– Graphs 1 Graph Categories Strong Components Example of Digraph

1 Main Index Contents1 Main Index Contents

Graph Categories

Example of Digraph

Connectedness of Digraph

Adjacency Matrix

Adjacency Set

vertexInfo Object

Vertex Map and Vector vInfo

VtxMap and Vinfo Example

Breadth-First Search Algorithm (2 slides)

Dfs()

– – GraphsGraphsStrong Components

Graph G and Its Transpose GT

Shortest-Path Example (2 slides)

Dijkstra Minimum-Path Algorithm (2 slides)

Minimum Spanning Tree Example

Minimum Spanning Tree: vertices A and B

Completing the Minimum Spanning-Tree

with Vertices C and D

Summary Slides (4 slides)

2 Main Index Contents

Königsberg Bridge ProblemKönigsberg Bridge Problem

AD

a

b

cd

e

f

g

B

C

River

Land

Land

A and D are two islands in the river;

a, b, c, d, e, f, and g are bridges.

Question: Starting at one land area, is it possible to walk across all the bridges exactly once and return to the starting area?


Euler’s ModelEuler’s Model

C

B

A D

c

a

d

b

e

f

g


Graph definitions and notationsGraph definitions and notations A graph G is a pair, G = (V, E), where V is a finite nonempty set of

vertices of G and E V x V., i.e., the elements of E are pair of elements of V and E is called edges.

Let V(G) denote the set of vertices, and E(G) denote the set of edges of G.

– if elements of E(G) are ordered pairs, G is called directed graph or digraph; otherwise, G is called an undirected graph in which (u, v) and (v, u) represent the same edge.

Let G be a graph, a graph H is called a subgraph of G if V(H) V(G) and E(H) E(G); i.e., every vertex of H is a vertex of G and every edge in H is an edge in G.

A graph can be shown pictorially: a vertex is drawn as a circle, and an edge is drawn as a line. In a directed graph, the edges are drawn using arrows. Additional terms will be defined in the following slides.


Graph CategoriesGraph Categories

( a : C o n n e c t e d) ( c : C o m p le t e )( b: D isc o n n e c t e d)

A graph is connected if each pair of vertices have a edge or path between them

A complete graph is a connected graph in which each pair of vertices are linked by an edge


Example of DigraphExample of Digraph

Vert ices V = { A , B , C , D , E }E d ges E = { (A ,B ), (A ,C ), (A ,D ), (B ,D ), (B ,E ), (C ,A ), (D ,E )}

A

DE

B

C

Graph with ordered edges are called directed graphs or digraphs


Connectedness of DigraphConnectedness of Digraph

A

C

B

E

D

A

DC

B

C

A

ED

B

(a) (b ) (c)

N o t St ro ngly o r W eak ly C on nect ed(N o p at h E t o D o r D t o E )

St ron gly C o nn ect ed W eak ly C o n n ect ed(N o p at h fro m D t o a v ert ex)

Strongly connected if there is a path from any vertex to any other vertex. Weakly connected if, for each pair of vertices vi and vj,

there is either a path P(vi, vj) or a path P(vi,vj).


Representation of a graph: Representation of a graph: Adjacency MatrixAdjacency Matrix

( a)

D

A

C

E

B

( b )D

B

A

E

C

4

2 7

3

6

4

1 2

An m by m matrix (where m is the number of vertices in the graph), called an adjacency matrix, identifies the edges. An entry in row i and column j corresponds to the edge e = (vi,, vj). Its value is the weight of the edge, or –1 (or 0) if the edge does not exist.


Another way of Representing a Another way of Representing a graph: Adjacency listgraph: Adjacency list

( a )

D

A

C

E

B

A

E

D

C

B

Vert ices Set o f N eighb o rs

B 1 C 1

C 1

B 1

D 1B 1

A

E

D

C

B

Vert ices Set o f N eigh b ors(b )

D

B

A

E

C

4

2 7

3

6

4

1 2

C 1

C 7B 4

A 2

B 3

E 4

E 2

D 6


Definition of a node in a graphDefinition of a node in a graphtemplate <class vType)class nodeType {public:

vType vertex;nodeType<vType> *link;

};


Operations on a GraphOperations on a Graph Common operations

– Create a graph (demonstration based on adjacent list with a linked list)

– Clear the graph– print a graph– Determine if the graph is empty– Traverse the graph– Others (depending on applications)


Review: Linked list ADTReview: Linked list ADTtemplate <class Type>class nodeType {public:

type info;nodetype<Type> *link;

};Template<class Type>class linkedListType {public:

const linkedListtype<Type> &operator=(const linkedListtype<Type> &);void initializedList( );bool isEmpty( );void print( );int length( );void destroyList( );


