1. Lecture 12
CSE 331
Sep 22, 2014

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

3. n vertex tree has n-1 edges
Proof idea (recap):
Root tree T at vertex r r
Direct edges from child to parent

4. n vertex tree has n-1 edges
Proof idea (recap):
tail
Claim 1: r is not the head of any directed edge. r
Claim 2: Every directed edge has a unique head
head

5. Questions?

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

7. Checking by inspection

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

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

10. Algorithm motivation
all

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

12. Questions?

13. Breadth First Search (BFS)
Is s connected to t?
Build layers of vertices connected to s
Lj : all nodes at distance j from s
L0 = {s}
Assume L0,..,Lj have been constructed
Lj+1 set of vertices not chosen yet but are connected to Lj
Stop when new layer is empty

14. Exercise for you
Prove that Lj has all nodes at distance j from s

15. BFS Tree BFS naturally defines a tree rooted at s Add non-tree edges
Lj forms the jth “level” in the tree
u in Lj+1 is child of v in Lj from which it was “discovered” 1 7 1 L0 2 3 L1 2 3 8 4 5 4 5 7 8 L2 6 6 L3