Graph 1

Sunawar Khan
Islamia University of Bahawalpur
Bahawalnagar Campus
Analysis o f Algorithm

‫خان‬ ‫سنور‬ Algorithm Analysis
DFS and BFS
Graph Traversal
Representation of Graph
Minimum Spanning Tree
Topological Sort
Contents

Topics
• Graph Terminology
• Graph Representation
• Matrix Representation
• Linked List Representation
• Depth First Search Algorithm(DFS)
• Strategy
• Application
• Analysis
• Breadth First Search Algorithm(BFS)
• Strategy
• Application
• Analysis

What is Graph
• A graph G = (V, E) is a set of vertices V and a set of edges E, where
• E ⊆ V x V(Subset of cross-product of vertices called binary relation)
• Example: Let G = (V, E) where
• V = {a, b, c, d}
• E = {(a, b) , (a, d) , (b, c) , (b, a) , (c, d) , (b, d) , (d, d)}
• The number of vertices and edges are given by the cardinalities |V| and |E| of
the corresponding sets. The sample graph consists of four vertices and seven
edges. Thus,
|V| = 4, |E| = 7
• Graph are usually represented by pictorial diagrams. The vertices can be shown
as labeled CIRCLE or RECTANGLES. The edges are depicts arcs lines. Except for
some applications, in which distances among vertices are important, the
positions of the vertices are in general, immaterial. Thus, graphs can be shown
pictorially in several ways.

Basic Terminologies
• Mathematical graphs can be represented in data structure. We can
represent a graph using an array of vertices and a two-dimensional array
of edges. Before we proceed further, let's familiarize ourselves with some
important terms −
• Vertex
• Edges
• Adjacency
• Path

Vertex
• Each node of the graph is represented as a vertex. In the following
example, the labeled circle represents vertices. Thus, A to G are vertices.
We can represent them using an array as shown in the following image.
Here A can be identified by index 0. B can be identified using index 1 and
so on.

Edges
• Edge represents a path between two vertices or a line between two
vertices. In the following example, the lines from A to B, B to C, and so on
represents edges. We can use a two-dimensional array to represent an
array as shown in the following image. Here AB can be represented as 1 at
row 0, column 1, BC as 1 at row 1, column 2 and so on, keeping other
combinations as 0.

Adjacency
• Two node or vertices are adjacent if they are connected to each other
through an edge. In the following example, B is adjacent to A, C is adjacent
to B, and so on.

Path
• Path represents a sequence of edges between the two vertices. In the
following example, ABCD represents a path from A to D.

Graph – Undirected Graph
• A graph G = (V,E) with vertex set V = {v1,
v2, v3, . . . , vn} is called undirected if (vi, vj)
= (vj,vi) for i ≠ j
• An undirected graph is sometimes referred
to as undigraph.
• Inn pictorial representation of undirected
graph, the edges are not assigned any
direction.
• Example:- Figure shows an undirected
graph.
• G=(V. E)
• V ={a, b, c, d, e, f}
• E = {(a, b) , (a, d) , (a, f) , (b, c) , (b, e) , (c, f) ,
(c, d) , (d, e) , (e, f)}

Undirected Graph – Complete
• A graph which has links among all of the
vertices in a graph is referred t as
complete.
• Example: The figure shows complete
graph
• An undirected complete graph, with n
vertices, has n(n-1)/2 edges. The spaces
complexity graph is O(n2).
• A complete graph is dense. By contrast, a
graph with space complexity O(n log n) is
called sparse

Graph – Directed Graph
• A graph G = (V,E) with vertex set V = {v1,
v2, v3, . . . , vn} is called undirected if (vi,
vj) = (vj,vi) for i ≠ j
• In other words, the edges (vi, vj) and (vj,
vi), associated with any pair of verties vi,
vj are considered distinct. In pictorial
representation show here wit arrows.
• The directed graph is often referred to
digraph or network.

Directed Graph - Complete
• A complete directed graph has links
among all of the vertices.
• Example:- The figure shows a directed
complete graph.
• A complete directed graph with n
vertices has n(n-1) edges.

Weighted Graphs- Undirected
• A graph in which labels or values w1,
w2, w3, . . . are associated with edges
is called weighted graph. Weights are
typically costs or distances in different
application of graphs.
• Example:- The figure shows an
example of weighted undirected
graph.

Weighted Graph- Directed
• A weighted graph can also be directed.
The weights attached to the edges to
the edges between the same pair of
vertices may, in general, be different
• Example :- The figure below shows an
example of the directed weighted
graph. Note that weight of edged from
vertex a to vertex b is 73 and that from
vertex b too vertex a is 35.

