CS322

1. Week 11 - Monday CS322

2. Last time • What did we talk about last time? • Binomial theorem and Pascal's triangle • Conditional probability • Bayes’ theorem

3. Questions?

4. Logical warmup • As I was going to St. Ives • I crossed the path of seven wives • Every wife had seven sacks • Every sack had seven cats • Every cat had seven kittens • Kittens, cats, sacks, wives • How many were going to St. Ives?

5. Bayes' Theorem • Let sample space S be a union of mutually disjoint events B1, B2, B3, … Bn • Let A be an event in S • Let A and B1 through Bn have non-zero probabilities • For Bk where 1 ≤ k ≤ n

6. Applying Bayes' theorem • Bayes' theorem is often used to evaluate tests that can have false positives and false negatives • Consider a test for a disease that 1 in 5000 people have • The false positive rate is 3% • The false negative rate is 1% • What's the probability that a person who tests positive for the disease has the disease? • Let A be the event that the person tests positively for the disease • Let B1 be the event that the person actually has the disease • Let B2 be the event that the person does not have the disease • Apply Bayes' theorem

7. Independent events • If events A and B are events in a sample space S , then these events are independent if and only if P(A B) = P(A)∙P(B) • This should be clear from conditional probability • If A and B are independent, then P(B|A) = P(B)

8. Graphs: Definitions and Representations Student Lecture

9. Graphs

10. Graphs • A graphG is made up of two finite sets • Vertices: V(G) • Edges: E(G) • Each edge is connected to either one or two vertices called its endpoints • An edge with a single endpoint is called a loop • Two edges with the same sets of endpoints are called parallel • Edges are said to connect their endpoints • Two vertices that share an edge are said to be adjacent • A graph with no edges is called empty

11. The purpose of graphs • Graphs can be used to represent connections between arbitrary things • Streets connecting towns • Links connecting computers in a network • Friendships between people • Enmities between people • Almost anything…

12. Graph representation • We can represent graphs in many ways • One is simply by listing all the vertices, all the edges, and all the vertices connected by each edge • Let V(G) = {v1, v2, v3, v4, v5, v6} • Let E(G) = {e1, e2, e3, e4, e5, e6,e7} • Edges connect the following vertices: • Draw the graph with the given connections

13. Drawing graphs • Graphs can (generally) be drawn in many different ways • We can label graphs to show that they are the same • Label these two graphs to show they are the same:

14. Special graphs • A simple graph does not have any loops or parallel edges • Let n be a positive integer • A complete graph on n vertices, written Kn, is a simple graph with n vertices such that every pair of vertices is connected by an edge • Draw K1, K2, K3, K4, K5 • A complete bipartite graph on (m, n) vertices, written Km,n is a simple graph with a set of m vertices and a disjoint set of n vertices such that: • There is an edge from each of the m vertices to each of the n vertices • There are no edges among the set of m vertices • There are no edges among the set of n vertices • Draw K3,2 and K3,3 • A subgraph is a graph whose vertices and edges are a subset of another graph

15. Degree • The degree of a vertex is the number of edges that are incident on the vertex • The total degree of a graph G is the sum of the degrees of all of its vertices • What's the relationship between the degree of a graph and the number of edges it has? • What's the degree of a complete graph with n vertices? • Note that the number of vertices with odd degree must be even… why?

16. Paths and Circuits

17. Königsberg • Used to be Königsberg, Prussia • Now called Kaliningrad, Russia • On the Pregel River, including two large islands

18. Seven Bridges of Königsberg • In 1736, the islands were connected by seven bridges • In modern times, there are only five

19. The Challenge • After a lazy Sunday and a bit of drinking, the citizens would challenge each other to walk around the city and try to find a path which crossed each bridge exactly once

20. Euler’s Solution • What did Euler find? • The same thing you did: nothing • But, he also proved it was impossible • Here’s how: North Shore Center Island East Island South Shore

21. Graph Theoretical View • By simplifying the problem into a graph, the important features are clear • To arrive as many times as you leave, the degrees of each node must be even (except for the starting and ending points) North Shore Center Island East Island South Shore

22. Definitions • A walk from v to w is a finite alternating sequence of adjacent vertices and edges of G, starting at vertex v and ending at vertex w • A walk must begin and end at a vertex • A path from v to w is a walk that does not contain a repeated edge • A simple pathfrom v to w is a path that does contain a repeated vertex • A closed walk is a walk that starts and ends at the same vertex • A circuit is a closed walk that does not contain a repeated edge • A simple circuit is a circuit that does not have a repeated vertex other than the first and last

