1. CS 312: Algorithm Design & Analysis Lecture #17: Connectedness in Graphs This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License Slides by: Eric Ringger, adapting figures from Dasgupta et al.

2. Announcements  Due now:  HW #13  Project #4  Wednesday: Whiteboard Experience  Friday: Early Day  Virtual Monday = Tuesday: Due  Technical Career Fair: Thursday 9am-3pm in Wilkinson Student Center  Preview: Wednesday 4:00pm in 1170 TMCB  This Week  According to schedule!

3. Objectives  Wrap up connectedness in Undirected Graphs  DFS on Directed Graphs  Understand how to linearize a DAG

4. Many Kinds of Connectedness

5. DFS Analysis “Non-committal Case”: Our generic analysis covers best and worst cases.

6. DFS Yields a Forest

7. Directed Graphs  Let’s switch gears to directed graphs now

8. DFS on Directed Graphs

11. Edge Types  Tree edges: part of the DFS forest  Forward edges: lead to a non- child descendant in the DFS tree.  Back edges: lead to an ancestor in the DFS tree.  Cross edges: lead to a node that has already been completely explored (that is, already post-visited)  lead to neither descendant nor ancestor

12. Classifying Edge Types Pre Post ?

13. Classifying Edge Types Pre Post

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17. Classifying Edge Types Pre Post <

18. Classify Edge

19. Cycles  How do you detect a cycle in a directed graph?  Property: A directed graph has a cycle if and only if its depth-first search reveals a back edge.  Otherwise: a Directed Acyclic Graph (DAG)

20. Sources, Sinks, Linearization DFS Search Forest: Linearization of the DAG == Topological Ordering of the DAG: 0. Define order; 1. DFS; 2. Read off post-order values in reverse order Analysis: Input DAG:

21. Sources, Sinks, Linearization DFS Search Forest: Linearization of the DAG == Topological Ordering of the DAG: 0. Define order; 1. DFS; 2. Read off post-order values in reverse order Analysis: Input DAG:

22. Sources, Sinks, Linearization DFS Search Forest: Linearized DAG: 0. Define order; 1. DFS; 2. Read off post-order values in reverse order Analysis:

23. Sources, Sinks, Linearization  In our DAG, what do you notice about sources, sinks and post-order numbers?

24. Assignment  HW 13.5  Whiteboard Experience for Project #4