1. Lecture 11
CSE 331
Sep 19, 2014

2. HW 2 due today I will not accept HWs after 1:15pm
Place Q1, Q2 and Q3 in separate piles
I will not accept HWs after 1:15pm

3. Other HW related stuff HW 3 has been posted online: see piazza
Solutions to HW 2 at the END of the lecture
Graded HW 1 available from Monday onwards

4. Out of town next week
Andrew Hughes will cover the lecture for me

5. Graphs Graph G = (V,E) Directed vs Undirected (default) u w
No “self loops”

6. Paths , , Sequence of vertices connected by edges Connected
Path length 3 ,

7. Connectivity
u and w are connected iff there is a path between them
A graph is connected iff all pairs of vertices are connected

8. Connected Graphs
Every pair of vertices has a path between them

9. Cycles
Sequence of k vertices connected by edges, first k-1 are distinct ,

11. Formally define everything

12. Rest of Today’s agenda Prove n vertex tree has n-1 edges
Formal definitions of paths, cycles, connectivity and trees
Prove n vertex tree has n-1 edges
Algorithms for checking connectivity

13. HW 2 due today I will not accept HWs after 1:15pm
Place Q1, Q2 and Q3 in separate piles
I will not accept HWs after 1:15pm

14. Tree
Connected undirected graph with no cycles

15. Rooted Tree

16. A rooted tree AC’s child=SG Pick any vertex as root SG’s parent=AC
Let the rest of the tree hang under “gravity”

17. Rest of Today’s agenda Prove n vertex tree has n-1 edges
Algorithms for checking connectivity

18. Checking by inspection

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

20. 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

21. Algorithm motivation
all

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

23. Questions?

24. 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

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

26. 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