Intro to Graphs

CSIT 402

Data Structures II

CSIT 402 Graph Introduction 2

Graphs

• Graphs are composed of› Nodes (vertices)

• Can be labeled

› Edges (arcs)• Directed (arrowhead)• Undirected• Can be labeled

node

edge

CSIT 402 Graph Introduction 3

Graphs: Data Structure Progression

Consider previous data structures•Linked List: Nodes with 1 incoming edge and 1 outgoing edge

•Binary Trees/Heaps: Nodes with 1 incoming edge and 2 outgoing edges

•General Trees: Nodes with 1 incoming edge and any number of outgoing edges

Idea: All of the above are graphsMore examples on next seven slides…

10

96 99

94

97

Value Nextnode

Value Nextnode

CSIT 402 Graph Introduction 4

Course Prerequisites

321143

142

322

326

341370

378

401

421Nodes = coursesDirected edge = prerequisite

373

410

413

415

417

461

CSIT 402 Graph Introduction 5

Representing a Maze

S

Nodes = locationsEdge = paths between locations

S

E

B

E

Directed or Undirected?

CSIT 402 Graph Introduction 6

Representing Electrical Circuits

Nodes = battery, switch, resistor, etc.Edges = connections

Battery Switch

Resistor

Directed or Undirected?

CSIT 402 Graph Introduction 7

Program statements

x1=q+y*zx2=y*z-q

Naive:

commonsubexpression

eliminated:

y z

*

-

q

+

q *

x1 x2

y z

-

q

+

q *

x1 x2

Nodes = symbols/operatorsEdges = relationships

y*z calculated twice

Directed or Undirected?

CSIT 402 Graph Introduction 8

PrecedenceS1 a=0;

S2 b=1;

S3 c=a+1

S4 d=b+a;

S5 e=d+1;

S6 e=c+d;

3

1 2

6

5

4Which statements must execute before S6?

S1, S2, S3, S4

Nodes = statementsEdges = precedence requirements

Directed or Undirected?

Values

CSIT 402 Graph Introduction 9

Information Transmission in a Computer Network

Seattle

New York

L.A.

Tokyo

Sydney

Seoul

Nodes = computersEdges = transmission rates

128

140

18130

16

56Directed or Undirected?

Values

CSIT 402 Graph Introduction 10

Traffic Flow on Highways

Nodes = citiesEdges = # vehicles on connecting highway

UW

Directed or Undirected?

CSIT 402 Graph Introduction 11

Formal Definition

• A graph G is a tuple (V, E) where› V is a set of vertices or nodes › E is a set of edges

• An Edge e is a pair (v1,v2), where vi ε V› Example:

• V = {A, B, C, D, E, F}• E = {(A,B), (A,D), (B,C), (C,D), (C,E), (D,E)}

A

BC

ED

F

• If the order of edge pairs (vi, vj) is important (e.g prerequisites), the graph is directed (also called a digraph): (vi, vj) (vj, vi) , i j

• If the order of edge pairs (vi, vj) is unimportant (e.g. maze), the graph is called an undirected graph (never called an undigraph): In this case, (vi, vj) = (vj, vi) , i j

CSIT 402 Graph Introduction 12

Directed vs Undirected Graphs

vivj

vivj

CSIT 402 Graph Introduction 13

Undirected Graph Terminology

• Two vertices u and v are adjacent in an undirected graph G if {u,v} is an edge in G› edge e = {u,v} is incident (in contact) with vertex u and vertex v

• The degree of a vertex in an undirected graph is the number of edges incident with it› a self-loop counts twice (both ends count)

› denoted with deg(v)

A

BC

ED

F

Deg(D) = 3 Deg(F) = 0

(A,B) is incidentto A and to B

Self-loop

B is adjacent to C and C is adjacent to B

CSIT 402 Graph Introduction 14

Directed Graph Terminology

• Vertex u is adjacent to vertex v in a directed graph G if (u,v) is an edge in G› vertex u is the initial vertex of (u,v)

• Vertex v is adjacent from vertex u› vertex v is the terminal (or end) vertex of (u,v)

• Degree› in-degree is the number of edges with the vertex

as the terminal vertex› out-degree is the number of edges with the vertex

as the initial vertex

CSIT 402 Graph Introduction 15

Directed Graph Terminology

A

BC

ED

F

In-degree = 2Out-degree = 1

In-degree = 0Out-degree = 0

B adjacent to C and C adjacent from B

CSIT 402 Graph Introduction 16

Handshaking Theorem

• Let G=(V,E) be an undirected graph with |E|=e edges. Then

• Every edge contributes +1 to the degree of each of the two vertices it is incident with › number of edges is exactly half the sum of deg(v)› the sum of the deg(v) values must be even

Vv

deg(v)2e Add up the degrees of all vertices.

CSIT 402 Graph Introduction 17

• Space and time are analyzed in terms of:

• Number of vertices = |V| and

• Number of edges = |E|

• There are at least two ways of representing graphs:

• The adjacency matrix representation

• The adjacency list representation

Graph Representations

CSIT 402 Graph Introduction 18

A B C D E F

0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0

M(v, w) = 1 if (v, w) is in E

0 otherwise

A

B

C

D

E

F

Space = |V|2

A

BC

EDF

Adjacency Matrix

CSIT 402 Graph Introduction 19

A B C D E F

0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

A

B

C

D

E

F

Space = |V|2

M(v, w) = 1 if (v, w) is in E

0 otherwise

A

BC

EDF

Adjacency Matrix for a Digraph

CSIT 402 Graph Introduction 20

B D

B D

C

A C E

D

E

A C

A

B

C

D

E

F

A

BC

EDF

Space = |V| + 2 |E|

For each v in V, L(v) = list of w such that (v, w) is in E a b

Adjacency List

list ofneighbors

CSIT 402 Graph Introduction 21

B D

E

D

C

a b

A

B

C

D

E

F

E

A

BC

EDF

For each v in V, L(v) = list of w such that (v, w) is in E

Space = |V| + |E|

Adjacency List for a Digraph

CSIT 402 Directed Graphs

22

Paths and Cycles

• Given a directed graph G = (V,E), a path is a sequence of vertices v1,v2, …,vk such that:

› (vi,vi+1) in E for 1 < i < k• path length = number of edges in the path• path cost = sum of costs of each edge (if edges have costs)

• A graph contains a cycle if, there exists a path v1,v2, …,vk, k > 1 such that vk = v1

• G is acyclic if it has no cycles.