1. Week 6 - Wednesday
CS 113

2. Last time What did we talk about last time? Exam 1 post-mortem
Programming practice

3. Project 2

4. Graph a sine curve We can show a sine curve
Using a while loop, we can make a lot of x values between -10 and 10
We can use the sin() function to find the corresponding y location
Then we put a small red sphere at each location

5. Graph Theory
A completely different kind of graph than what we just talked about!

6. What is a graph?
Vertices (Nodes)
Edges

7. What can graphs represent?
Friendships on Facebook
Steps in a task
Nodes: Subtasks
Nodes: People
Edges: Decisions
Edges: Friendship
Routes between cities
6 degrees of Kevin Bacon
Nodes: Cities
Edges: Streets
Nodes: Actors
Edges: Whether or not they've been in a movie together

8. Lots of flavors of graphs
Labeled
Weighted
Colored
Multigraphs E A D B F C A 5 6 4 3 2 3 E 5 B D F C 7 1

9. No Triangle Inequality
When a weighted graph obeys the triangle inequality, the direct route to a node is always fastest 1 4 2 3 5
No Triangle Inequality 1 2 4 3 5
Triangle Inequality

10. Directed graphs Some graphs have edges with direction
Example: One way streets
Reachability?
ONE WAY E A D B F C

11. Connected graphs Often we talk about connected graphs
But, not all graphs have to be connected

12. The other extreme Complete graphs
Every node is connected to every other
How many edges?
|E| = ½(n(n – 1)) = ½(n2 – n) is O(n2)

13. Subgraphs We can talk about a part of a graph
For example, what is the largest complete subgraph in this graph? A E B D F C

14. Paths A path is a sequence of connected nodes
The cost or weight of the path is usually the sum of the edge weights
This path from A to C costs 5 A 5 6 4 3 2 3 E B D 3 F 7 C

15. Cycles
A cycle is a path that starts at a node and comes back to the same node
How many cycles of length 3 does this graph have?
What about 4? 5? 6? A D B A E B D F C

16. Tours
A tour is a path that visits every node and (usually) returns to its starting node
In other words, it's a cycle that visits every node
This tour costs 24 A 5 6 4 3 2 3 E B D 3 F 7 C

17. Matching

18. Bipartite graphs
A bipartite graph is one whose nodes can be divided into two disjoint sets X and Y
There can be edges between set X and set Y
There are no edges inside set X or set Y
A graph is bipartite if and only if it contains no odd cycles

19. Bipartite graph X A B C D E F Y A B C D E F

20. Maximum matching
A perfect matching is when every node in set X and every node in set Y is matched
It is not always possible to have a perfect matching
We can still try to find a maximum matching in which as many nodes are matched up as possible

21. Matching algorithm Come up with a legal, maximal matching
Take an augmenting path that starts at an unmatched node in X and ends at an unmatched node in Y
If there is such a path, switch all the edges along the path from being in the matching to being out and vice versa
If there is another augmenting path, go back to Step 2

22. X Y Match the graph Anna Becky Caitlin Daisy Erin Fiona A B C D E F A
Adam
Ben
Carlos
Dan
Evan
Fred

23. Quiz

24. Upcoming

25. Next time… Stable marriage Euler paths and tours
Minimum spanning trees
Lab 6

26. Reminders Start working on Project 2 Read Python Chapter 5
Think about what you want to do for your Final Project
Proposal due by 3/31