Unit ix graph

Unit – VIII
Graph
Prepared By:
Dabbal Singh Mahara
1

2
Contents
a) Introduction
b) b) Representation of Graph
• Array
• Linked list
c) Traversals
• Depth first Search
• Breadth first search
d) Minimum spanning tree
• Kruskal's algorithm

Introduction
• Graphs are a formalism for representing
relationships between objects.
– a graph G is represented as G = (V, E),
where,
• V is a set of vertices
• E is a set of edges
Balkhu
kalanki
Bafal
V = {Balkhu, Kalanki, Bafal}
E = {(Bafal, Kalanki),
(Balkhu, Kalanki),
(Kalanki, Balkhu)}
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Graph
• A graph G = (V,E) is composed of:
V: set of vertices
E: set of edges connecting the vertices in V
• An edge e = (u,v) is a pair of vertices
• Example:
a b
c
d e
V= {a,b,c,d,e}
E= {(a,b),(a,c),(a,d),
(b,e),(c,d),(c,e),
(d,e)}
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Graph Terminology:
• Node
Each element of a graph is called node of a graph
• Edge
Line joining two nodes is called an edge.
It is denoted by e=[u,v] where u and v are adjacent vertices.
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 Loop
An edge of the form (u, u) is said to be a loop. Here in figure [v2,v2]
 Multiedge
If x was y’s friend several times over, we can model this relationship
using multiedges. In figure e1 and e3.
Loop
e1
V1
V2e2V2
node
edge
e3

Adjacent and Incident
• If (v0, v1) is an edge in an undirected graph,
• v0 and v1 are adjacent
• The edge (v0, v1) is incident on vertices v0 and v1
• If <v0, v1> is an edge in a directed graph
• v0 is adjacent to v1, and v1 is adjacent from v0
• The edge <v0, v1> is incident on v0 and v1
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• The degree of a vertex is the number of edges incident to that
vertex
• A node with degree 0 is known as isolated node.
• A node with degree 1 is known as pendant node.
• 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
• if di is the degree of a vertex i in a graph G with n vertices and e
edges, the number of edges is
Degree of a Vertex
8

0
1 2
3 4 5 6
G1 G2
3
2
3 3
1 1 1 1
directed graph
in-degree
out-degree
0
1
2
G3
in:1, out: 1
in: 1, out: 2
in: 1, out: 0
0
1 2
3
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3
Examples
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Path
• Path: A sequence of
vertices v1,v2,. . .vk such
that consecutive vertices vi
and vi+1 are adjacent.
• Example: {1, 4, 3, 5, 2}
1
4 5
3
2
a b
c
d e
a b
c
d e
a b e d c b e d c
• Length of a path: Number of edges on the path

• simple path: The path with no repeated vertices
• cycle: simple path, except that the last vertex is the same as the
first vertex
a b
c
d e
b e c
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• subgraph: subset of vertices and edges forming a graph
• connected component: maximal connected subgraph. E.g., the graph below has
3 connected components.
connected not connected
•connected graph: any two vertices are connected by some path
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Types
• Graphs are generally classified as,
• Directed graph
• Undirected graph
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Un Directed graph
• A graphs G is called directed graph if each edge has no
direction.
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• Simple Graph
A graph in which there is no loop and no multiple edges between two nodes.
• Multigraph
The graph which has multiple edges between any two nodes but no loops is called
multigraph.
Types of Graph

Directed graph
• A graphs G is called directed graph if each edge has a
direction.
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• Pseudograph
A graph which has loop is called pseudograph.

• Complete graph
A graph G is called complete, if every nodes are adjacent
with other node
• Weighted graph
If each edge of graph is assigned a number or value, then it
is weighted graph. The number is weight.
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Why Use Graphs?
• Graphs serve as models of a wide range of objects:
– A roadmap
– A map of airline routes
– A layout of an adventure game world
– A schematic of the computers and connections that make up the
Internet
– The links between pages on the Web
– The relationship between students and courses
– A diagram of the flow capacities in a communications or transportation
network

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Representations of Graphs
• To represent graphs, you need a convenient way to store the
vertices and the edges that connect them
• Two commonly used representations of graphs:
– The adjacency matrix
– The adjacency list

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Adjacency Matrix
• If a graph has N vertices labeled 0, 1, . . . , N – 1:
– The adjacency matrix for the graph is a grid G with N rows and N columns
– Cell G[i][ j] = 1 if there’s an edge from vertex i to j
• Otherwise, there is no edge and that cell contains 0
• These are the simplest ways for representing graphs.
• Space requirement: O(n2)
• Adding and deleting edge: O(1)
• Testing an edge : O(1)

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Adjacency Matrix (continued)
• If the graph is undirected, then four more cells are occupied by 1:
• If the vertices are labeled, then the labels can be stored in a
separate one-dimensional array

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Adjacency List
• If a graph has N vertices labeled 0, 1, . . . , N – 1,
– The adjacency list for the graph is an array of N linked lists
– The ith linked list contains a node for vertex j if and only if there is an
edge from vertex i to vertex j
– It is suitable for sparse graphs i.e. graphs with the few edges
– Space required: O(V+E)
– Time for
• Testing edge to u O(dge(u))
• Finding adjacent vertices: O(deg(u))
• Insertion and deletion : O(deg(u))

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Exercise:
Construct Adjacency List and Adjacency Matrix
i. ii.
1
5 4
3
2

