M. Suganya assistant professor St Joseph's college for ...ijaema.com/gallery/89-ijaema-mar-3575.pdf · M. Suganya assistant professor St Joseph's college for women tirupur G. Bhuvanaswari

ABSTACT

The use of mathematics is quite visible in every area of computer science, in artificial

intelligence, software of development environments and tools, software architecture and

design multiprocessing, automatic control, distributed and concurrent algorithms etc.

mathematics helps in the design, implementation and analysis of algorithms for scientific and

engineering applications. It also improves the effectiveness and applicability of existing

methods and algorithms.

Graph theory is an important area in mathematics, this paper explores the use of

graphs for modelling communication networks, generally vertices will represent computers,

processors and switches, and edges will represent wires, fiber, or other transmission lines

through which data flows, for some communication networks, like the internet, the

corresponding graph is enormous and largely chaotic. It represents the communication

networks as butterfly networks and Benes networks.

The two representation have been compared or their diameter, switch size, switch count and

congestion.

KEYWORD:2-D array, switch size, congestion,diameter.

UNIT I

INTRODUCTION

The communication networks can be represented using the various mathematical

structures which also help us to compare the various representations based on congestion,

switch size and switch count. Graphs have an important application in modelling

communications networks. Generally, vertices in graph represent terminals, processors and

edges represent transmission channels like wires, fibres etc. through which the data flows.

Thus, a data packet hops through the network from an input terminal, through a sequence of

switches joined by directed edges, to an output terminal.

Data transmission in wireless networks, especially in wireless mesh networks

(WMN), is done via multicast broadcast. However, it is impossible for a node to receive

simultaneously from two different neighbours. Transmissions must be done in different time

periods to avoid collisions. Some have been conducted to overcome these problems, and

make wireless communication more efficient by allowing simultaneous transmission between

nodes even if they are hidden or exposed relative to one another. These studies converge on

the use of network coding in wireless networks to allow sending simultaneously several

The International journal of analytical and experimental modal analysis

Volume XII, Issue III, March/2020

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COMPARISON OF BUTTERFLY AND BENES NETWORKM. Suganya assistant professor St Joseph's college for women tirupur

G. Bhuvanaswari student, St Joseph's college for women tirupur

symbols from single or several sources. In traditional networks, the nodes copy and

disseminate information. This operation is known as “copy-and-forward”. While in network

coding scheme, node received information before broadcasting it. This operation is known as

“copy, and forward code”

For some communication networks, like the internet, the corresponding graph is

enormous and largely chaotic. However, there do exist more organizing networks, such as

certain telephone switching networks and the communication networks inside parallel

computers. For these, the corresponding graphs are highly structured. In this lecture, we’ll

look at some of the nicest and most commonly used communication networks.

This paper expose the use of graphs, for modelling communication network with

represent the communication networks as butterfly and Benes network, the 2 representation

have been compared are the diameter, switch size, switch count, and congestion and the

application of each network indifferent field are discussed.



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UNIT II

SOME BASIC DEFINITIONS

MAXIMUM and MINIMUM DEGREE:

The maximum of the degree of all the vertices is called the maximum degree of the

graph and it is denoted by ∆ (G) or ∆.

The minimum of the degree of all the vertices is called the minimum degree of the

graph and it is denoted by ẟ (G) or ẟ.

ISOLATED VERTEX:

A vertex having no incident edge is called isolated vertex.

PENDENT VERTEX:

Any vertex of degree one is called a pendent vertex.

NULL GRAPH:

A graph without any edge is called a null graph.



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COMPLETE GRAPH:

A simple graph in which there exists an edge between every pair of vertices is called a

complete graph.

CLIQUE:

A clique in an undirected graph G= (V, E) is a subset of the vertex set C≤V, such that

for every two vertices in C, there exists an edge connecting the two. This is equivalent to

saying that the sub graph induced by C is complete.

