I've just found the internet
History of the internet
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Digital computers: 1950s
ARPA, ARPANET: developed in 1960s
1980s: NSF funds CSNET, TCP/IP developed
Late 1980s: commercial ISPs; WWW born
1990: Arpanet decommissioned
1990s: email, primitive VoIP
2000s: social networks
2010s: cloud computing
2020s: IoT (50 billion devices)
…Embedded devices?
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e= mc2
How does information travel across
the internet?
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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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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
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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
These are all equivalent
Al qaeda graph
Euler and the Konigsberg bridges
Types of graphs
• Eulerian: circuit that traverses each edge
exactly once
• Which graphs possess Euler circuits?
Problem: does this graph have an Euler
cycle?
Theorem: If every vertex has even degree
then there is an Eulerian path
Clicker question:
The following graph is Eulerian
A) True
B) False
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 of the
twenty vertices exactly once
Hamiltonian graph
• A graph is said to be
Hamiltonian if, starting
from a vertex v, it is
possible to visit each
vertex of the graph
exactly once, and end up
back at v
• Such a path is called a
Hamiltonian cycle
Hamilton’s puzzle: find a path in the
dodecahedron graph that traverses each vertex
exactly once
Hamiltonian graph
Clicker Question:
Is the following graph
Hamiltonian?
A) Yes
B) No
Rhetorical question:
Is the following graph
Hamiltonian?
Fullerenes
Petersen graph: symmetry
Other types of graphs
Other properties
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Diameter
Girth
Chromatic number
etc
Graph colorings
Graph coloring and map coloring
• The four color problem
Which continent is this?
[Clicker Question: What continent
does this graph represent? ]
A.
B.
C.
D.
E.
F.
G.
[Asia]
[Africa]
[Europe]
[North America]
[South America]
[Australia]
[Antarctica]
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
Worker
A
B
C
D
E
F
Doesn’t
like
B,C
A,C
A,B,D,E
C,F
C,F
D,E
Other examples of problems whose
solutions are simplified using graph
theory
What does this graph have to do with the Boss’s
dilemma?
Complementary graph
Complete subgraph
• Subgraph: vertices subset of vertex set, edges
subset of edge set
• Complete: every vertex is connected to every
other vertex.
Complementary graph
Clicker question: How many men
are in the room
• 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?
Clicker question
• 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?
• A) 15
• B) 8
• C) 20
• D) 6
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?
Clicker Question:
Can Samantha do this?
• A) Yes
• B) No
Solution:
• Think of the “continents” as vertices of a
graph
• Think of the strings as edges
• Is it possible to have a graph with seven
vertices each with degree 3?
• No: Each edge joins two vertices, so
contribute one to each vertex degree. The
sum of the vertex degree over all vertices
equals twice the number of edges, so has to
be even.
7persons
Each limb
that is
connected to
another
represents an
edge. Some
have four
connections,
some have
three.
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?
Clicker question: Is this possible
• A) Yes
• B) No
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?