Lecture 2.3.1 Graph.pptx

DISCOVER . LEARN . EMPOWER
GRAPHS
UNIVERSITY INSTITUTE OF
ENGINEERING
DEPARTMENT OF COMPUTER SCIENCE
AND ENGG.
Bachelor of Engineering (Computer Science & Engineering)
DATA STRUCTURES CST-231
Prepared by: Er. Hatesh

• https://www.thecrazyprogrammer.com/2
017/08/difference-between-tree-and-
graph.html
2
GRAPHS
CO
Number
Title Level
CO1 Able to develop applications using the graph data
structure.
Applying
CO2 Able to compare between different types of tree
data structures and use an appropriate data
structure for a design.
evaluating
CO3 Identify the strengths and weaknesses of different data
structures.
Analyzing
COURSE OUTCOMES
CO1

 A graph G consists of two sets
– a finite, nonempty set of vertices V(G)
– a finite, possible empty set of edges E(G)
– G(V,E) represents a graph
 An undirected graph is one in which the pair of vertices in a
edge is unordered, (v0, v1) = (v1,v0)
 A directed graph is one in which each edge is a directed pair of
vertices, <v0, v1> != <v1,v0>
A Graph is a non-linear data structure consisting
of nodes and edges.
The nodes are sometimes also referred to as
vertices and the edges are lines or arcs that
connect any two nodes in the graph. ...
Graphs are used to represent networks.
3
GRAPHS
https://tekslate.com/graphs-in-data-structures

in Facebook, each person is represented with a vertex or a
node. Each node is a structure and contains the information
like user id, user name, gender etc.
Graph 1:
V = {A, B, C, D, E, F}
E = {(A, B), (A, C), (B, C), (B, D), (D, E), (D, F), (E, F)}
Graph 2:
V = {A, B, C, D, E, F}
E = {(A, B), (A, C), (B, D), (C, E), (C, F)}
Graph 3:
V = {A, B, C}
E = {(A, B), (A, C), (C, B)}
4
Examples of
GRAPHS
https://www.tutorialride.com/data-structures/graphs-in-data-
structure.htm
Set of Vertices V
Set of Edges E

Graph is a data structure that consists of following
two components:
1. A finite set of vertices also called as nodes.
2. A finite set of ordered pair of the form (u, v)
called as edge.
The pair is ordered because (u, v) is not same as (v,
u) in case of a directed graph(di-graph). The pair of
the form (u, v) indicates that there is an edge from
vertex u to vertex v. The edges may contain
weight/value/cost.
s.
5
Graph
Representations
http://btechsmartclass.com/data_structures/graph-
representations.html/

An adjacency matrix is a square matrix used to
represent a finite graph.
In the special case of a finite simple graph, the
adjacency matrix is a (0,1) with zeros on its diagonal.
The adjacency matrix should be distinguished from the
incidence for a graph, a different matrix representation
whose elements indicate whether vertex–edge pairs are
incident or not, and degree.
6
Adjacency Matrix
https://www.ebi.ac.uk/training/online/course/network-
analysis-protein-interaction-data-
introduction/introduction-graph-theory/graph-0
• .

Adjacency list representations of graphs take a more
vertex-centric approach.
There are many possible implementations of adjacency
lists representation of graph.
Here we will implement it in 2 ways:- one using array of
vectors and another using vector of lists.
The representation of graph is implemented using
adjacency list. It makes use of STL(Standard Template
Library of C++)
7
Adjacency List
https://www.kodefork.com/articles/adjacency-list-graph-
representation-using-c-stl-in-competitive-programming/
• .