REGULAR GRAPH:

A graph G in which all vertices are of equal degree is called a regular graph.



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COMPONENTS:

The maximal connected sub graphs of G are called its components.

CONNECTED:

A graph G is connected if for every u, v Ԑ G there exists a uv-path in G. Otherwise G

is called disconnected.

SUPER GRAPH:

If G and H are two graphs with vertex sets V(H),V(G) and edge sets E(H) and

E(G)respectively such that V(H)≤V(G) and E(H)≤E(G) then we call H as a sub graph of G or

G as a super graph of H.



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UNIT III

GRAPHICAL REPRESENTATION OF COMMUNICATION NETWORK

1.1 As Butterfly

All the terminals and switches in the network are arranged in N rows. In particular,

input I is at the left end of row I, and output I is at the right end of row i. Rows are labelled in

binary, thus, the label on row I is the binary number 𝑏1𝑏2……….𝑏log 𝑁 that represents integer

i. Between the inputs and the outputs, there are log (N) +1 levels of switches, numbered from

0 to log N. Each level consists of a column of N switches, one per row. Thus, each switch in

the network is uniquely identified by a sequence (𝑏1 , 𝑏2, … 𝑏log 𝑁, L), where 𝑏1, 𝑏2, … 𝑏log 𝑁 is

the switch’s row in binary and L is the switch’s level. There are directed edges from switch

(𝑏1, 𝑏2, … 𝑏log 𝑁, L) to two switches in the next level. One edge leads to the switch in the same

row, and the other edge leads to the switch in the row obtained by inverting bit L+1.

1.1.1. Diameter: Between the inputs and the outputs, there are log (N) +1 levels of switches,

numbered from 0 to logN. Each level consists of a column of N switches, one per row.

Therefore, the diameter for this case is log (N) +1.

1.1.2. Switch size: The switch size is 2ꓫ2 as visible from fig. 3.



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1.1.3. Switch count: As the network consists log (N) +1 level of switches and each level has

N switches. Therefore, total switch count is N (log (N) +1)

1.1.4. Congestion: There is unique path from each input to each output, so the congestion is

given by the maximum number passing through a vertex for any routing. If V is a vertex in

column i of the butterfly network, there is a path from exactly 2iinput vertices to V and a path

from V to exactly 2n-I output vertices. Therefore, congestion of the butterfly network turns

out to be around √N if N is an even power of 2 and √N/2 if N is an odd power of 2.

2.2 Benes network

In the 1960’s, a researcher at bell labs named Benes had a remarkable idea. He

noticed that by placing two butterflies

Back-to-back, he obtained a marvellous communication network.

2.2.1. Diameter: The inputs and outputs, there are 2log (N) +2 levels of switches, numbered

from 0 to logN. Each level consists of a column of N Switches, one per row. Therefore the

diameter for this case is 2logN+1.

2.2.2. Switch size: The switch size is 2ꓫ2 as visible from fig, 4.

2.2.3. Switch count: As the network consists 2N log N level of switches and each level has N

switches. Therefore, total switch count is 2N log N.

2.2.4. Congestion: There is a unique path from each input to each output, so the congestion is

given by the maximum number of messages passing through a vertex for any routing, If V is

a vertex column n of the Benes network, there is a path from exactly 2i input vertices to V

and a path from V to exactly 2n-n output vertices. Therefore, congestion of the Benes

network turns out to be around 1, if 1 is an evenpower of 2 and if 1 is an odd power of 1.



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UNIT IV

COMPARION OF VARIOUS REPRESENTATION

The Table 1 shows that butterfly network has lower congestion then the complete

binary tree and it uses fewer switches and has lower diameter then 2-D array. The congestion

for 2-D array does not depend on the number of inputs and outputs and is always fixed while

this is not the case for binary tree and 2-D array. The structure of binary tree, which is

otherwise simpler then butterfly, becomes bigger and complex with the increase in number of

inputs and outputs. The root acts as a bottleneck for binary tree representation. In spite of the

complexity of butterfly networks, the way to route a packet from input to output is very

simple due to the labelling of rows in binary. One bit is corrected at each level.

