CPS 1: Introduction to Graph Theory
Instructor: Nihshanka Debroy ([email protected])
What is a Graph ?
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Means of representation
Set of vertices and
edges
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Vertices (blue circles)
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Edges (black lines)
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Vertices (also called
nodes) connected to
each other by edges
Largest example: the
Internet!
Internet and Graphs
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Vertices/Nodes: machines/servers/routers
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Edges: physical connections (wires)
Internet
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Packets of information (bits) get routed on the
Internet
- Message gets divided into packets on client
(your machine)
- Packets sent out towards destination
- Packets may take different routes to
destination
Another example: Computer File
System
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A tree (type of graph)
Each file contained
within only 1 folder
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Purple nodes: folders
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Blue nodes: files
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Parent node of a
node X: node just
above X
Child node: node just
below
Navigating through the File System
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Referencing pictures in our
HTML code -> located files by
file path
File path: how to start from
one file, go through the
folders , and reach another
file
index.html to page2.html:
index.html -> public_html ->
cps1 -> page2.html
Oracle of Bacon: another example
Concept: Link an actor/actress with Kevin Bacon in the following
manner:
• Pick any film actor in history.
• Link the actor you've chosen to Bacon via the films they've
shared with other actors until you end up with Kevin Bacon
himself.
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Finding smallest number of links can be difficult
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Can be modeled using a graph
Let's try it out!
Social Networks: Examples
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Friendship Networks (Facebook)
- vertices: people in the world
- edges/links: have met in person/on friends list
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Scientific Collaboration
- vertices: Math and Comp. Sci. researchers
- edges/links: b/w coauthors on a published
paper
- Erdos number: distance to Paul Erdos
- Erdos was a hub: 507 coauthors!
Terminology
A
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B
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C
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Path: sequence of vertices
(v0, v1, v2, ...,vn) such that
vertices vk and vk+1 are
connected by an edge (For
eg., A-B-C)
Cycle: path of vertices and
edges that forms a
connected loop (is there
one here ?)
Trees: do not have cycles
Degree of a node: number
of edges adjacent to it
Connected Graphs: Path
b/w any pair of vertices
Is this Connected ? Is it a Graph ?
A
B
C
Directed Graphs
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Directed Edge: has
direction
Directed Graph: edges
are all directed
Path: can only have
directed edges in this
case
Is there a cycle ?
Edge weights
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Edges could have
weights
Mailman example
Two Standard Traversals
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BFT (breadth-first):
explore neighboring
nodes first
1
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DFT (depth-first):
explore as far as
possible before backtracking
4
2
8
7
11
3
6
5
9
10
12
Interesting questions
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Shortest path from A to any other
vertex ?
Color each circles in the graph,in
such a way that if two circles have
a line between them then they do
not have the same color. What is
the fewest number of colors you
can use to do this?
Visit all vertices without repeating ?
Edges ?
Given 2 vertices, A and B, is there
more than one way of getting from
A to B ?