CS322

1. Week 11 - Friday CS322

2. Last time • What did we talk about last time? • Combinations

3. Questions?

4. Logical warmup • A butcher goes to the market with $100 • He has to buy exactly 100 animals (for no especially good reason) • There are cows, geese and chicken for sale • Cows are $15 each • Geese are $1 each • Chickens are $0.25 each • He has to buy at least one of each animal and has to spend all his money • What does the butcher buy?

5. Bionomial Theorem and Pascal's Triangle Student Lecture

6. Binomial Matters

7. Pascal's Triangle • Hopefully, you are all familiar with Pascal's Triangle, the beginning of which is: • If we number rows and columns starting at 0, note that the value of row n, column r is exactly

8. Pascal's Formula • Pascal's Triangle works because of Pascal's Formula: • We can easily show its truth:

9. Binomial Theorem • a + b is called a binomial • Using combinations (or Pascal's Triangle) it is easy to compute (a + b)n • We could prove this by induction, but you probably don't care

10. Binomial Example • Compute (1 – x)6 using the binomial theorem

11. More on Probability

12. Probability axioms • Let A and B be events in the sample space S • 0 ≤ P(A) ≤ 1 • P() = 0 and P(S) = 1 • If A  B = , then P(A  B) = P(A) + P(B) • It is clear then that P(Ac) = 1 – P(A) • More generally, P(A  B) = P(A) + P(B) – P(A  B) • All of these axioms can be derived from set theory and the definition of probability

13. Union probability example • What is the probability that a card drawn randomly from an Anglo-American 52 card deck is a face card (jack, queen, or king) or is red (hearts or diamonds)? • Hint: • Compute the probability that it is a face card • Compute the probability that it is red • Compute the probability that it is both

14. Expected value • Expected value is one of the most important concepts in probability, especially if you want to gamble • The expected value is simply the sum of all events, weighted by their probabilities • If you have n outcomes with real number values a1, a2, a3, … an, each of which has probability p1, p2, p3, … pn, then the expected value is:

15. Expected value: Roulette • A normal American roulette wheel has 38 numbers: 1 through 36, 0, and 00 • 18 numbers are red, 18 numbers are black, and 0 and 00 are green • The best strategy you can have is always betting on black (or red) • If you bet $1 on black and win, you get $1, but you lose your dollar if it lands red or green • What is the expected value of a bet?

16. Conditional probability • Given that some event A has happened, the probability that some event B will happen is called conditional probability • This probability is:

17. Conditional probability example • Given two, fair, 6-sided dice, what is the probability that the sum of the numbers they show when rolled is 8, given that both of the numbers are even?

18. 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

19. 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

20. 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)

21. Graphs

22. 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

23. 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…

24. 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

25. 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:

26. 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

27. 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?

28. Paths and Circuits

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

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

31. 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

32. 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

33. 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

34. 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

35. 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 edges, then a sequence of vertices uniquely determines the walk e2 e1 e4 v1 v2 v3 e3

36. 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

37. 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

38. 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

39. Upcoming

40. Next time… • Matrix representations of graphs • Directed graphs • Graph isomorphism

41. Reminders • Work on Homework 8 • Due tonight before midnight • Keep reading Chapter 10