chapter seven graphs

8/3/2019 Chapter Seven Graphs

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Click to edit Master subtitle style5/3/12

Chapter Seven:

trees & graphs


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Graphs


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What is Graphs?

Agraph is a mathematical abstraction used to represent"connectivity information".

A graph G consists of a set of V or vertices (nodes) and a set ifedges (arcs).

We write G=(V,E).

V is a finite non empty set of vertices.

E is a set of pairs of vertices. These pairs are called edges


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What is Graphs...

An edge e = (v,w), is a pair of vertices v and w, and is said to be incidentwith v and w.

V(G)= { 1,2,3,4,5}

E(G)={ (1,2), (2,3), (3,4), (4,5), (1,5), (1,3), (3,5) }

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Type of Graph

There is two type of graphs

1. undirected graph.

. In an undirected graph, pair of vertices representing any edge

is unordered.. Thus (v,w) and (w,v) represent the same edge.

. V(G)= { 1,2,3,4,5}

.

E(G)={ (1,2), (2,3), (3,4), (4,5), (1,5), (1,3), (3,5) }.

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Type of Graph

2. A Directed Graph

Each edge is an ordered pairs of vertices, i.e. each edge isrepresented by a directed pair.

If e = (v,w) then v is tail or initial vertex and w is head of finalvertex.

Subsequently (v,w) and (w,v) represent two different edges.

The direction is indicated by arrow.

V(G) = {1,2,3,4,5}

E(G) = { (1,2), (2,3), (3,4), (5,3), (5,4), (5,1) }

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Graphs

Adjacent Vertices

Vertex v1 is said to be adjacent to vertex v2 if there is an edge (v1,v2) or (v2, v1).

Ex:

Vertices adjacent to node 3 are 1,5,6,4

Vertices adjacent to node 2 are 1 and 7

Finite Graph:

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Graphs

Loop:

An edge with identical end points is called a loop.

Multiple edges:

The edges connected the same end points.

Multi-Graph:

A graph with multiple edges. For multi graphs, even though, thereare finite number of nodes, the edges may not be finite.


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Graphs

Path:

A path from vertex v to vertex w is a sequence of vertices, eachadjacent to the next.

1,3,4 is a path1,3,6 is a path

1,2,7 is a path

1,3,4,7 is a path

We may notice that there is a path which starts at vertex 1 andfinishes at vertex 1. i.e. path 1,3,4,7,2,1. Such a path is called acycle.

Cycle:- A cycle is a path in which first and last vertices are thesame.

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Graphs

Weighted graphs:

Sometimes, we include a "weight" (number) with each edge.

Weight can signify length (for a geometric application) or

"importance".


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Graphs

Connected Graph

A graph is called 'connected' , if there exists a path betweenany two of its nodes. The above graph is a connected graph.

A weakly connected graph

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Graphs

Strongly Connected

A digraph is called strongly connected if there is a directed pathfrom any vertex to any other vertex.

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Strongly Connected Graph


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Graphs

In a digraph, the path is called a directed pathand a cycle iscalled as directed cycle.

1,2 is a directed path

1,3,5,7,6 is a directed path

1,4,5 is not a directed path There is no directed cycle in the above graph.

Simple Graph: is a graph with the following properties:

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Graphs

Degree

There is no limitation of number of edges incident on one vertex.

It could be none, one or more.

The number of edges incident on a vertex determines its degree.

The number of edges containing a node is called the degree ofthat node.

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Graphs

In a digraph we attach an indegree and an outdegree to each ofthe vetice.

Indegree:

The Indegree of a node 'u' is the number of edges ending at 'u' . Outdegree:

The outdegree of a node 'u' is the number of edges starting from 'u'.

The indegree of vertex 5 is 2

The Outdegree of vertex 5 is 1 54

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Graphs

Tree Graph: It is a special type of graph.

It is connected and

There are no cycles in the graph.1

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Application of Graphs

History:

Maze-searching.

Euler's crossing problem: the Konigsberg bridges


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Application of Graphs

Applications:

Fundamental mathematical construct to represent"connectivity".

Appears in thousands of problems.

Source of many classic problems: travelingsalesman, routing, spanning trees.

Many "graph-structured" applications: networks,transportation-systems, electronic circuits,molecules.

Graphs are the essential data structure in networka lications.


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Graph Representation

The picture with circles (vertices) and lines (edges) is only adepiction=> a graph is purely a mathematical abstraction.

Graph is a mathematical structure and finds its application in

many areas of interest in which problems need to be solvedusing computers.

Thus this mathematical structure must be represented as somekind of data structures.

Two such representations are commonly used. They are,

Adjacent matrix and

Adjacency list representation

The choice of representation depends on the application and


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Graph Representation ...Adjacency Matrix

The adjacency matrix A for graph G = (V,E) with n vertices, isan nXn matrix of bits, such that,

Aij = 1 if there is an edge from vi to vj and

Aij = 0 if there is no such edgeVertice 1 2 3 4 5

1 0 1 1 0 1

2 1 0 1 0 0

3 1 1 0 1 1

4 0 0 1 0 1

5 1 0 1 1 0

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Graph Representation ...Adjacency Matrix

The total number of 1s account for the number of edges in thedigraph.