Review: Linked list ADT Review: Linked list ADT continuedcontinued

void retrieveFirst(Type &firstElement);void search(const Type &searchItem); // displys “Item is/is not found”void insertFirst(const Types &newItem);void insertLast(const Types &newItem);void deleteNode(const types &deleteItem);linkedListType( ); // initialize first = last = NULLlinkedListType(const linkedListtype<type> &otherList); // copy constructor~linkedListType( ); // delete all nodes in the listprotedted:

nodeType<Type> *first;nodeType<Type> *last;

};


Review: TemplateReview: Template// specifies an array data member with generic element// type elemType and its size with the following templatetemplate<class elemType, int size>class listType {public:

…private:

int maxSize;int length;elemType listelem[size];

};

To create a list of 100 int elements:

listType<int, 100> inList;


Graph ADT: Adjacent list Graph ADT: Adjacent list approach – the interfaceapproach – the interface

template<class vType, int size>class graphType {public:

bool isEmpty( );void createGraph( );void clearGraph( ); // vertices deallocatedvoid printGraph( );graphtype( ); // default constructor; gsize = 0 and maxSize = size~graphType( ); // storage deallocated

protected:int maxSize; // max number of verticesint size; // current number of verticeslinkedListGraph<vType> *graph; // array of pointers to create// the adjacency list

};


Graph ADT: Adjacent list Graph ADT: Adjacent list approach – the implementationapproach – the implementation

template<class vType, int size>bool graphType<vtype, size>::isEmpty( ){

return (gsize == 0);}To implement the createGraph( ) funciton: assuming the user

is prompted to enter at run time 1) number of vertices, 2) each and every vertex and its adjacent vertices, e.g.,

50 2 4 … -991 0 3 … -992 …Note that –99 is the sentinel value used to terminal the input

loop.



template<class vType, int size>bool graphType<vtype, size>::createGraph( ){

vtype vertex, adjacentVertex;if (gSize != 0)

clearGraph( );cout << “Enter # vertices followed by each individual vertices and “

<< “its adjacent vertices ending with –99: ”;cin >> gSize;for (int i = 0; i < gSize; i++) {

cin >> vertex;cin >> adjacentVertex;while (adjacentVertex != -99) {

graph[vertex].insertLast(adjacentVertex);cin >> adjacentVertex;

}}

}



template<class vType, int size>bool graphType<vtype, size>::clearGraph( ){

int i;for (i = 0; i < gSize; i++)

graph[i].destroyList( );gSize = 0;

}


The following slides for The following slides for reference onlyreference only


vertexInfo ObjectvertexInfo Object

v t x M a p L o c

e dge s ( se t o f n e igh bo r s)

in D e gr e e

o c c up ie d

c o lo r

Id en t ifies Vert ex

U s ed t o bu ildan d u s e a grap h

da t a Va lue

p a r e n t

v

v t xM ap L o c

Vert ex v in a m apv ert exIn fo O b ject

A d jacen cy Set

d es t w eig h t

d es t w eig h t

A vertexInfo object consists of seven data members. The first two members, called vtxMapLoc and edges, identify the vertex in the map and its adjacency set.


Vertex Map and Vector vInfoVertex Map and Vector vInfo

v e r t e xm I t e r

( it e r a t o rlo c a t io n )

in de x

. . .in de x

v t x M a p L o c

e dge s

in D e gr e e

o c c up ie d

c o lo r

da t a Va lue

v t x M a p

dist a n c e

v In fo

To store the vertices in a graph, we provide a map<T,int> container, called vtxMap, where a vertex name is the key of type T. The int field of a map object is an index into a vector of vertexInfo objects, called vInfo. The size of the vector is initially the number of vertices in the graph, and there is a 1-1 correspondence between an entry in the map and a vertexInfo entry in the vector


VtxMap and Vinfo ExampleVtxMap and Vinfo Example

v t x M a p

A 0

lo c AB 1

lo c BC 2

lo c CD 3

lo c D0 21 3

v I n f o

1 ( B ) 1

3 ( D ) 1

0 ( A ) 12 ( C ) 1

2 ( C ) 1

lo c A

e dge s

in D e gr e e = 1

. . . .

lo c B

e dge s

in D e gr e e = 1

. . . .

lo c C

e dge s

in D e gr e e = 2

. . . .

lo c D

e dge s

in D e gr e e = 1

. . . .


Breadth-First Search AlgorithmBreadth-First Search Algorithm

E

D

A

CB

F

G

B

vis it Q u eu e

v is it Set A

(a)

B C G v is it Q u eu e

v is it Set A

(b )

C G D vis it Q u eu e

v is it Set A

(c)

G D

CB


Breadth-First Search… (Cont.)Breadth-First Search… (Cont.)