Graph Paths
• A path is a sequence of andjacent
vertices. Two vertices are called
adjacent if they are link oor
connecting edge. The path is denoted
by enclosing the vertices is a pair of
square brackets. In path, the vertices
may repeated.
• Example: The figure below depicts a
path P = [a, b, g, d, e, i, h ]in a sample
graph. The path is shown in bold red
lines.
• The number of edges connecting the vertices in a path is called
path length. The path length in the above example is 6.

Graph Path – Simple Path
• A path is called simple if no vertices are repeated; otherwise, the path is
referred to as non-simple.
• Example: The figures below show simple and non-simple paths in a graph.
The path P1=[ a, c, g, d, f ] is simple. The path P2=[a, c, g, d, f, c, b, f ] is
non-simple, because it passes through vertices c, f twice.

Graph Path - Loops
• A loop is special path that originates and
terminates at a single node, and does not
pass through other vertices.
• Example: The figure depicts two loops in a
graph, at vertices a and c
• The loops are important in certain some
applications. For example, loops represent
certain states in Finite State Automat

Cyclic and Acyclic Graph
• A path that originates and terminates at the same vertex, and links two or
more vertices, is called cycle. If a graph contains a cycle it is called cyclic.
By contrast, a graph which contains no cycles is known as acyclic.
• Example: In diagram (i), the path P=[b, c, d, b] is a cycle. The diagram (ii)
represents a acyclic graph.

Graph Representation – Adjacency Matrix
• One of the standard ways of representing a graph uses a matrix to denote links
between pairs of vertices in a graph. The matrix is known as adjacency matrix.
• The adjacency matrix can be defined mathematically. Let G=(V,E) be a graph,
with V={v1,v2,……vn}, and E={(v1,v2),(v1,v3)….}where n=|V|. The adjacency
matrix A=[aij] as
• 𝑎𝑖𝑗 = ቊ
1 𝑖𝑓 𝑣𝑡, 𝑣𝑗 ∈ 𝐸
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• The definition implies that entry in the ith row and jth column of the matrix is 1
• if a link exists between the ith and jth vertex; otherwise, it is 0.
• The size adjacency matrix is with vertex set V is |V|x|V| . Thus space
complexity is θ(|V|2)

Graph Representation – Adjacency Matrix
• Example : Figure (i) shows a sample directed graph. Figure (ii) shows the
adjacency matrix for the graph.

Graph Representation- Adjacency Link List
• A graph can also be represented using linked lists. The list representation
consists of an array of linked lists, each corresponding to one vertex. The
list stores all of the vertices that are linked to a given vertex.
• Let G=(V,E) be a graph, and Adj(u) be the linked list corresponding to
vertex u, then for all v,
v ∈Adj(u), if (u ,v ) ∈E.
• The vertices belonging to Adj(u) are called neighbors of vertex u, or
adjacent vertices.

Graph Representation – Linked List
• Example:- The diagram below show a sample
graph and its linked list representation.

Graph Traversal
• In graph traversal, all vertices are visited once. Depending on the
application, the data and keys are accessed and processed further. For
example, the traversal procedure can be used to list keys stored in the
vertices The two important procedures for graph traversal are:
1) Depth First Search (DFS)
2) Breadth First Search (BFS)
• The DFS and algorithms can be used to traverse both directed and
undirected graphs

Depth First Search
• The Depth First Search (DFS) systematically visits all vertices of a graph.
Initially, a start vertex is chosen, and its neighbors are explored. Then
neighbors and of neighbors are explored. This process is continued till the
remotest neighbor of the start vertex is reached, which is called the dead
end. The algorithm then backtracks to the start vertex and on the way
back lists all of the vertices in depth first search order. The depth first
search algorithm belongs to the class of backtracking algorithms.
• A stack is used to implement the DFS procedure. The stack keeps track of
vertices traversed . For this purpose , the three states of vertices are
designated as ready, wait and processed, defined as follows :
• All vertices are initially in the ready state
• A vertex pushed on to the stack, is in the wait state
• A vertex popped off the stack is in the processed state

Algorithm
• The Depth First Search (DFS) for a connected graph consists of the
following steps:
• Step #1: Initialize all vertices to mark as unvisited (ready state)
• Step #2: Select an arbitrary vertex, push it to stack (wait state)
• Step #3: Repeat steps # 4 through Step #5 until the stack is empty
• Step #4: Pop off an element from the stack. Process and mark it as visited
(processed)
• Step #5: Push to stack adjacent vertices of the processed vertex. Go toStep
# 3.
• The same procedure is applied to a subgraph, which may contain any
unvisited vertices.

