1

Abstract Data Type

We have discussed:

List

Tree

Today we will talk about Graph

2

Introduction to Graphs

Several problems in the real-world can be formulated as objects and connections between them.e.g.

What is the fastest way to get from city A to city B?

What is the cheapest way to get from city A to city B?

• We need to represent :

cities and distances and

cities and costs respectively.

3

Northwest Airline Flight

Boston

Hartford

Atlanta

Minneapolis

Austin

SF

Seattle

Anchorage

4

Introduction to Graphs

Graphs are so important as structures that they deserve their own field called graph theory

Graph theory is a branch of combinatorial mathematics and has been studied for hundreds of years.

Our goal is to study graphs from the algorithmic point of view.

5

Introduction to Graphs

Campus map Traveling salesperson Circuit layout

Project scheduling Oil flow

In many cases we are faced with a problem that is defined in terms of a number of objects that have some relationship among them. We then try to answer interesting questions.

Graphs are the basic mathematical formulation we use to tackle such problems.

6

Introduction to Graphs Are all the circuits elements in a computer connected?

• This can be answered by looking at the model representing the circuits (objects) and the wiring of the circuit (connections)

Job scheduling. Some tasks have to be processed in some pre-defined order

• Graphs can be used to model the order jobs are processed

7

Computer Network Or Internet

MCI

Regional NetworkComposed of other

networks

IntelCampusUmass

AT&T

8

Application

Salesman’s Route Postman’s Route

Shortest route for salesman, we call it the Traveling Salesman Problem

Start

9

Course Prerequisites

CSC122

Freshman Sophomore Junior Senior

CSC242

10

Nomenclature of Graphs

A graph is a collection of vertices (nodes) and edges (arcs)

Nodes are objects that have names and other properties

Edges normally don't have properties but sometimes they are assigned a value – e.g. a weight.

A graph can be drawn by marking points and linking them with a line

11

Applications --- Communication Network

Vertex = city, edge = communication link.

1

23

810

4 5 9 11

67

Vertex

Edge

12

Graph Definition

G = (V, E) Graph = (Vertices, Edges)

V is the vertex set

Vertices are also called nodes or points

E is the edge set

Each edge connects two vertices

13

Concepts of Graphs

edges (weight)

node or vertex

14

Two representations of the same graph

A B

C D

E

F

A

B

C

D E

F

A graph is an object that is independent of its representation (in the computer or a pictorial)

15

Graphs and their shape

In some applications the shape is used to better express the graph.

If we're representing cities as they appear in the map,

keep the shape of the map

Still, changing it will not interfere in the solution if you do.

Connected Nodes

Solutions base themselves on the

list of pairs of nodes pairs of nodes

that are connectedconnected in the graph

16

17

Number of Edges

V is the number of nodes (vertices) in a graph and

E represents the number of edges

The number of edges in a graph can range from

0 to V(V-1)/2.

We call a sparse graph a graph in which E is much less than V(V-1)/2.

18

Sparse Graph

For example: if

N = 4 - then

E = N-1= O(E)

(E = 3)

Where E is the # of Edges

• There are a very small number of edges in a

sparse graph where N is the number of vertices.

GRAPHS Definitions

We call a dense a graph in which the number of edges is close to the maximal number of edges.

We call a complete graph a graph in which E is exactly

V(V-1) / 2V(V-1) / 2

19

20

Complete Graph

Total number of edges in a complete graph:

E = N(N-1)/2

if N = 4, E = 6

• There is an edge between any two vertices

21

Definitions

A path from node A to node B in a graph is a list of successive connected nodes in the graph

A graph is connected if there is a path from every node to every other node in the graph

A graph that is not connected is made up of several connected components

A B

C D

E

F

A path from C to FA path from C to F

22

What is a Graph?

A Graph G consists of a set V of vertices or nodes and

a set E of edges that connect the vertices.

We write G=(V,E).

v1

v5

v4

v2

v3

e1

e4

e3

e2

G=(V,E)V={v1,v2,v3,v4,v5}E={e1,e2,e3,e4}e1=(v1,v2)e2=(v2,v3)e3=(v2,v4)e4=(v3,v5)

23

v1

v5v4

v2

v3 e2

e1

G=(V,E)

V={v1,v2,v3,v4,v5}

E={e1,e2}

e1=(v2,v3)e2=(v2,v4)

Graphs --- Examples

Note that not all vertices are connected

24

v1

v4

v2

v3

e1

e5

e4

e2

e3

e6

A Complete Graph V(V-1)/2

Fully connected

25

Directed Graphs

In some cases we want the edges to have directions associated with them:

we call such a graph a directed graph or a digraph.

