1. Lecture 12 CSE 331 Sep 28, 2015

2. Upcoming Important Dates Tuesday Oct 8 (~1.5 weeks) Mini Project group compositions Monday Oct 12 (2 weeks) Quiz 1 Monday Oct 19 (3 weeks) Mid-term exam 1

3. Delay in handing back HW 1

4. 5 min ask me anything Q&A Feel free to stop by and chat about non-331 stuff with me Director of UG Studies OH: Tuesdays, 11:30am-1:30pm

5. Today’s agenda Finish Proving n vertex tree has n-1 edges Algorithms for checking connectivity

6. n vertex tree has n-1 edges r r Root tree T at vertex r Proof idea (recap): Direct edges from child to parent tail head Every edge has exactly one non-root head

7. Questions?

8. Rest of Today’s agenda Finish Proving n vertex tree has n-1 edges Algorithms for checking connectivity

9. Checking by inspection

10. What about large graphs? s t Are s and t connected?

11. Brute-force algorithm? List all possible vertex sequences between s and t Check if any is a path between s and t n n such sequences

12. Algorithm motivation all

13. Distance between u and v Length of the shortest length path between u and v Distance between RM and BO? 1

14. Questions?

15. Breadth First Search (BFS) Is s connected to t? Build layers of vertices connected to s L 0 = {s} Assume L 0,..,L j have been constructed L j+1 set of vertices not chosen yet but are connected to L j Stop when new layer is empty L j : all nodes at distance j from s

16. Exercise for you Prove that L j has all nodes at distance j from s

17. BFS Tree BFS naturally defines a tree rooted at s L j forms the jth “level” in the tree u in L j+1 is child of v in L j from which it was “discovered” 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 1 1 2 2 3 3 L0L0 L1L1 4 4 5 5 7 7 8 8 L2L2 6 6 L3L3 Add non- tree edges