DFS – Spanning Tree
• Apart from extracting information stored in vertices, the DFS can be used
to construct special tree called DFS Spanning trees The Spanning trees are
useful in investigating important properties of the graphs and subgraphs
• During the DFS, a spanning tree is constructed by linking together a vertex
that is visited for the first time to the vertex from which it is reached. The
links are called

Example
• Consider the sample
directed graph shown
below. The steps involved in
Depth First Search are
depicted in the next set of
diagrams As the algorithm
proceeds, the depth first
spanning tree is also
generated.

Solved Example

Applications
• Some important applications of DFS are
(i) Checking connectivity of a graph
(ii) Checking accessibility of vertices from a given vertex
(iii) Checking cyclicity/ acyclicity of a graph
• Graph connectivity and cyclicity are illustrated by the following examples

Graph Connectivity
• An undirected graph is not connected if the DFS algorithm terminates
before all of the vertices have been visited. The unvisited vertices form one
or more subgraphs
• Example: Figure (i) shows a sample graph. Figure (ii) provides a DFS tree.
The vertices M.P,R,Q,D remain unvisited and constitute a disconnected
subgraph.

Subgraph Accessibility
• In a directed graph, all the vertices may have links, but may not be
accessible from given vertex. The inaccessible vertices can be explored by
using DFS
• Example: Figure (i) shows a sample directed graph. Figure (ii) provides a
DFS tree. The vertices L,G,A remain unvisited and constitute subgraph
whose vertices are not accessible from initial vertex M

Acyclic Graph
• An undirected graph is acyclic ( has no cycles) if the DFS spanning has no
back edges
• Example: Figure (i) shows a sample graph . Figure (ii) contains a DFS
Spanning Tree, which has no back edges. The graph is therefore acyclic

Cyclic Graph
• An undirected graph is cyclic if the DFS spanning has back edges
• Example: Figure (i) shows a sample graph . Figure (ii) contains the DFS
spanning tree, which has back edges shown in blue color. The graph is
therefore cyclic

Implementation
• The DFS implements depth-first-search procedure by using a stack S. The
address s of first selected vertex is passed to the method. The
implementation uses an array state[] which stores status of vertices as
READY, WAIT, PROCESSED. The graph is represented by adjacency list Adj .
The notation Adj[u] denotes a neighbor of vertex u

Algorithm
DFS(G, s) ► s is address of starting vertex
1. for each vertex u ∈ V[G] –{ s } ►V[G] is vertex set of graph G. V[G]-{s} is the
difference set that excludes initial vertex s.
2. do state[u] ← READY ►All unvisited vertices are assigned READY status
3. state[s] ← WAIT ► Vertex s is pushed to stack and assigned WAIT status
4. P ←φ ► Stack P is initialized as empty
5. PUSH(S,s) ► The starting vertex s is added to the stack
6. while S ≠φ. ►A loop is created to pop off vertices .
7. do u←POP(S) ► u is the popped vertex
8. for each v ∈ Adj[u] ►All vertices v that are adjacent to vertex u are explored
9. do if state[v] = READY ►If the neighbor vertex v is not visited
10. then state[v]← WAIT ►Then it is assigned WAIT state
11. PUSH(S,v) Pushed on to the stack S
12. state[u] ←PROCESSED ►The vertex u popped off the stack is marked visited
and assigned PROCESSED state

Analysis of DFS – Running Time
• The running time for DFS consists of following major components
Ti = Initializing vertices T
Ts = Scanning Adjacency list
• The Graph G(V,E) has |V| vertices and |E| edges. Each vertex is initialized once.
• Therefore,
Ti= |V|
• The scanning of Adjacency list takes place for each vertex. The total time is
• proportional to the number of edges scanned. Since |E| is the number of edges
in G, in
• worst case the algorithm explores all edges , therefore,
Ts = |E|
• Thus, worst case running time is given by time
TDFS= |V|+ |E|= O(|V|+|E|)

Breadth First Strategy - Strategy
• The Breadth First Search(BFS) is one of the standard graph traversal methods.
Each node is visited once only. The method proceeds by visiting all immediate
neighbors ( vertices one edge away), then visiting neighbor’s neighbors
(vertices two edges away), and so forth, until all reachable vertices have been
visited.
• If a graph has any subgraph(which is not reachable from the initial vertex), the
procedure may be re-started by choosing a vertex from the unvisited subgraph.
• A queue is used to implement the BFS procedure. It keeps track of vertices to
be visited next. A vertex pushed to the queue is said to be in ready state.
• An enqueued vertex is said be in wait state. The dequeued vertex is referred to
as
• .
• The queue is initialized with the address of a starting vertex.

