Graphs 9/14/2004 2:50 PM

1

Graphs 1

Graphs

ORD

DFW

SFO

LAX

80

2

1743

1843

1233

33

7

Graphs 2

Outline and Reading

Graphs (§6.1)� Definition

� Applications

� Terminology

� Properties

� ADT

Data structures for graphs (§6.2)� Edge list structure

� Adjacency list structure

� Adjacency matrix structure

Graphs 3

GraphA graph is a pair (V, E), where� V is a set of nodes, called vertices

� E is a collection of pairs of vertices, called edges

� Vertices and edges are positions and store elements

Example:� A vertex represents an airport and stores the three-letter airport code

� An edge represents a flight route between two airports and stores the mileage of the route

ORDPVD

MIADFW

SFO

LAX

LGA

HNL

849

80

2

13871743

1843

10991120

1233

33

7

2555

142

Graphs 4

Edge TypesDirected edge� ordered pair of vertices (u,v)

� first vertex u is the origin

� second vertex v is the destination

� e.g., a flight

Undirected edge� unordered pair of vertices (u,v)

� e.g., a flight route

Directed graph

� all the edges are directed

� e.g., route network

Undirected graph� all the edges are undirected

� e.g., flight network

ORD PVDflight

AA 1206

ORD PVD849

miles

Graphs 5

John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.net

qwest.net

math.brown.edu

cslab1bcslab1a

ApplicationsElectronic circuits

� Printed circuit board

� Integrated circuit

Transportation networks

� Highway network

� Flight network

Computer networks

� Local area network

� Internet

� Web

Databases

� Entity-relationship diagram

Graphs 6

TerminologyEnd vertices (or endpoints) of an edge

� U and V are the endpoints of a

Edges incident on a vertex� a, d, and b are incident on V

Adjacent vertices

� U and V are adjacent

Degree of a vertex� X has degree 5

Parallel edges� h and i are parallel edges

Self-loop

� j is a self-loop

XU

V

W

Z

Y

a

c

b

e

d

f

g

h

i

j

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2

Graphs 7

P1

Terminology (cont.)

Path� sequence of alternating

vertices and edges

� begins with a vertex

� ends with a vertex

� each edge is preceded and followed by its endpoints

Simple path� path such that all its vertices

and edges are distinct

Examples� P1=(V,b,X,h,Z) is a simple path

� P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple

XU

V

W

Z

Y

a

c

b

e

d

f

g

hP2

Graphs 8

Terminology (cont.)Cycle

� circular sequence of alternating vertices and edges

� each edge is preceded and followed by its endpoints

Simple cycle

� cycle such that all its vertices and edges are distinct

Examples

� C1=(V,b,X,g,Y,f,W,c,U,a,↵) is a

simple cycle

� C2=(U,c,W,e,X,g,Y,f,W,d,V,a,↵)is a cycle that is not simple

C1

XU

V

W

Z

Y

a

c

b

e

d

f

g

hC2

Graphs 9

PropertiesNotation

n number of vertices

m number of edges

deg(v) degree of vertex v

Property 1

ΣΣΣΣv deg(v) = 2m

Proof: each endpoint is counted twice

Property 2In an undirected

graph with no self-loops and no multiple edges

m ≤≤≤≤ n (n −−−− 1)////2Proof: each vertex

has degree at most (n −−−− 1)

Example

� n = = = = 4

� m = = = = 6

� deg(v) = 3

Graphs 10

Connected GraphsA (non-directed) graph is connected if there exists a path

∀ u, v ∈ V.

u

Connected components

G

Graphs 11

Main Methods of the Graph ADT

Vertices and edges

� are positions

� store elements

Accessor methods

� aVertex()

� incidentEdges(v)

� endVertices(e)

� isDirected(e)

� origin(e)

� destination(e)

� opposite(v, e)

� areAdjacent(v, w)

Update methods

� insertVertex(o)

� insertEdge(v, w, o)

� insertDirectedEdge(v, w, o)

� removeVertex(v)

� removeEdge(e)

Generic methods

� numVertices()

� numEdges()

� vertices()

� edges()

There could be other methods …..

Graphs 12

Representations

Edge ListAdjacency ListAdjacency MatrixIncidence Matrix

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Graphs 13

Edge List Structure (example)

l9l8l7l6l5l4l3l2l1

v2

v1

v6

v3

v4

v5

l1

l2

l3

l4 l

5l6

l7 l

8l9

Space: n + m

v6v5v4v3v2v1

Graphs 14

Edge List StructureVertex object

� element

� reference to position in vertex sequence

Edge object

� element

� origin vertex object

� destination vertex object

� reference to position in edge sequence

Vertex sequence

� sequence of vertex objects

Edge sequence� sequence of edge objects

v

u

w

a c

b

a

zd

u v w z

b c d

Graphs 15

5

1

4

2

3

5

4

3

2

1 2 4

2

5

1 2

Adjacency List (example)