Graph Terminology
Vertex
A individual data element of a graph is called as Vertex. Vertex is also known as node. In
above example graph, A, B, C, D & E are known as vertices.
Edge
An edge is a connecting link between two vertices. Edge is also known as Arc. An edge is
represented as (starting Vertex, ending Vertex). For example, in above graph, the link between
vertices A and B is represented as (A,B). In above example graph, there are 7 edges (i.e., (A,B), (A,C),
(A,D), (B,D), (B,E), (C,D), (D,E)).
Edges are three types.
Undirected Edge - An undirected edge is a bidirectional edge. If there is a undirected edge
between vertices A and B then edge (A , B) is equal to edge (B , A).
Directed Edge - A directed edge is a unidirectional edge. If there is a directed edge between
vertices A and B then edge (A , B) is not equal to edge (B , A).
Weighted Edge - A weighted edge is an edge with cost on it.
8

Graph Terminology
Degree
Total number of edges connected to a vertex is said to be degree of that vertex.
Indegree
Total number of incoming edges connected to a vertex is said to be indegree of that vertex.
Outdegree
Total number of outgoing edges connected to a vertex is said to be outdegree of that vertex.
Parallel edges or Multiple edges
If there are two undirected edges to have the same end vertices, and for two directed edges to have the
same origin and the same destination. Such edges are called parallel edges or multiple edges.
Self-loop
An edge (undirected or directed) is a self-loop if its two endpoints coincide.
Simple Graph
A graph is said to be simple if there are no parallel and self-loop edges.
Path
A path is a sequence of alternating vertices and edges that starts at a vertex and ends at a vertex such that
each edge is incident to its predecessor and successor vertex.
http://www.csl.mtu.edu/cs2321/www/newLectures/24_Graph_Terminology.html
9

A vertex is incident to an edge if the vertex is one of
the two vertices the edge connects. Two distinct
incidences and are adjacent
10
Adjacent and
Incident
http://www.personal.kent.edu/~rmuhamma/GraphTheory/
MyGraphTheory/defEx.htm
• .

 The degree of a vertex is the number of edges
incident to that vertex
 For directed graph,
– the in-degree of a vertex v is the number of
edges
that have v as the head
– the out-degree of a vertex v is the number of
edges
that have v as the tail
11
Degree
https://www.slideshare.net/aakashdeepsinghal/lecture-8-e
• .

 A connected component of an undirected graph
is a maximal connected subgraph.
 A tree is a graph that is connected and acyclic.
 A directed graph is strongly connected if there
is a directed path from vi to vj and also
from vj to vi.
 A strongly connected component is a maximal
subgraph that is strongly connected.
.
12
Connected
Component
https://www.slideshare.net/angellaandrea/advanced-
algorithms-1-unionfind-on-disjointset-data-structures
• .

 A simple path is a path in which all vertices,
except possibly the first and the last, are distinct
 A cycle is a simple path in which the first and
the last vertices are the same
 In an undirected graph G, two vertices, v0 and v1,
are connected if there is a path in G from v0 to v1
 An undirected graph is connected if, for every
pair of distinct vertices vi, vj, there is a path
from vi to vj
13
Simple Path
http://www.algolist.net/Data_structures/Graph
• .

Graph traversal is technique used for searching a
vertex in a graph. The graph traversal is also used to
decide the order of vertices to be visit in the search
process. A graph traversal finds the egdes to be used
in the search process without creating loops that
means using graph traversal we visit all verticces of
graph without getting into looping path.
There are two graph traversal techniques and they
are as follows...
• DFS (Depth First Search)
• BFS (Breadth First Search)
14
Graph Traversal
https://medium.com/basecs/going-broad-in-a-graph-bfs-
traversal-959bd1a09255
• .

Depth-first search is an algorithm for traversing or
searching tree or graph data structures.
The algorithm starts at the root node and explores as far
as possible along each branch before backtracking.
DFS for a graph is similar to Depth First Transversal Of A
tree. The only catch here is, unlike trees, graphs may
contain cycles, so we may come to the same node again.
15
Depth First Search
https://www.youtube.com/watch?v=iaBEKo5sM7w
• .