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 One of the most fundamental graph problems is to traverse every edge and
vertex in a graph. Applications include:
 Printing out the contents of each edge and vertex.
 Counting the number of edges.
 Identifying connected components of a graph.
 Graph traversal algorithms visit the vertices of a graph, according to some
Strategy.
 Given G=(V,E) and vertex v, find all wV, such that w connects v
 Depth First Search (DFS): preorder traversal
 Breadth First Search (BFS): level order traversal
Graph Traversals

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DFS
The basic idea is:
• Start from the given vertex and go as far as possible
i.e. search continues until the end of the path if not visited
already,
• Otherwise, backtrack and try another path.
• DFS uses stack to process the nodes.
Algorithm:
DFS(G,S)
{
T = { S };
Traverse(S);
}
Traverse(v)
{
for each w adjacent to v not yet in T
{
T = T U {w}; // add edge {v,w} in T
Traverse(w);
}
}

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2
1
6
7
5
3
4
Example: DFS Tracing
Starting Vertex : 1
2
1
6
7
5
3
4
Visited Nodes: T = { }
2
1
6
7
5
3
4
T = { 1 }
T = { 1, 2 }

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2
1
6
7
5
3
4
T = { 1, 2, 3 }
backtrack from 3
Visit next branch from 2
2
1
6
7
5
3
4
T = { 1, 2, 3, 4 }
2
1
6
7
5
3
4
T = { 1, 2, 3, 4, 7 }

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T = { 1, 2, 3, 4, 7, 5 }
2
1
6
7
5
3
4
2
1
6
7
5
3
4
T = { 1, 2, 3, 4, 7, 5, 6 }
2
1
6
7
5
3
4
Final DFS tree.

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Analysis:
• The complexity of the algorithm is greatly affected by
Traverse function we can write its running time in terms of
the relation,
T(n) = T(n-1) + O(n),
• At each recursive call a vertex is decreased and for each
vertex atmost n adjacent vertices can be there. So, O(n).
• Solving this we get, T(n) = O(n2). This is the case when
we use adjacency matrix.
• If adjacency list is used, T(n) = O(n + e), where e is
number of edges.

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BFS
• This is one of the simplest methods of graph searching.
• Choose some vertex as a root or starting vertex.
• Add it to the queue. Mark it as visited and dequeue it from the queue.
• Add all the adjacent vertices of this node into queue.
• Remove one node from front and mark it as visited. Repeat this process until all the
nodes are visited.
BFS (G, S)
{
Initialize Queue, q = { }
mark S as visited;
enqueue(q, S);
while( q != Empty)
{
v = dequeue(q);
for each w adjacent to v
{
if w is not marked as visited
{
enqueue(q, w)
mark w as visited.
}
}
}
}

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Analysis:
• This algorithms puts all the vertices in the queue and they
are accessed one by one.
• for each accessed vertex from the queue their adjacent
vertices are looked up for O(n) time (for worst case).
• Total time complexity , T(n) = O(n2) , in case of adjacency
matrix.
• T(n) = O(n +e), in case of adjacency list.

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2
1
6
7
5
3
4
Example: BFS Algorithm Tracing
Solution:
2
1
6
7
5
3
4
 Starting vertex : 1
Visited: { 1 }
2
1
6
7
5
3
4
Visited: { 1,2, 3, 7, 5, 6 }
1
2 3 7 5 6Queue :
Queue :

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2
1
6
7
5
3
4
 Dequeue front of queue. Add its all unvisited
adjacent nodes to the queue.
Queue 3 7 5 6 4
Visited : { 1, 2, 3, 7, 5, 6, 4 }
2
1
6
7
5
3
4
Queue
Visited : { 1, 2, 3, 7, 5, 6, 4 }

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2
1
6
7
5
3
4
Final BFS Tree.

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Spanning Tree
A spanning tree of a connected undirected graph G is a sub graph
T of without cycle G that connects all the vertices of G.
i.e. A spanning tree for a connected graph G is a tree containing
all the vertices of G
• A minimum spanning tree in a connected weighted graph is a
spanning tree that has the smallest possible sum of weights of its
edges.
• It represents the cheapest way of connecting all the nodes in G.
• It is not necessarily unique.
• Any time you want to visit all vertices in a graph at minimum cost
(e.g., wire routing on printed circuit boards, sewer pipe layout,
road planning…)
Minimum Spanning Trees

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• Two algorithms that are used to construct the minimum
spanning tree from the given connected weighted graph of
given graph are:
 Kruskal's Algorithm
 Prim's Algorithm
MST Generating Algorithms
Kruskal's Algorithm
We have V as a set of n vertices and E as set of edges of graph G. The idea behind
this algorithm is:
• The nodes of the graph are considered as n distinct partial trees with one
node each.
• At each step of the algorithm, two partial trees are connected into single
partial tree by an edge of the graph.
• While connecting two nodes of partial trees, minimum weighted arc is
selected.
• After n-1 steps MST is obtained.

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Algorithm:
Kruskal_MSt( G )
{
T = { V } // forest of n nodes
S = Set of edges sorted in non-decreasing order of weight
while( |T| < n-1 AND S != Empty)
{
select edge (u,v) from S in order
S = S – (u,v)
if (u,v) does not form cycle in T
T = T U {(u,v)}
}
}
Complexity Analysis
To form the forest of n trees takes O(n) time, the creation of S takes O(E.log E)
time and while loop executes O(n) time and the steps inside loop take almost
linear time.
So, total time – O(n) + O (E logE) +O(n log n)

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2
1
6
7
5
3
4
18
7
10
15
8
9
11
13
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Exercise: Trace Kruskals algorithm.

Unit ix graph

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