G=(V,E)V={v1,v2,v3,v4}E={e1,e2,e3,e4}e1=(v2,v1)e2=(v1,v3)e3=(v1,v4)e4=(v4,v3)e5=(v3,v4)

v1

v4

v2v3

e1

e4

e2

e3

ordered pair(predecessor, successor)

26

Undirected vs. Directed Graph

Undirected Graph

– no oriented edge

Directed Graph

– every edge has oriented vertex

27

Weighted Graphs

In some cases, we want to associate a weight with each edge in the graph. Such a graph is known as a weighted graph.

Directed graphs(as well as undirected) can also be weighted (directed weighted graphs).

G=(V,E)V={v1,v2,v3,v4}E={e1,e2,e3,e4,e5}…………………..…………………..

v1

v4

v2

v3

40

63

15

50

75

55

28

Weighted Graph

Weighted graph: a graph with numbers assigned to its edges

Weight: cost, distance, travel time, hop, etc.

0

1

3

2

20 10

1

54

29

Sub-graph

Sub-graph: subset of vertices and edges

MORE TERMS

A simple path is a path in which no nodes are repeated(A C E)

A cycle is a path that is simple except that the first and the last nodes are the same. (ABDA)

It is a path from a node back to itself.

30

31

Simple Path

A simple path traverses a node no more than once ABCD is a simple path - its length is 3

B

C

D

A

path

32

Cycle –

A cycle is a path that starts and ends at the same point

CBDC is a simple cycle.

B

C

D

A

33

More Definitions

A graph with no cycles is called a tree

A spanning tree of a graph is a connected subgraph that contains all the nodes of the graph but has no cycles (a tree by definition)

A

B

C

D E

F

A

B

C

D E

F

GraphGraph

SpanningTree

SpanningTree

CycleCycle

34

More Graph Terminology

A vertex vj is said to be adjacent to a different vertex vi if an edge connects vi to vj, i.e.,

e=(vi,vj).

The length of a path is the number of edges in it.

A graph with no cycles is called an acyclic graph.

A directed acyclic graph is called a DAG.

35

More Graph Terminology

Two different vertices are connected if there is a path between them.

The degree of a vertex is the number of edges connected to it.

C

D

A

The degree of C is 2 – it has 2 edges

36

Connected vs. Unconnected Graph

Connected Graph Unconnected Graph

37

Directed Acyclic Graph

Directed Acyclic Graph (DAG) : directed graph without cycle

Examples Course Requirement Graph: DAG

38

Directed Acyclic Graph

D

C

B

A

This is NOT a DAG(Directed Acyclic Graph)

ABCD is a cycle in a directed graph

39

Graph Representation- Matrix

There are several ways to represent a graph in a computer program.

The two most common ways are using a• matrix or a list:

An Adjacency-matrix representation:

• Used when the graph is dense

Adjacency-matrix representation:

Adjacency-matrix representation:

•In the representation of a graph G=(V,E),

• it is assumed that nodes are numbered 1,2,3,...,V.

•The matrix representation consists of a matrix Ai,j

•This matrix is normally symmetric along the main diagonal

40

41

Adjacency Matrix

We have N = 20 nodes in graph

Use a Matrix A[0…N-1][0…N-1] or A[N][N]

if vertex i and vertex j are adjacent in graph, A[i][ j ] = 1,

otherwise A[i][ j ] = 0

if vertex i has a loop, A[ i ][ i ] = 1

if vertex i has no loop, A[ i ][ i ] = 0

42

Adjacency Matrix--Example 1

1

4

5

6

3

2

8

7

0

0 0 1 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 1

0 0 0 0 0 0 0 0

0 1 2 3 4 5 6 7

0

1

2

3

4

7

5

6

0

0

0

0

0

1

0

0

8

0 0 0 0 0 0 0 08 0

Arrows mark connections

43

Example of Adjacency Matrix

0

1

3

2

A[i][j] 0 1 2 3

0 0 1 1 0

1 1 0 1 1

2 1 1 0 1

3 0 1 1 0

So, Matrix A =

0 1 1 0

1 0 1 1

1 1 0 1

0 1 1 0

44

Adjacency Matrix --- Example 2

8 6

8 9 9 6

0 1 2 3

0

1

2

323

0

16

8

9

The matrix is symmetric for undirected graphs.