23. Notation • We can always pin down a walk unambiguously if we list each vertex and each edge traversed • How would we notate a walk that starts at v1 and ends at v2 and visits every edge exactly once in the following graph? • However, if a graph has no parallel edges, then a sequence of vertices uniquely determines the walk e2 e1 e4 v1 v2 v3 e3

24. Connectedness • Vertices v and w of G are connected iff there is a walk from v to w • Graph G is connected iff all pairs of vertices v and w are connected to each other • A graph H is a connected component of a graph Giff • H is a subgraph of G • H is connected • No connected subgraph of G has H as a subgraph and contains vertices or edges that are not in H • A connected component is essentially a connected subgraph that cannot be any larger • Every (non-empty) graph can be partitioned into one or more connected components

25. Euler circuits • What if you want to find an Euler circuit of your own? • If a graph is connected, non-empty, and every node in the graph has even degree, the graph has an Euler circuit • Algorithm to find one: • Pick an arbitrary starting vertex • Move to an adjacent vertex and remove the edge you cross from the graph • Whenever you choose such a vertex, pick an edge that will not disconnected the graph • If there are still uncrossed edges, go back to Step 2

26. Hamiltonian circuits • An Euler circuit has to visit every edge of a graph exactly once • A Hamiltonian circuit must visit every vertex of a graph exactly once (except for the first and the last) • If a graph G has a Hamiltonian circuit, then G has a subgraphH with the following properties: • H contains every vertex of G • H is connected • H has the same number of edges as vertices • Every vertex of H has degree 2 • In some cases, you can use these properties to show that a graph does not have a Hamiltonian circuit • In general, showing that a graph has or does not have a Hamiltonian circuit is NP-complete (widely believed to take exponential time) • Does the following graph have a Hamiltonian circuit? a c b e d

27. Matrix Representations of Graphs

28. Matrices • As you presumably know, a matrix is a rectangular array of elements • An m x n matrix has m rows and n columns

29. Graph representations • There are many, many different ways to represent a graph • If you get tired of drawing pictures or listing ordered pairs, a matrix is not a bad way • To represent a graph as an adjacency matrix, make an n x n matrix, where n is the number of vertices • Let the nonnegative integer at aij give the number of edges from vertex i to vertex j • A simple graph will always have either 1 or 0 for every location

30. Graph to matrix examples • What is the adjacency matrix for the following graph? • What about for this one? v1 v3 v2 v1 v3 v2

31. Matrix to graph example • Draw a graph corresponding to this matrix

32. Another graph to matrix • What's the adjacency matrix of this graph? • Note that the matrix is symmetric • In a symmetric matrix, aij = aji for all 1 ≤ i ≤ n and 1 ≤ j ≤ n • All undirected graphs have a symmetric matrix representation v1 v2 v4 v3

33. Matrix multiplications • To multiply matrices A and B, it must be the case that A is an m x k matrix and that B is a k x n matrix • Then, the ith row, jth column of the result is the dot product of the ith row of A with the jth column of B • In other words, we could compute element cij in the result matrix C as follows:

34. Matrix multiplication practice • Multiply matrices A and B

35. A few points about matrix multiplication • Matrix multiplication is associative • That is, A(BC) = (AB)C • Matrix multiplication is not commutative • AB is not always equal to BA (for one thing, BA might not even be legal if AB is) • There is an n x n identity matrix I such that, for any m x n matrix A, AI = A • I is all zeroes, except for the diagonal (where row = column) which is all ones • We can raise square matrices to powers using the following recursive definition • A0 = I, where I is the n x n identity matrix • Ak = AAk-1, for all integers k ≥ 1

36. Finding powers of a matrix • Is A symmetric? • Compute A0 • Compute A1 • Compute A2 • Compute A3

37. Matrix powers for graphs • We can find the number of walks of length k that connect two vertices in a graph by raising the adjacency matrix of the graph to the kth power • Raising a matrix to the zeroth power means you can only get from a vertex to itself (identity matrix) • Raising a matrix to the first power means that the number of paths of length one from one vertex to another is exactly the number of edges between them • The result holds for all k, but we aren't going to prove it

38. Upcoming

39. Next time… • Graph isomorphism • Trees

40. Reminders • Keep reading Chapter 10