13-graphs - Spring 2009 CS216 Program and Data...

Graphs
CS216: Program and Data Representation
University of Virginia Computer Science
Spring 2009 Aaron Bloomfield
www.cheswick.com/ches/ map /index.html

2
Topic Coverage
• Graph terminology
– Lots of it!
• Ways of representing graphs and costs &
benefits
– Adjacency matrix
– Adjacency list
• Solving problems with graphs
– Topological sort
– Traveling salesperson

3
Graph examples
• Google Maps, of course
– Which actually uses MapQuest data
– But we’ll call it Google Maps data
• Others?

4
Airline maps
From http://www.travelmood.com
/site/seatsale/QF/images/
MR_Domestic-April2005.jpg

5
Flowcharts

6
Course
pre-req
graphs

7
Graphs
G = ( V , E )
V are the vertices ; E are the edges .
Edges are of the form (v, w) , where v, w ∈ V.
• ordered pair: directed graph or digraph
• unordered pair: undirected graph
v
9
v
8
v
10
v
3
v
6
v
7
v
4
v
5
v
1
v
2

8
How big are these graphs?
• These are educated guesses, by the way
• Airport codes
– There are probably 3,000 world-wide airports
– Assume you can fly to 25 airports from each
– That’s 3,000*25 = 75,000 edges
• Google maps
– There are probably 10 million vertices in the US
– Assume each one connects to three others
– That’s 3 * 10 million = 30 million edges

9
Terminology
• A weight or cost can be associated with
each edge.
• w is adjacent to v iff (v, w) ∈ E.
• path : sequence of vertices w
1
, w
2
, w
3
, … ,
w
N
such that (w
i
, w
i+1
) ∈ E for 1 ≤ i < N.
• length of a path: number of edges in the
path.
• simple path : all vertices are distinct.

10
How to weight a graph…
• For Google maps?
• For airline routes?

11
More terminology
cycle :
directed graph : path of length ≥ 1 such that w
1
= w
N
.
undirected graph : same, except all edges are
distinct.
connected : there is a path from every
vertex to every other vertex.
loop : (v, v) ∈ E.
complete graph : there is an edge between
every pair of vertices.

12
Digraph terminology
directed acyclic graph : no cycles. “DAG”
strongly connected : there is a path from every
vertex to every other vertex.
weakly connected : the underlying undirected
graph is connected.
C
D
A
B
E
F
G

13
End of lecture on Wed, Apr 8
• Also did the last 13 slides of 12-huffman

14
Representation
• adjacency matrix :
1 2 3 4
1
2
3
4
1
3
4
2

15
Representation
• adjacency list:
1
2
3
4
2 3 4
3
1 2
1
3
4
2

16
Representation in the real world
• Two types of representation
– Adjacency matrix
– Adjacency graph
• How does Google maps probably store it?
• How do airline routes probably store it?

Topological Sort

18
Topological Sort
• Given a directed acyclic graph, construct an
ordering of the vertices such that if there is a
path from v
i
to v
j
, then v
j
appears after v
i
in the
ordering.
– The result is a linear list of vertices
• indegree of v: # of edges (u, v).