The number of 1s in each row tells the out degree of thecorresponding vertex.

The total number of 1's in each column tells the in degree of the

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Vertex 1 2 3 4 5 6 7

1 0 1 1 1 0 0 02 0 0 0 0 0 0 03 0 0 0 0 1 0 04 0 0 0 0 0 1 0

5 0 0 0 1 0 0 16 0 0 0 0 0 0 07 0 0 0 0 0 1 0


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Graph Representation ...Adjacency List Representation

we store a graph as a linked structure.

We store all the vertices in a list and then for each vertex, wehave a linked list of its adjacent vertices.

v1

V3V2

V4

V5

V6

V1

V2

V3

V4

V5

V6

V2

V3

V4

V1

V4

V6V5

V3


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Graph Representation ...Adjacency List Representation


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Graph Traversal

A graph traversal means visiting all the nodes of the graphexactly once.

Two graph traversal methods are commonly used. These are,

Depth First Search (DFS) Breadth First Search (BFS)


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Graph Traversal Depth First Search

In graphs, we do not have any start vertex or any special vertexsignaled out to start traversal from.

Therefore the traversal may start from any arbitrary vertex.

We start with vertex v. An adjacent vertex is selected and a depth first search is initiated

from it.

i.e. V1,V2,.Vk are adjacent vertices to vertex v.

We may select any vertex from this list.

Say we select v1.

Now all the adjacent vertices to v1 are identified and all of thoseare visited.

Next v2 is selected and all its adjacent vertices visited and so on.


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Graph Traversal

Depth First Search

Let us start with V1.

V4

V8

V7V6V5

V2 V3

V1

G


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Graph Traversal

Depth First Search

Its adjacent vertices are V2, V8, and V3.

Let us pick on V2.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

Its adjacent vertices are V1, V4, V5.

V1 is already visited.

Let us pick on V4.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

Its adjacent vertices are V2, V8.

V2 is already visited.

Let us pick on V8.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

Its adjacent vertices are V4, V5, V1, V6, V7.

V4 and V1 are already visited.

Let us pick on V5.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search


Both are already visited.

Therefore we back track.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

We have V6 and V7 unvisited in the list of V8.

We may visit any. We visit V6.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

Its adjacent are V8 and V3.

Obviously the choice is V3.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search


We visit V7.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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Graph Traversal

Depth First Search

We may implement the depth first search method by using astack,

pushing all unvisited vertices adjacent to the one just visited and

popping the stack to find the next vertex to visit.

Algorithm

Iterative version:

During the execution of these algorithms, each node will be inone of the following states, called the status of node n.

status=1(Ready state) :- Initial state of the node 'n'.

status=2(waiting state):- 'n' is on queue or stack.

G h T l


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Graph Traversal

Depth First Search

Step1: Initialize all nodes to the ready state i.e. status=1.

Step 2: Push the starting node A onto stack and change itsstatus=2.

Step 3: Repeat while(stack !=empty)a) Pop the top node 'n' of the stack. Process 'n' and change

its

status to the processed state(i.e status=3)

b) Push onto stack all the neighbours of 'n' that are still inthe

ready state and change their status to the waitingstate(i.e.

=

G h T l


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Graph Traversal Breadth First Search

In DFS we pick on one of the adjacent vertices; visit all of theadjacent vertices and back track to visit the unvisited adjacentvertices.

In BFS we first visit all the adjacent vertices of the start vertex

and then visit all the unvisited vertices adjacent to these and soon.

V4

V8

V7V6V5

V2 V3

V1

G h T l


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We start with V1. Its adjacent vertices are V2, V8, V3. We visitall one by one.

We pick on one of these, say V2. The unvisited adjacentvertices to V2 are V4, V5. We visit both.

V1

V2 V3

V8

V1

V5V4

V3V8V2

V1

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We go back to the remaining unvisited adjacent vertices of V1and pick on one of those say V3.

The unvisited adjacent vertices are V6, V7.

There are no more unvisited adjacent vertices of V8, V4, V5,V6 and V7.

Thus the sequence so generated is

V7V6V5V4

V3V8V2

V1

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Here we need a queue instead of a stack to implement it.

We add unvisited vertices adjacent to the one just visited, at the rearand

read at front to find the next vertex to visit.

Abstract view of BFS Algorithm:

Step1: Initialize all nodes to the ready state i.e. status=1.

Step 2: Put the starting node A onto queue and change its

status=2.

Step 3: Repeat while(queue !=empty)

a) Remove the front node 'n' of the queue. Process 'n' andchange

G h T l


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Detailed view of BFS algorithm:

bfs (vertex v)

{

vertex w;

queue q;

visited [v] = true;

initialise (q);

addqueue (q,v)

while (! Emptyqueue(q))

{

deletequeue (q,v);

for all vertices w adjacent to v

if (!visited [w])


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Shortest Path Problem


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Minimum Spanning tree


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Thank you!!!

chapter seven graphs

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