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Recall: Digraphs

This overview introduces directed graphs (digraphs) as graphs with directed edges, ideal for modeling applications such as task scheduling and navigation. We explore how depth-first search (DFS) works in digraphs, identifying four edge types: discovery, back, forward, and cross edges. Additionally, we define directed acyclic graphs (DAGs) and the concept of topological ordering, essential for sequential task management. Algorithms for topological sorting are detailed, highlighting their efficiency and practical applications.

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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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