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I've just found the internet. How does information travel across the internet?. TCP/IP TCP wiki IP wiki Request generated by user (“click”) Response sent as set of packets with time stamps Receipt acknowledged Response regenerated if ack not received. Bandwidth.
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How does information travel across the internet? • TCP/IP • TCP wiki • IP wiki • Request generated by user (“click”) • Response sent as set of packets with time stamps • Receipt acknowledged • Response regenerated if ack not received.
Bandwidth • Packets seek shortest/fastest path • Determined by number of hops • Queues form at hubs; bottlenecks can occur • Repeat requests can add to traffic
Main problem • Determining the shortest path • Presumes: lookup table of possible routes • Presumes: knowledge of structure of internet • Mathematical structure: directed, weighted graph. • Other related problems: railroad networks, interstate network, google search problem, etc.
Graph theory • A graph consists of: • set of vertices • A set of edges connecting vertex pair • Incidence matrix: which edges are connected
The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and (v,e)=1 iff vertex v is incident upon edge e
Types of graphs • Eulerian: circuit that traverses each edge exactly once • Which graphs possess Euler circuits?
Theorem: If every vertex has even degree then there is an Eulerian path
What is a theorem? • A statement that no one can understand • A statement that only a mathematician can understand • A statement that can be verified from “first principles” • A statement that is “always true”
Heuristic argument • An argument that appeals to intuition, but may not be compelling by itself. • In the case of the Eulerian graph theorem, think of the vertex as a room and the edges as hallways connecting rooms. • If you leave using one hallway then you have to return using a different one. • “Induction argument”
Hamilton’s puzzle: find a path in the dodecahedron graph that traverses each vertex exactly once
Other properties • Diameter • Girth • Chromatic number • etc
Graph coloring and map coloring • The four color problem
Boss’s dilemna • Six employees, A,B,C,D,E,F • Some do not get along with others • Find smallest number of compatible work groups
Other examples of problems whose solutions are simplified using graph theory
Complete subgraph • Subgraph: vertices subset of vertex set, edges subset of edge set • Complete: every vertex is connected to every other vertex.
Handshakes, part 2 • There are several men and 15 women in a room. Each man shakes hands with exactly 6 women, and each woman shakes hands with exactly 8 men. • How many men are in the room?
Visualize whirled peas • Samantha the sculptress wishes to make “world peace” sculpture based on the following idea: she will sculpt 7 pillars, one for each continent, placing them in circle. Then she will string gold thread between the pillars so that each pillar is connected to exactly 3 others. • Can Samantha do this?
Some additional exercises in graph theory • There are 7 guests at a formal dinner party. The host wishes each person to shake hands with each other person, for a total of 21 handshakes, according to: • Each handshake should involve someone from the previous handshake • No person should be involved in 3 consecutive handshakes • Is this possible?
Camelot • King Arthur and his knights wish to sit at the round table every evening in such a way that each person has different neighbors on each occasion. If KA has 10 knights, for how long can he do this? • Suppose he wants to do this for 7 nights. How many knights does he need, at a minimum?