.Meyer
 Computer Science

Click to edit the document details