A[2][1] =A[1][2]

45

Undirected vs. Directed

Undirected graph

adjacency matrix is symmetric

A[i][j]=A[j][i]

Directed graph

adjacency matrix may not be symmetric

A[i][j] A[j][i]

46

Directed Graph Matrix RepresentationNot symmetric

A[i][j] 0 1 2 3

0 0 1 1 1

1 0 0 0 1

2 0 0 0 1

3 0 0 0 0

0

1

3

2

So, Matrix A =

0 1 1 1

0 0 0 1

0 0 0 1

0 0 0 0

A[2][3] A[3][2]

47

Weighted Graph – Use weights in cells

A[i][j] 0 1 2 3

0 0 10 20 1

1 10 0 0 4

2 20 0 0 5

3 1 4 5 00

1

3

2

20 10

1

54

So, Matrix A =

0 10 20 1

10 0 0 4

20 0 0 5

1 4 5 0

48

Graph Representation

Another way to represent a graph uses an array of linked lists:

Adjacency-list representation:

This is preferred when the graph is sparse.

Adjacency-list representation:

The adjacency list for a graph with N vertices consists of

N Linked Lists.

The ith linked list has a node for vertex j if and only if the graph contains an edge from vertex i to vertex j.

In other words, AdjList[i] is a list of all nodes adjacent to i in the graph

49

50

Adjacency List --- Example 1

1

4

5

6

3

2

8

7

0

0

1

2

3

4

7

5

6

8

2 5

6

6

4

5

3 7

2 8

AdjList[]

51

Adjacency List

An adjacency list is as an array of linked lists

the ith element of the array is a list of vertices that

connect to vertex i

.

52

Example of Adjacency List

0

1

3

2

0

1

2

3

1 2 3

3

3

vertex 0 connect to vertex 1, 2 and 3

vertex 1 connects to 3

vertex 2 connects to 3

53

Weighted Graph

Weighted graph: extend each node with an additional field:

the weight

0

1

3

2

20 10

1

54

0

1

2

3

1 10 2 20 3 1

0 10 3 4

0 20 3 5

0 1 1 4 2 5

54

Adjacency List -Weighted --- Example 2

23

0

16

8

9

0

1

2

3

1 8 3 6

0 8 2 9

1 9

0 6

55

Space Requirement of Representations

Which one is better?

Memory space: adjacency matrix O(N2) adjacency list O(E) where E is # of

edges

Sparse graph adjacency list is better

Dense graph same running time for both

56

Comparison Of Representations where degree is the number of connected nodes

Cost Adjacency Matrix Adjacency List

Given two vertices u and v: Do a LookUP -

find out whether u and v are adjacent

O(1)

Find M[u][v]

degree of node

O(N)

Given a vertex u:

enumerate all neighbors of u

O(N)

Go through whole row

degree of node

O(N)

For all vertices:

enumerate all neighbors of each vertexO(N2)

Summations of all node degrees

O(E)

57

Which is Better?

Operation 1: Is there an edge from vertex i to vertex j? Operation 2: Find all vertices adjacent to vertex i.

Time:

Operation 1 Edge?

Operation 2 Find?

O(1) Search ListO(d)

Traverse ListO(d)

Traverse rowO(n)

Matrix List

Determine which operation is most frequent.

58

Which is Better?

Space:

O(n2)Matrix:

List: n + O(|E|) where E is number of edges AND N IS # OF NODES

O(n+|E|) = O(|V| + |E|)

OR THE SUM OF THE # OF EDGES AND NODES

59

Graph Traversal Algorithms

Two algorithms

Breadth First Traversal - uses a queue

Depth First Traversal - uses a stack

60

Breadth-first Search (BFS)

The breadth-first search in graphs will make use of shadings to identify nodes that have been visited

White nodes are nodes not yet visited

Gray nodes have been discovered but not visited, that is, not considered in the order of the breadth-first traversal

Black nodes have been visited

In the end of the breadth first search, a tree is formed.

This tree is called BFS tree.

61

Breadth-first Search (Start = A ) (Queue)

A B

C D

E

F

G

H

A B

C D

E

F

G

H

62

A B

C D F

G

H

63

Breadth-first Search-First In - is – First out

A B

C D

E

F

G

H

AAQUEUE

Take out A and put in its children- C,D,B

Put in A to start - we will use a Queue

OUTPUT: A

64

Breadth-first Search- First in First out

A B

C D

E

F

G

H

CCQUEUE DD BB

Take out C and put in its children: E,F

OUTPUT: A C

65

Breadth-first Search First In - is – First out

A B

C D

E

F

G

H

DDQUEUE BB EE FF

Take out D and put in its children- all have been visited.