Algorithm
The Breadth First Search (BFS) for a connected graph consists of following
steps:
Step #1: Initialize all vertices to mark as unvisited (ready state)
Step #2: Select an arbitrary vertex, add it to queue (wait state)
Step #3: Repeat steps # 4 through Step #5 until queue is empty
Step #4: Remove an element from the queue. Process and mark it as visited
(processed state)
Step #5: Enqueue all adjacent vertices of the processed vertex. Go to Step #
3.
The same algorithm may be applied to a subgraph, which contains any
unvisited vertices

Example
• Consider the sample directed graph shown below. The steps involved in
performing breadth first search on the graph are depicted in the next set
of diagrams. The execution of algorithm also generates BFS Spanning Tree,
which is shown in bold red lines.

Solved Example
• (1) Initially, all the
vertices are in ready
state. For the execution
of BFS algorithm, an
auxiliary data structure
queue is used, which
holds the vertices in
wait state

Solved Example
• (2) The vertex a is chosen as
the start vertex . The
neighbors b, n, k of a are
pushed to the queue. The
current status of all of the
vertices is given by the legend
placed at bottom. The vertex
traversed so for is a

Solved Example
• (3) The vertex b is removed
from the queue. The vertex c,
neighbor of b, is pushed to the
queue The vertices traversed
so for are: a,b

Solved Example
• (4) The vertex k is removed
from the queue. The neighbor j
of vertex k is pushed to the
queue The vertices traversed
so for are: a,b,k

Solved Example
(5) The vertex n is removed
from the queue. The neighbor l
of vertex n is pushed to the
queue .
• The vertices traversed so for
are: a,b,k,n

Solved Example
(6) The vertex c is removed from
the queue. The neighbor d of
vertex c is pushed to the queue .
are: a,b,k,n,c

Solved Example
(7) The vertex j is removed from
the queue. The neighbor i of
vertex j is pushed to the queue.
are: a,b,k,n,c,j

Solved Example
(8)The vertex l is removed from
the queue. The neighbors e, m of
vertex l are pushed to the queue.
are: a,b,k,n,c,j,l

Solved Example
(9)The vertex d is removed from
the queue. The vertex d has no
unvisited neighbor.
are: a,b,k,n,c,j,l,d

Solved Example
(10)The vertex i is removed
from the queue. The vertex i has
no unvisited neighbor.
are: a,b,k,n,c,j,l,d,i

Solved Example
(11)The vertex e is removed
from the queue. The neighbor f
of vertex e is pushed to the
queue.
are: a,b,k,n,c,j,l,d,I,e.

Solved Example
(12) The vertex m is removed
from the queue. The neighbor g
of vertex m is pushed to the
queue.
are: a,b,k,n,c,j,l,d,I,e,

Solved Example
(13) The vertex f is removed
from the queue. The vertex f has
no unvisited neighbor
are: a,b,k,n,c,j,l,d,I,e,f

Solved Example
(14)The vertex g is removed
from the queue. The vertex h
neighbor of g is pushed to the
queue.
are: a,b,k,n,c,j,l,d,I,e,f,g

Solved Example
(15)The vertex h is removed
from the queue. The vertex h
has no unvisited neighbors.
Further, the queue is empty.
The breadth first search
algorithm terminates

Implementation
• The BFS implements breadth-first-search procedure by using a queue Q.
The address s of first selected vertex is passed to the method. The
implementation uses an array state[] to keep track of the state of vertices:
READY, WAIT, PROCESSED. The graph is represented by adjacency.

Algorithm
BFS(G, s) ► s is address of starting vertex
1. for each vertex u ∈ V[G] –{ s } ►V[G] is vertex set of graph G. V[G]-{s} is the
difference set that excludes initial vertex s.
2. do state[u] ← READY ►All unvisited vertices are assigned READY status
3. state[s] ← WAIT ► Vertex s is pushed to Queue and assigned WAIT status
4. Q ←φ ► Queue is initialized as empty
5. ENQUEUE(Q,s) ► The starting vertex s is added to the Queue
6. while Q ≠φ. ►A loop is created to remove vertices from Queue .
7. do u←DEQUEUE(Q) ► u is the removed vertex
8. for each v ∈ Adj[u] ►All vertices v that are adjacent to vertex u are explored
9. do if state[v] = READY ►If the neighbor vertex v is not visited
10. then state[v]← WAIT ►Then it is assigned WAIT state
11. ENQUEUE(Q,v) And added on to the Queue
12. state[u] ←PROCESSED ►The vertex u removed from the Queue is marked visited
and assigned PROCESSED state

