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Recall: Digraphs

E. D. C. B. A. Recall: Digraphs. A digraph is a graph whose edges are all directed Short for “directed graph” Applications one-way streets flights task scheduling. Digraph Application. Scheduling: edge (a,b ) means task a must be completed before b can be started. ics21. ics22.

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Recall: Digraphs

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  1. E D C B A Recall: Digraphs • A digraph is a graph whose edges are all directed • Short for “directed graph” • Applications • one-way streets • flights • task scheduling

  2. Digraph Application • Scheduling: edge (a,b) means task a must be completed before b can be started ics21 ics22 ics23 ics51 ics53 ics52 ics161 ics131 ics141 ics121 ics171 The good life ics151

  3. DFS • In digraphs we traverse edges only along their direction • In the directed DFS algorithm, we can distinguish four types of edges • discovery edges • back edges • forward edges • cross edges • A directed DFS starting a a vertex s determines the vertices reachable from s E D C B A

  4. Reachability • DFS tree rooted at v: vertices reachable from v via directed paths E D E D C A C F E D A B C F A B

  5. DAGs and Topological Ordering D E • A directed acyclic graph (DAG) is a digraph that has no directed cycles • A topological ordering of a digraph is a numbering v1 , …, vn of the vertices such that for every edge (vi , vj), we have i < j • Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints Theorem A digraph admits a topological ordering if and only if it is a DAG B C A DAG G v4 v5 D E v2 B v3 C v1 Topological ordering of G A

  6. Topological Sorting • Number vertices, so that (u,v) in E implies u < v 1 A typical student day wake up 3 2 eat study computer sci. 5 4 nap more c.s. 7 play 8 write c.s. program 6 9 work out make cookies for professors 10 11 sleep dream about graphs

  7. Algorithm for Topological Sorting • Running time: ? MethodTopologicalSort(G) H G // Temporary copy of G n G.numVertices() whileH is not emptydo Let v be a vertex with no outgoing edges Label v  n n  n - 1 Remove v from H

  8. Topological Sorting Algorithm using DFS • Run DFS • Topological sort = Reverse order of Finish Order List • O(n+m) time.

  9. Topological Sorting Example

  10. Topological Sorting Example 9

  11. Topological Sorting Example 8 9

  12. Topological Sorting Example 7 8 9

  13. Topological Sorting Example 6 7 8 9

  14. Topological Sorting Example 6 5 7 8 9

  15. Topological Sorting Example 4 6 5 7 8 9

  16. Topological Sorting Example 3 4 6 5 7 8 9

  17. Topological Sorting Example 2 3 4 6 5 7 8 9

  18. Topological Sorting Example 2 1 3 4 6 5 7 8 9

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