E Fv is it Q u eu e

v is it SetA

(e)

DG

CB

Dv is it Q u eu e

v is it SetA

(d )

G

CB

E

F

v is it Q u eu e

v is it SetA

(g)

DG

CB

Fv is it Q u eu e

v is it SetA

(f)

G

CB

D E


dfs()dfs()

AB

C

v = A , w = BF irs t call t o d fs Vis it () s t art s at D .Secon d call t o d fs Vis it () s t art s at A .d fs Lis t : A D C B

D

(a)

B

D

C

v S = A , v = B , w = Cd fs Vis it () s t art s at B , d is co v ersn eigh bo r D , an d t h en n eigh b o r C .d fs L is t : A B D C

(b )

A

DC

v S = A , v = B , w = Cdfs Vis it () s t art s at A , d is co vers C ,d is co v ers D , and finally d is co v ersB . T h e n eigh b or o f B (C ) is G R A Y(a b ack edge).d fs L is t : < em p t y >p at h C D B C is a cy cle

B

A C

B

v S = A , v = B , w = Cd fs Vis it () s t art s at A an d p ro ceed salon g t he p at h fro m B t o C .d fs L is t : A B C

(d )

A

(c)


Strong ComponentsStrong Components

F

E

C

A

BC o m p o n en tsA, B, CD , F , GED

G

A strongly connected component of a graph G is a maximal set of vertices SC in G that are mutually accessible.


Graph G and Its Transpose Graph G and Its Transpose GGTT

F

E

C

A

B

G

D

G rap h G

F

E

C

A

B

G

D

G rap h G T

The transpose has the same set of vertices V as graph G but a new edge set ET consisting of the edges of G but with the opposite direction.


Shortest-Path ExampleShortest-Path Example

A B

CD

E

F

The shortest-path algorithm includes a queue that indirectly stores the vertices, using the corresponding vInfo index. Each iterative step removes a vertex from the queue and searches its adjacency set to locate all of the unvisited neighbors and add them to the queue.


Shortest-Path ExampleShortest-Path Example

E 1 , F A 2 , D

v is it Q u eu e

(a)

v is it Q u eu e

(c)

v is it Q u eu e

(b )

D 1 , F A 2 , DE 1 , F

E 1 , F A 2 , D

v is it Q ueu e

(a)

v is it Q ueu e

(c)

v is it Q ueu e

(b )

D 1 , F A 2 , DE 1 , F B 3 ,A C 3 ,A

v is it Q u eu e

(d )

Example: Find the shortest path for the previous graph from F to C.


Dijkstra Minimum-Path Dijkstra Minimum-Path Algorithm From A to D ExampleAlgorithm From A to D Example

A B

CD

E

F

6

4

2 0 1 4

12

4

2

6

8 11

m in In fo (B ,4 ) m in In fo (E ,4 )m in In fo (C ,1 1 )

p rio rit y q u eu e


Dijkstra Minimum-Path Dijkstra Minimum-Path Algorithm From… (Cont…)Algorithm From… (Cont…)

m in In fo (C ,1 0 ) m in In fo (E ,4 )m in In fo (C ,1 1 )


m in In fo (D ,1 2 )

m in In fo (C ,1 0 ) m in In fo (C ,1 1 )






Minimum Spanning Tree Minimum Spanning Tree ExampleExample

B

A

H

E

G

F

D

C

5 0 25

462 5

23

5 5

3 29 8

35

6 7

B

A

H

E

G

F

D

C

25

4 62 5

23

5 5

3 2

35

M in im u m am o u n t o f cab le = 2 41

M in im u m s p ann in g t reeN et w o rk o f H u b s


Minimum Spanning Tree: Minimum Spanning Tree: Vertices A and BVertices A and B

A

B C

D

5

A

B

(a) (b )

Sp an n in g t ree w it h v ert ices A , Bm in Sp an T reeSiz e = 2 , m in T reeW eigh t = 2


Completing the Minimum Spanning-Tree Completing the Minimum Spanning-Tree Algorithm with Vertices C and DAlgorithm with Vertices C and D

C

7D

A

B

(a)

Sp ann in g t ree w it h v ert ices A , B , Dm in Sp an T reeSiz e = 3 , m in T reeW eigh t = 7

2

5

D

A

B

(b )

Sp ann in g t ree w it h v ert ices A , B , D , Cm in Sp an T reeSiz e = 4 , m in T reeW eigh t = 1 4

2

5


Summary Slide 1Summary Slide 1

§- Undirected and Directed Graph (digraph) - Both types of graphs can be either weighted or

nonweighted.



§- Breadth-First, bfs() - locates all vertices reachable from a starting vertex

- can be used to find the minimum distance from a starting vertex to an ending vertex in a graph.



§- Depth-First search, dfs() - produces a list of all graph vertices in the reverse

order of their finishing times.

- supported by a recursive depth-first visit function, dfsVisit()

- an algorithm can check to see whether a graph is acyclic (has no cycles) and can perform a

topological sort of a directed acyclic graph (DAG)

- forms the basis for an efficient algorithm that finds the strong components of a graph



§- Dijkstra's algorithm - if weights, uses a priority queue to determine a path

from a starting to an ending vertex, of minimum weight

- This idea can be extended to Prim's algorithm, which computes the minimum spanning tree in an undirected, connected graph.

– Graphs 1 Graph Categories Strong Components Example of Digraph

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