Analysis of DFS – Running Time
• The running time for DFS consists of following major components
Ti = Initializing vertices T
Ts = Scanning Adjacency list
• The Graph G(V,E) has |V| vertices and |E| edges. Each vertex is initialized once.
• Therefore,
Ti= |V|
• The scanning of Adjacency list takes place for each dequeued vertex. In BFS all
links are examined. The total time is proportional to the number of edges
scanned. Since |E| is the number of edges in G, in worst case the algorithm
explores all edges , therefore,
Ts = |E|
• Thus, worst case running time is given by time
TDFS= |V|+ |E|= O(|V|+|E|)

Topological sorting
The vertices of the graph may represent tasks to be performed, and the edges
may represent constraints that one task must be performed before another;
in this application, a topological ordering is just a valid sequence for the
tasks. A topological ordering is possible if and only if the graph has no
directed cycles, that is, if it is a directed acyclic graph (DAG).
A closely related application of topological sorting algorithms was first
studied in the early 1960s in the context of the PERT (Program evaluation and
review technique) which was designed to analyze and represent the tasks
involved in completing a given project.

Kahn’s Algorithm
One of these algorithms, first described by Kahn (1962) works by choosing vertices
in the same order as the eventual topological sort.
• L ← Empty list that will contain the sorted elements S ← Set of all nodes with
no incoming edge
• while S is non-empty do remove a node n from S add
n to tail of L
• for each node m with an edge e from n to m do remove edge e from the
graph
• if m has no other incoming edges then insert m into S
• if graph has edges then
• return error (graph has at least one cycle) else
• return L (a topologically sorted order)

Why . . .
• Topological Sorting is a sorting process of items over which partial
ordering is defended.
• Topological sort of a directed acyclic graph(DAG) G is an ordering of the
vertices of G such that for every edge (ei, ej) of G we have i<j {ei appear
before ej in the graph}
• Traverse vertices in increasing order.
• There are two methods to solve it.
• Adjacency Matrix
• Using DFS

Example using Adjacency Matrix
1 2 3 4 5 6 7
1 1 1 1
2 1 1 1
3 1 1
4
5 1
6 1
7
2 3
4
7 1
6 5

Example using Adjacency Matrix
1 2 3 4 5 6 7
1 1 1 1
2 1 1 1
3 1 1
4
5 1
6 1
7
2 3
4
7 1
6 5
1 2 3 4 5 6 7

Example using Adjacency Matrix - Prove
1 2 3 4 5 6 7
1 1 1 1
2 1 1 1
3 1 1
4
5 1
6 1
7
2 3
4
7 1
6 5
1 2 3 4 5 6 7
• 1 should appear before 4, 5, 6 so it is true
• 2 should appear before 3, 5 and 6. it is also true

Example Using DFS(Depth First Search)
Visited SetA B
E D
F G G
C
Sorted Set

Visited Set
H
E
A B
E D
H F G
C
Sorted Set
H
Start From Any Node, In our Example We are starting From E

Visited Set
G
F
H
E
A B
E D
F G
C
Sorted Set
G
H

Visited Set
G
F
H
E
A B
E D
F G
C
Sorted Set
F
G
H

Visited Set
G
F
H
E
A B
E D
F G
C
Sorted Set
E
F
G
H

Visited Set
C
A
G
F
H
E
A B
E D
F G
C
Sorted Set
C
E
F
G
H

Visited Set
C
A
G
F
H
E
A B
E D
F G
C
Sorted Set
A
C
E
F
G
H

Visited Set
D
B
C
A
G
F
H
E
A B
E D
F G
C
Sorted Set
D
A
C
E
F
G
H

Visited Set
D
B
C
A
G
F
H
E
A B
E D
F G
C
Sorted Set
B
D
A
C
E
F
G
H

Visited Set
D
B
C
A
G
F
H
E
A B
E D
F G
C
Sorted Set
B
D
A
C
E
F
G
H
Sorted List B D A C E F G H

Applications of this type arise in instruction scheduling (is a compiler
optimization, which improves performance on machines with instruction
pipelines) when re-computing formula values in spreadsheets, determining
the order of compilation tasks to perform in makefiles. It is also used to
decide in which order to load tables with foreign keys in databases.
Applications

Graph 1

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