14. GRAPH in data structures and algorithm.ppt

CONTENTS:
• Introduction to Graph
• Graph Terminologies
• Representation of Graph
• Graph Traversal

Introduction to Graph
• A graph is a non-linear data structure, which
contains a set of vertices (or nodes) and edges
which connect themselves.
• In case of Tree structure, there is a
hierarchical relationship between parent and
child node i.e One parent and many children.
• On the other hand, in case of a graph, the
relationship among the vertices is many
parents to many children.

Introduction to Graph
Mathematically, A graph can be defined as : G = { V, E }
Where V = Set of Vertices and E = Set of Edges
E:g: In the above graph:
Here, V = { A,B,C,D,E}
E = { (A,B),(B,C),(A,D),(B,D),(D,E),(C,E) }

Graph Terminologies
• Undirected Graph
• Directed Graph
• Weighted Graph
• Adjacent Vertices
• Self Loop
• Parallel Edges
• Cyclic Graph
• Acyclic Graph
• Isolated Vertex
• Degree of Vertex
• Pendant Vertex

Undirected Graph
It is a graph such that G = { V,E} where V is set of
vertices and E is the set of unordered pair of vertices.
Here,
V = { A,B,C,D}
E={(A,B),(B,C),(C,D),(A,D),(A,C),(B,D)}
Note: Unordered pair means same edge can be represented
in any combination. For example, the edge (A,B) can be
represented as (B,A).

Directed Graph
It is a graph such that G = {V , E} where V is set of
vertices and E is the set of ordered pair of vertices.
Here,
V = { A,B,C,D}
E={(A,B),(B,C),(C,D),(D,A),(C,A),(B,D)}
Note: If an ordered pair (A,B) is in set E, then there is an
edge directed from A to B (indicated by arrow sign)

Weighted Graph
A graph is termed as weighted if all the edges are
labeled with some weights. A weighted graph can be
undirected or directed.

Adjacent Vertices
In case of directed graph, a vertex X is adjacent to Y, if
there is an edge from X to Y.
Here, Vertex A is adjacent to Vertex B,
Vertex D is adjacent to Vertex A

Self Loop
If there is an edge exist, where starting and end
vertex are same i.e (A,A) is an edge, then it is called a
self loop.
Here, there is a self loop at vertex A

Parallel Edges
If there are more than one edge exist between the
same pair of vertices, then they are known as parallel
edges.
Here, parallel edge exist between Vertex B and Vertex C.

Cyclic & Acyclic Graph
Cycle: If there is path containing edges which start from
vertex A and ends vertex A, then the path is called Cycle.
A graph which has one or more cycles is called Cyclic graph.
Otherwise, it is called Acyclic graph.
Cyclic Graph
Here, three cycles exist.
The path is : (A,B),(B,C),(C,A)
(A,B),(B,D),(D,A)
(B,C),(C,D),(D,A),(A,B)

Isolated Vertex
• A Vertex is isolated, if there is no edge connected
from any other vertex to that vertex.
• Here Vertex E is an isolated vertex as there is no
edge connected to any other vertex of the graph.

Degree of a Vertex ( Undirected Graph)
For undirected graph, the number of edges
connected with a vertex X is degree of that
vertex X.
Here, Degree(A) = 2, Degree(B) = 3
Degree(C) = 2, Degree(D) = 3

Degree of a Vertex ( Directed Graph)
For directed graph, there are two types of
degrees. They are:
1.In-degree : In-degree of a vertex A is the
number of edges incident into A(i.e. the
number of edges ending at A)
2.Out-degree : Out-degree of vertex A is the
number of edges eminating from A (i.e.
number of edges beginning at A )

Degree of a Vertex ( Directed Graph)
In the above directed graph,
Indegree(A) = 2 Outdegree(A) = 1
Indegree(B) = 1 Outdegree(B) = 2
Indegree(C) = 1 Outdegree(C) = 2
Indegree(D) = 2 Outdegree(D) = 1

Pendant Vertex
A vertex X is called pendant if in-degree(X) is 1
and out-degree(X) is 0.
Here, Vertex C, D, E are pendant vertex

Representation of Graph
The graph can be represented using :
1.Sequential representation ( Using Matrix)
 Adjacency Matrix
 Incidence Matrix
2. Linked representation (Using Linked List)

Matrix Representation of Graph
Adjacency Matrix: For representing a graph
“G” with “N” number of vertices, a matrix of
NxN size can be considered.
Assume A[N][N] is matrix representing
Adjacency matrix.
If “i” represents row position and “j”
represents column position, then if there is an
edge from Vi to Vj , Then A[i][j] = 1 else A[i]
[j] = 0 .

Adjacency Matrix: An Example
Represent the following graph in Adjacency
Matrix.
A B C D
A 0 1 0 0
B 0 0 1 1
C 1 0 0 1
D 1 0 0 0

Disadvantage of Adjacency Matrix
The Adjacency matrix has following
disadvantages:
1.It requires n2
space to represent a graph and
also takes n2
time solve most the graph
related problems.
2.It may be difficult to insert and delete nodes,
because the size of matrix need to be
changed.