Graphs 16

Adjacency List Structure Edge list structure

Incidence sequence for each vertex

� sequence of references to edge objects of incident edges

Augmented edge objects� references to

associated positions in incidence sequences of end vertices

u

v

w

a b

a

u v w

b

Graphs 17

5

1

4

2

3 000115

100004

000103

000002

010101

54321

G

5

1

4

2

3

G

If G is not-directed

010115

100014

000103

101012

110101

54321

symmetric matrix

Adjacency Matrix (examples)

Graphs 18

Space: n x n

Lots of waste space if the matrix is SPARSE …

0000100100

0000000110

0001000000

0010000010

1000000000

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

1

0

0

1

0

0

1

10000

01101

10000

00100

00001

Adjacency Matrix (observation)

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Graphs 19

Adjacency Matrix StructureEdge list structure

Augmented vertex objects

� Integer key (index) associated with vertex

2D-array adjacency array� Reference to edge

object for adjacent vertices

� Null for non nonadjacent vertices

u

v

w

a b

2

1

0

210

∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅

a

u v w0 1 2

b

Graphs 20

Incidence Matrix (1)

100-100000v6

-1-11000000v5

0

0

0

0

l9

1

0

0

0

l8

0

-1

0

0

l7

0

0

1

0

l6

-1

1

0

0

l5

0-100v4

-10-10v3

1001v2

011-1v1

l4l3l2l1

v2

v1

v6

v3

v4

v5

l1

l2

l3

l4 l

5l6

l7 l

8l9

Space: n x m

Graphs 21

Is (vi, vj) an edge?

Adjacency

Matrix:

Adjacency

List:

Edge

List:

[ ]i

j

O(1)

i …

O(deg(i))

l9l8l

7

l

6

l

5

l

4

l

3

l2l1

v6

v5

v4

v3

v2

v1 O(m)

Graphs 22

Which nodes are adjacent to vi?

Adjacency

Matrix:

Adjacency

List:

Edge

List:

[ ]i O(n)

i

O(deg(i))

l9l8l

7

l

6

l

5

l

4

l

3

l2l1

v6

v5

v4

v3

v2

v1 O(m)

Graphs 23

Mark all Edges

Adjacency

Matrix:

Adjacency

List:

[ ] O(n2)

1 2 ………… n

1

2

M

M

n

O(m)

1

n

l9l8l

7

l

6

l

5

l

4

l

3

l2l1

v6

v5

v4

v3

v2

v1

Edge

List: O(m)

Graphs 24

Add an Edge (vi, vj)

Adjacency

Matrix:

Adjacency

List (linked):

[ ] O(1)

O(1)i

i

j

1

l9l8l

7

l

6

l

5

l

4

l

3

l2l1

v6

v5

v4

v3

v2

v1

Edge

List: O(1)

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Graphs 25

Remove an Edge (vi, vj)

Adjacency

Matrix:

Adjacency

List :

[ ] O(1)

O(deg(i))ji

i

j

0

l9l8l

7

l

6

l

5

l

4

l

3

l2l1

v6

v5

v4

v3

v2

v1

Edge

List: O(1)

Graphs 26

O(deg(i))O(1)Remove edge

(vi, vj)

O(1)O(1)Add edge

(vi, vj)

O(m)O(n2)Mark all edges

O(deg(i))O(n)Which nodes are

adjacent to vi?

O(deg(i))O(1)Is (vi, vj) an

edge?

Adjacency List

Adjacency Matrix

O(deg(i)) = OUT-degree of node vi

G is directed

What are the

predecessors of vi??

Graphs 27

Performance

n2n + mn + mSpace

n2deg(v)mremoveVertex(v)

111insertEdge(v, w, o)

n211insertVertex(o)

111removeEdge(e)

1min(deg(v), deg(w))mareAdjacent (v, w)

ndeg(v)mincidentEdges(v)

Adjacency Matrix

AdjacencyList

EdgeList

n vertices

m edges

no parallel edges

no self-loops

Graphs 28

Special GraphsRegular Graphs

∀vi, vj ∈ V d(vi) = d(vj)

Bipartite Graphs Planar Graphs

Cannot have

Graphs 29

n – 1 ≤ m ≤

1 ≤ deg(i) ≤ n – 1

2

)1( −nn

connected,

non-directed

degree

n – 1 ≤ m ≤ n(n – 1)

1 ≤ deg(i) ≤ n – 1

connected,

directed

OUT-degree

n = | V | m = | E |

Graphs 30

—Ring —

— Complete Graph —

— Hypercube —

m = n

di = 2 ∀i

Tree

m = O(n)

m =

di = n – 1 ∀i

m = O(n2)2

)1( −nn

dim 0 dim 1 dim 2

Some Regular Graphs

dim 3 dim 4

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Graphs 31

Hypercube

12x2+8 = 3216h4

4x2+4 = 128h3

1x2+2 = 44h2

12h1

01h0

mn

n0 = 1 ni = 2i

ni = 2 ni–1 mi = i•2i–1

hi:

m =

d = log n

m = O(n log n)2

lognn

Graphs 32

—Grid —

— Torus —

Not regular

m = O(n)

deg(i) = 4 vi = internal

deg(i) = 3 vi = border

deg(i) = 2 vi = corner

m = O(n)

deg(i) = 4 ∀i