The Benes network is small, compact, and completely eliminates congestion. The

Holy Grail of routing networks is in hand!

UNIT V

APPLICATIONS OF BUTTERFLY AND BENES NETWORK:

BUTTERFLY NETWORK:

SIMULATION:

In this section we perform simulation for construction of butterfly networks in the

wireless mesh network. We generate 80 nodes in 800m x 800m area. The nodes are randomly

deployed with a transmission range of 250m. We implement the RBC algorithm in Mat lab.

The objective of the simulation is to construct butterfly effects in a mesh network. Remember

that it is more likely to find a butterfly in a mesh network than in a mobile and hoc wireless

network. In fact, in a mesh network, all nodes are more populated. This increases the number

of links in a network. Consequently, the number of butterfly candidate links increases also.

Network Diameter Switch size Switch count Congestion

Butterfly LogN+1 2ꓫ2 N(log(n)+1) √N or √N/2

Benes 2logN+2 2ꓫ2 2N log N 1



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Note that the source and the destination are chosen randomly from the set of the WMN

nodes.

For evaluating RBC algorithm, the simulation has been performed 10 times. Each

simulation allows creating one primary and one backup butterfly network. Figure 3 illustrates

butterfly networks constructed in two different wireless mesh network topologies, and with

two different source and destination couple.

The existence of a butterfly effect requires finding at least three disjoints paths

connecting the connecting the children and the grand-children of the source and the

destination. These paths from the core of the butterfly effect.

As shown figures 3 and 4, the RBC algorithm calculates three disjoints paths for each

butterfly effect. One path is defined between each pair source’s and destination’s children.

One additional path is found between the source’s and the destination’s grand-children. RBC

uses Dijkstra0 algorithm to find the different shortest path. Furthermore, our results show that

RBC is able to find not only three distinct paths to the same butterfly network but also

completely disjoint networks butterfly. The purpose is mainly to enable application of the

load balancing in the WMN, by using more than one butterfly effect for routing packets from

the source to the destination.

As mentioned above, the simulation results show the RBC algorithm allows finding

more than one butterfly effect in a WMN. As WMN is more populated with nodes, there is

higher probability of finding more butterfly effects. Figure 4 shows a couple of butterfly

effects. The first one is called the primary butterfly. It is illustrated with solid lines. The

second one is called the backup butterfly, and illustrated with dotted lines. The two butterfly

created by our simulation allow the topology recovery without calculating a new routes

between the source and the destination. Therefore, our solution is reliable since RBC allows

finding a set of backup butterfly effects atany time, to insure the topology recovery in case of

butterfly effect fail. The other important result of our simulation is that RBC allows applying

the load balancing in WMN in order to increase the throughout. In fact, more than one

butterfly effects may collaborate to transmit data by sharing the load within the same WMN.

The application of the load balancing mechanism is presented in figure 5. The source

node uses two paths to transmit the data. The packets sent through the first path, represented

by solid arrows, are different from those sent through the second path, illustrated by dashed

ones. The application of the load balancing will than increase the throughput in the network

since more than one path are used to transmit data packets.



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Figure describes the topology recovery mechanism invoked by a butterfly fail. The

primary butterfly is represented by solid lines and the backup one is illustrated with dotted

lines. At first, the transmission is initiated by the source and performed through the primary

butterfly network. The packets transmission is shown by the solid arrows. Once one more

butterfly network to transmit data. This transmission is represent by the dashed arrows in the

fig .

REAL LIFE APPLICATION OF BUTTERFLY NETWORK:

BENES NETWORK:

We wish to comply with any type of interleaving for a turbo-decoding application, we

need an interconnection network which supports all the possible permutations of its inputs

with its outputs. Moreover, this network must offer path diversity in order to reduce the

conflicts between packets as much as possible.