Matrix Representation of Graph
Incidence Matrix: An Incidence matrix consists
of a row for every vertex and a column for
every edge.
For representing a graph “G” with “N” number
of vertices and “E” number of edges, a matrix
of NxE size can be considered.
Assume A[N][E] is matrix representing
Incidence matrix.
Here, if there is an edge Ek exist from Vi to Vj
then ViEk = 1, VjEk = -1 and other at Ek = 0

Incidence Matrix: An Example
Represent the following graph in Incidence
Matrix.
E1 E2 E3 E4 E5
A 1 0 -1 0 0
B -1 -1 0 0 1
C 0 0 0 -1 -1
D 0 1 1 1 0

Linked Representation of Graph
In the linked representation, an adjacency list
is used to store the Graph into the computer's
memory.
The structure of node will be as follow:
1.VERTEX : It consists the name of the Vertex
2.NEXT : It is a pointer to next vertex
3.LIST : It points to 1st
node of adjacency list
VERTEX NEXT LIST

Linked Representation : An Example
Represent the following graph Using Linked List
VERTEX ADJACENCY LIST
A B
B C,D
C E
D A
E D

Linked Representation : An Example

Graph Traversal
A graph can be traversed in two standard ways. They
are:
1. Depth First Search (DFS)
2. Breadth First Search (BFS)
In DFS, we will take the help of a Stack and in BFS,
we will take the help of a Queue to hold the nodes
for future processing.
During execution of above traversal algorithms, we
consider three states for each vertex :
1. Ready State : It is the initial state of each vertex
2. Waiting State : When the vertex is within Stack or Queue
3. Processed State : When the vertex is visited

Depth First Search (DFS)
• Depth first search (DFS) algorithm starts with the
initial node of the graph G, and then goes to deeper
and deeper until we find the node which has no
children.
• The algorithm, then backtracks from the dead end
towards the most recent node that is yet to be
completely unexplored.
• The data structure which is being used in DFS is
stack.

Depth First Search (DFS) Algorithm
Step 1: SET STATUS = 1 (ready state) for each node in G
Step 2: Push the starting node A on the stack and set its STATUS = 2
(waiting state)
Step 3: Repeat Steps 4 and 5 until STACK is empty
Step 4: Pop the top node N. Process it and set its STATUS = 3
(processed state)
Step 5: Push on the stack all the neighbours of N that are in the
ready state (whose STATUS = 1) and set their STATUS = 2
(waiting state)
[END OF LOOP – Step 3]
Step 6: EXIT

Depth First Search : An Example
Consider the graph G along with its adjacency list, given in the
figure below. Calculate the order to print all the nodes of the
graph starting from node H, by using depth first search (DFS)
algorithm.

Push H onto the stack
STACK : H
Pop the top element of the stack i.e. H, print it and push all the
neighbours of H onto the stack that are is ready state.
Print H
STACK : A
Pop the top element of the stack i.e. A, print it and push all the
neighbours of A onto the stack that are in ready state.
Print A
STACK: B, D
Pop the top element of the stack i.e. D, print it and push all the
neighbours of D onto the stack that are in ready state.

Print D
STACK : B, F
Pop the top element of the stack i.e. F, print it and push all
the neighbours of F onto the stack that are in ready state.
Print F
STACK : B
Pop the top of the stack i.e. B and push all the neighbours
Print B
STACK : C
Pop the top of the stack i.e. C and push all the neighbours.

Print C
STACK : E, G
Pop the top of the stack i.e. G and push all its neighbours.
Print G
STACK : E
Pop the top of the stack i.e. E and push all its neighbours.
Print E
STACK :
Hence, the stack now becomes empty and all the nodes of
the graph have been traversed.
The printing sequence of the graph will be :
H → A → D → F → B → C → G → E

Breadth First Search (BFS)
• Breadth first search is a graph traversal algorithm
that starts traversing the graph from root node and
explores all the neighbouring nodes.
• Then, it selects the nearest node and explore all the
unexplored nodes. The algorithm follows the same
process for each of the nearest node until it finds the
goal.
• The data structure which is being used in DFS is
queue.

Breadth First Search (BFS) Algorithm
Step 1: SET STATUS = 1 (ready state) for each node in G
Step 2: Insert the starting node A into queue and set its STATUS = 2
(waiting state)
Step 3: Repeat Steps 4 and 5 until QUEUE is empty
Step 4: Delete node N and Process it and set its STATUS = 3
(processed state).
Step 5: Delete all the neighbours of N that are in the ready state
(whose STATUS = 1) and set their STATUS = 2 (waiting state)
[END OF LOOP]
Step 6: EXIT

Breadth First Search : An Example
Consider the following graph G shown below , calculate the
minimum path p from node A to node E. Given that each edge
has a length of 1.

Minimum Path P can be found by applying breadth first search
algorithm that will begin at node A and will end at E.
The algorithm uses a QUEUE which holds all the nodes that are
to be processed.
Lets start examining the graph from Node A.
1. Add A to QUEUE.
QUEUE = {A}
2. Delete the Node A from QUEUE and insert all its neighbours.
Process the node A.
QUEUE = {B, D}
PRINT = {A}

3. Delete the node B from QUEUE and insert all its neighbours.
Process node B.
QUEUE = {D, C, F}
PRINT = {A, B}
4. Delete the node D from QUEUE and insert all its neighbours.
Since F is the only neighbour of it which has been inserted, we
will not insert it again. Process node D.
QUEUE1 = {C, F}
PRINT = { A, B, D}
5. Delete the node C from QUEUE1 and insert all its neighbours.
Process node C.
QUEUE = {F, E, G}
PRINT = {A, B, D, C}

6. Remove F from QUEUE and add all its neighbours. Since all of
its neighbours has already been added, we will not add them
again. Process node F.
QUEUE = {E, G}
PRINT = {A, B, D, C, F}
7. Remove E from QUEUE, all of E's neighbours has already been
added to QUEUE, therefore we will not add them again. All the
nodes are visited and the target node i.e. E is encountered.
QUEUE = {G}
PRINT = {A, B, D, C, F, E}
The minimum path will be A → B → C → E.

14. GRAPH in data structures and algorithm.ppt

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