# I've just found the internet - PowerPoint PPT Presentation

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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?

### 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”

### 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

### 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?