The Benes network is one of the already existing networks which has these

characteristics. Built from two butterflies put back-to-back, its diameter is almost the double

of that of butterfly: 2log2N-1. In addition, the latency is constant for all the couples (source,

destination) and it corresponds to the network diameter. However, this networks avoid the

conflicts if and only if all the paths have a different destination. But this is not the for the

turbo-decoding application because interleaving (respectively deinterleaving) ends in

potentials conflicts.

The suggested solution is to choose the packets to be transmitted so that for each

cycle, none is intended for the same network output port.



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On the basis of this constraint, the Benes topology was modified and gave the Benes 2N-N

network, illustrated in fig 8.

BENES NON-BLOCKING NETWORK:

The major performance metrics of the circuit designs networking include delay, area and

power consumption. The number of stages of a network is the key factor determining the

delay of the network. Generally, for networks built with the same type of logic units, more

stage means longer delay. The determining factor of area and power consumption is the

transits count.

The crossbar is a strictly non-blocking network, i.e., any permutation of inputs and

outputs can be realized without confliction. As each input port is connect to each output port

through a dedicated logic unit, which is composed of one configurable switch, the basic

component used in our circuit design. The number of logic units needed for an NxN crossbar

is𝑁2.

The number of stages traversed from one input output to one output port is only one.

However, the circuit complexity of crossbars quadratic ally with the crossbar size. The

resulted high power consumption and die area limits the use of crossbar for large-scale NoCs

The NxN Benes network basically is built with two symmetrical NxN butterfly

networks. Larger size Benes networks can be built with smaller Benes network recursively.

The basic logic unit is a 2x2 crossbar switch.

REAL LIFE APPLICATION OF BENES NETWORK:



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UNIT VI

CONCLUSION:

In this paper we have discussed the application of graph theory in communication

network.

The communication network has been represented as butterfly network and Benes network.

The two networks have been compared are their diameter, switch size, switch count,

and congestion. These applications of each networks are discussed in dually.

For some communication networks, like the internet, the corresponding graph is

enormous and largely chaotic. However, there do exist more organized networks, such as

certain telephone switching networks and communication networks inside parallel computers,

for these, the corresponding graphs are highly structured. In this lecture, we’ll look at some of

the nicest and mostly commonly used communication networks.

The butterfly has lower congestion than the complete binary tree. And it uses fewer

switches and has lower diameter then the array. However, the butterfly does not capture the

best qualities of each network, but rather is compromise somewhere between the two.

So our quest for the holy grail of routing networks goes on, and Benes noticed that by

placing two butterflies back-to-back. He obtained a marvellous communication network

called Benes network. This double number of switch size, switch count and diameter. It also

completely eliminates congestion problems.

Hence Benes network is small, compact, and completely eliminates, congestion.

BIBLIOGRAPHY

1. S.G.Shirinivas, S.Vetrivel, Dr.N.M. Elango “Application of graph theory an

overview” International Journal of Engineering Science and Technology vol.2(9), pp.

4610-4621,2010

2. John. P. Heyas,” A graph model for fault Tolerant Computing System”; IFEE

September 1976.



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3. H.J.A.M. Heijmans, Mathematical Morphology; a modern approach in image

processing, based on Algebra and Geometry SIAM review, vol,37 pp-36, 1995.

4. A.J. Baddeley and H.J.A.M. heijmans, Incidence and Lattice Calculus with

Applications to Stochastic Geometry and image analysis, Application Algebra in

Engineering, Communication networks and Computing vol.16, pp. 129-146, 1995.

5. Albert R.Meyer, “Mathematics for Computer science” chapter13; Communication

networks pp253-272, May 9, 2010.

6. C.Berrou, A.Glavieux; P.Thitimajshima, “Near Shannon Limit Error-Codding and

Decoding: Turbo-Codes”, in proc. 1993.

7. F.Harary, graph theory, Narosa Publishing House, (2013).



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