OUTPUT: A C D

66

Breadth-first Search

A B

C D

E

F

G

H

BBQUEUE EE FFTake out B and put in its children:G

OUTPUT: A C D B

67

Breadth-first Search

A B

C D

E

F

G

H

EEQUEUE FF GG

Take out E and put in its children – no unvisited children

OUTPUT: A C D B E

68

Breadth-first Search

A B

C D

E

F

G

H

FFQUEUE GG

Take out F and put in its children

OUTPUT: A C D B E F

69

Breadth-first Search

A B

C D

E

F

G

H

GGQUEUE HH

Take out G and put in its children

OUTPUT: A C D B E F G

70

Breadth-first Search

A B

C D

E

F

G

H

HHQUEUE Take out H and Queue is empty

OUTPUT: A C D B E F G H

71

Breadth-first Search

A B

C D

E

F

G

H

QUEUE is empty

DEPTH FIRST SEARCH

The other major graph traversal is Depth First Search Depth First Search which searches deeper into the graphdeeper into the graph.

Now the ADT used is a stack ADT used is a stack - with the last one in is the first one last one in is the first one out approach.out approach.

The children of the first node are put in the stack, then the last child added is removed and its children are put in.

Then last of the child’s children is taken out and its children are put in. And so on……..

72

73

Depth-first Search (Start = A)- Use Stack

A B

C D

E

F

G

H

OUTPUT: A

74

Depth-first Search

A B

C D

E

F

G

H

AASTACKTake out A and put in its children: C, D, B

OUTPUT: A

75

Depth-first Search

A B

C D

E

F

G

H

CCSTACK DD BB

Take out B and put in its unvisited children : G

OUTPUT: A B

76

Depth-first Search

A B

C D

E

F

G

H

CCSTACK DD GG

Take out G and put in its unvisited children :H

OUTPUT: A B G

77

Depth-first Search

A B

C D

E

F

G

H

CCSTACK DD HHTake out H and put in its unvisited children :F

OUTPUT: A B G H

78

Depth-first Search

A B

C D

E

F

G

H

CCSTACK DD FF

Take out F and put in its children- no unvisited children

OUTPUT: A B G H F

79

Depth-first Search

A B

C D

E

F

G

H

CCSTACK DDTake out D and put in its children – no unvisited children

OUTPUT: A B G H F D

80

Depth-first Search

A B

C D

E

F

G

H

CCSTACK

Take out C and ADD E

OUTPUT: A B G H F D C

81

Depth-first Search

A B

C D

E

F

G

H

EESTACK

Take out E and put in its children – no unvisited children and Stack is empty

OUTPUT: A B G H F D C E

82

Depth-first Search

A B

C D

E

F

G

H

STACK is emptyOUTPUT: A B G H F D C E

83

Graph Traversals --- Basic Algorithm

Depth First Search = structure used for storagewhile (An un-VISITED vertex exists) { Initialize vertex vI ;

mark it as VISITED

while ( stack is not empty) { remove a vertex vj from stack visit vj

add un-VISITED vertices adjacent to vj to stack; mark each as VISITED }}

84

Recursive Depth-First Algorithm

1

4

56

3

2

DepthFirstSearch(vi)

{ visit vj; mark vi as VISITED

for each un-VISITED vertex vj adjacent to vi

DepthFirstSearch (vj) }

85

Graph Traversal

List out all cities that United Airlines can reach from Hartford Airport

CHI

LA

SF

NYC

Hartford

W. DC

86

Demos For Traversal Algorithm

Step 1: { Hartford } find neighbors of Hartford { Hartford, NYC, CHI }

CHI

NYC

LA

SF Hartford

W. DC

87

Demos For Traversal Algorithm

Step 2: { Hartford, NYC, CHI } find neighbors of NYC, CHI NYC goes to LA, CHI goes to SF

{ Hartford, NYC, CHI, LA, SF }CHI

NYC

LA

SF Hartford

W. DC

88

Demos For Traversal Algorithm

Step 3: {Hartford, NYC, CHI, LA, SF } find neighbors of LA, SF no other new neighbors

CHI

NYC

LA

SF Hartford

W. DC

89

Demos For Traversal Algorithm

Finally we get all cities that United Airlines can reach from Hartford Airport {Hartford, NYC, CHI, LA, SF }

CHI

NYC

LA

SF Hartford

W. DC

90

Graph Traversals

Traversal: visit each vertex in the graph once and only onceTraversal: visit each vertex in the graph once and only once. This is an a matrix of vertices.

1

4

56

3

2

79

8

1

1

1 1

1 2 3

1

4

1

2

3

1

1

5

1

14 1

6

1 1 15 1

7 8

16 1

7 1

8 1

9

1

1

9 1 1

91

Elementary Graph Operations

Graph traversals provide the basis for many elementary graph operations:

Spanning trees on graphsSpanning trees on graphs

Graph cycles Graph cycles

Connected components of a graphConnected components of a graph