DFS
• DFS traversal of a graph, produces a spanning tree as final result. Spanning Tree is a graph without any loops.
• We use Stack data structure with maximum size of total number of vertices in the graph to implement DFS traversal of a
graph.
We use the following steps to implement DFS traversal...
• Step 1: Define a Stack of size total number of vertices in the graph.
• Step 2: Select any vertex as starting point for traversal. Visit that vertex and push it on to the Stack.
• Step 3: Visit any one of the adjacent vertex of the verex which is at top of the stack which is not visited and push it on to
the stack.
• Step 4: Repeat step 3 until there are no new vertex to be visit from the vertex on top of the stack.
• Step 5: When there is no new vertex to be visit then use back tracking and pop one vertex from the stack.
• Step 6: Repeat steps 3, 4 and 5 until stack becomes Empty.
• Step 7: When stack becomes Empty, then produce final spanning tree by removing unused edges from the graph
16

Breadth-first search (BFS) is an algorithm for traversing
or searching tree or graph data structures.
It starts at the tree root (or some arbitrary node of a
graph, sometimes referred to as a 'search key'), and
explores all of the neighbor nodes at the present depth
prior to moving on to the nodes at the next depth level.
It uses the opposite strategy as depth first search, which
instead explores the highest-depth nodes first before
being forced to backtrack and expand shallower nodes.
17
Breadth First
Search
https://artint.info/2e/html/ArtInt2e.Ch3.S5.SS1.html
• .

BFS
• BFS (Breadth First Search)
• BFS traversal of a graph, produces a spanning tree as final result. Spanning Tree is a graph without any loops. We
use Queue data structure with maximum size of total number of vertices in the graph to implement BFS traversal
of a graph.
We use the following steps to implement BFS traversal...
• Step 1: Define a Queue of size total number of vertices in the graph.
• Step 2: Select any vertex as starting point for traversal. Visit that vertex and insert it into the Queue.
• Step 3: Visit all the adjacent vertices of the verex which is at front of the Queue which is not visited and insert
them into the Queue.
• Step 4: When there is no new vertex to be visit from the vertex at front of the Queue then delete that vertex from
the Queue.
• Step 5: Repeat step 3 and 4 until queue becomes empty.
• Step 6: When queue becomes Empty, then produce final spanning tree by removing unused edges from the graph
18

Advanced topics on Graphs
• Distances in graphs
• Disjoint Set
• Perfect Graphs
• Uniquely Partially Orderable Graphs
• Comparability Graphs
19

Assessment Pattern
• MST 1 -36 MARKS
• MST 2 -36 MARKS
• ELEMNT 1- Quiz- 12 MARKS
• ELEMENT 2 – ST – 9 MARKS
• ELEMENT 3 – assignment – 12 MARKS
• ELEMNET 4 – TUTE – 9 MARKS
• 6 MARKS FOR 90% ATTENDANCE
20

Applications
• Facebook: Each user is represented as a vertex and two people are friends when there is an edge between two vertices.
Similarly friend suggestion also uses graph theory concept.
• Google Maps: Various locations are represented as vertices and the roads are represented as edges and graph theory is
used to find shortest path between two nodes.
• Recommendations on e-commerce websites: The “Recommendations for you” section on various e-commerce websites
uses graph theory to recommend items of similar type to user’s choice.
• Graph theory is also used to study molecules in chemistry and physics.
21

REFERENCES
1. Lipschutz, Seymour, “Data Structures”, Schaum's Outline Series, Tata McGraw Hill.
2. Data structure and algorithm by Narasimha Karumanchi.
3. www.tutorialspoint.com
4. www.geeksforgeeks.com
5. Lipschutz, Seymour, “Data Structures”, Schaum's Outline Series, Tata McGraw Hill.
6. Gilberg/Forouzan,” Data Structure with C ,Cengage Learning.
7. Augenstein,Moshe J , Tanenbaum, Aaron M, “Data Structures using C and C++”, Prentice Hall of India.
8. Goodrich, Michael T., Tamassia, Roberto, and Mount, David M., “Data Structures and Algorithms in C++”, Wiley Student
Edition.
9. Aho, Alfred V., Ullman, Jeffrey D., Hopcroft ,John E. “Data Structures and Algorithms”, Addison Wesley
22

THANK YOU
For queries
Email: hatesh.cse@cumail.in

Lecture 2.3.1 Graph.pptx

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