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International Journal of Artificial Intelligence and MechatronicsVolume 1, Issue 1, ISSN 2320 – 5121

Asymptotically Optimal Method for Connecting Nodesinto a Perfect Difference Network (PDN) Using NS-2

Madhuri M.PalM.E.(ESC)

Suresh Deshmukh College of EngineeringEmail ID : [email protected]

Abstract – PDN is an asymptotically optimal method forconnecting a set of nodes into a Perfect Difference Network(PDN) with diameter 2, so that any node is reachable fromany other node in one or two hops utmost. It is mainly basedon the mathematical notion “the Perfect Difference Sets”given by “Singer”. Perfect difference network is a robust,high-performance interconnection network for parallel anddistributed computation. PDNs have a diameter of 2 and anode degree of approximately 2, which place them close tocomplete networks in terms of routing performance andmuch lower with respect to implementation cost. The richconnectivity and small diameters of PDNs and relatednetworks make them good candidates for wireless/opticalnetwork technologies.

It show a simulated result of the data transmission in PDNand the bandwidth and latency explained by a graph.

Keyword - Asymptotically Optimal Method, PDN, NS 2.

I. INTRODUCTION

Networks evolved from the simplest ring, linear to a farbetter mesh, star to more complex and stronger hypercube,pancake and complete network architectures. All thesenetworks served their best in their ages, then why thenewer architectures should be introduced the answer iscommunication needs of applications and technologicalcapabilities. PDN was introduced as an approach to designa network that not only stands big in the performance listbut also rules out the limitations inherent in each of thearchitectures at a lesser cost. PDN is thus a step ahead inthe same path of achieving the goal of better networkservices1.1 Perfect Difference Sets

Set in mathematics is a collection of entities calledelements of the set, that may be real object or conceptualentities. Set theory has a wide applicability in not just thefield of mathematics but also in many other fields. PerfectDifference Set is an extension to it [3].Perfect DifferenceSets were first discussed by Singer in 1938. PerfectDifference Set is a specialized set of δ+1 non negativeinteger data {s0, s1,……, s δ} that can be uniquelyexpressed in the form si-sj, where i≠j, i,j=0,1.Ex: {1, 2,4}(mod 7), {1, 2, 5, 7}(mod 13), etc. necessary conditionfor a difference sets to exist is that n be of the form δ 2+ δ+1.and the sufficient condition is that δ be a prime power..

II. PERFECT DIFFERENCE NETWORK

Perfect Difference Network (PDN) based on the PDS{0,1,s2,…..,s δ } There are n= δ 2+ δ +1 nodes, numbered 0to n-1. Node i is connected via directed links to nodes i±1

and i≠sj (mod n), for 2 ≤j ≤ δ. The preceding connectivityleads to a chordal ring of in and out-degree d=2δ anddiameter D=2. Because, for each link from node i to nodej, the reverse link from node j to node i also exists, thenetwork can be drawn as an undirected graph. normal-form PDS contains 1 as a member. Therefore, PDNs basedon normal PDNs are special types of chordal rings. In theterminology of chordal rings, the links connectingconsecutive nodes i and i+1 are rings, the links connectingconsecutive nodes i and i+1 are ring links, while those thatconnect non consecutive nodes i and i+sj, 2 ≤j ≤ δ, are skiplinks or chords [1].

As seen in the figure below any two nodes in a PDN areeither connected by a link directly or via a path of length 2through an intermediate node [ 5]

Fig.1. PDN with n = 7 nodes based on the perfectdifference set {0, 1, 3}.

Perfect Difference Network (PDN) based on the PDS{0,1,s2,…..,s δ } There are n= δ 2+ δ +1 nodes, numbered0 to n-1. Node i is connected via directed links to nodesi±1 and i≠sj (mod n), for 2 ≤j ≤ δ. given that all indexexpressions in this paper are evaluated modulo n,henceforth, we will delete the qualifier “mod n” in ourpresentation. The preceding connectivity leads to a chordalring of in and out-degree d=2 δ and diameter D=2 (this isjustified later). Because, for each link from node i to nodej, the reverse link from node j to node i also exists, thenetwork can be drawn as an undirected graph[1].Everynormal-form PDS contains 1 as a member. Therefore,PDNs based on normal PDNs are special types of chordalrings. In the terminology of chordal rings, the linksconnecting consecutive nodes i and i+1 are rings, the linksconnecting consecutive nodes i and i+1 are ring links,while those that connect non-consecutive nodes i and i+sj,2 ≤j ≤ δ, are skip links or chords.2 .1 Steps for Developing PDN from PDS









I. INTRODUCTION

















I. INTRODUCTION









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1) Take the value of δ and calculate the value of n= δ 2+δ +1

2) Note the pds for the above value of δ from the pdstable specified above.

3) For constructing pdn follow the steps for nodeconnectivity.

4) Connection among the nodes is obtained as follows:5) For any node i connections is as follows i±1 and i

±sj (mod n) where j>= δThen we get the corresponding perfect difference

network for value of δ and corresponding no of nodes.

III. PDN FOR =3 AND NODES N=13 FOR PDS{0, 1, 3, 9} OF IMPLEMENTATION FOR =4

BASED ON THE PDS {0, 1, 4, 14, 16}

PDN with n = 13 nodes based on the perfect difference set{0, 1, 3, 9}.

1) As specified in the steps above take the value of δ=42) Calculate the no of nodes as n =δ 2+ δ +1 and obtain

the no of nodes = 213) Go through the pds table and note the corresponding

pds as {0,1,4,14,16}4) Calculate the each node connectivity by using the

formula specified in the steps for pds to pdnconversion.

5) Construct a routing table for each node connectivity.6) Finally construct the perfect difference network which

will be as follows.3.1 Node Connectivity for δ =4 based on the PDS {0,4, 14, 16}

Fig.4. PDN with N =21 and δ=4 based on the PDS{0,1,4,14,16}

Network Simulator (Version 2), widely known as NS2,is simply an event driven simulation tool that has proveduseful in studying the dynamic nature of communicationnetworks. Simulation of wired as well as wireless networkfunctions and protocols (e.g., routing algorithms, TCP,UDP) can be done using NS2. In general, NS2 providesusers with a way of specifying such network protocols andsimulating their corresponding behaviours. Due to itsflexibility and modular nature, NS2 has gained constantpopularity in the networking research community since itsbirth in 1989. Ever since, several revolutions and revisionshave marked the growing maturity of the tool, thanks tosubstantial contributions from the players in the field.Among these are the University of California and CornellUniversity who developed the REAL network simulator,1the foundation which NS is based on. Since 1995 theDefense Advanced Research Projects Agency (DARPA)supported development of NS through the VirtualInterNetwork Testbed (VINT) project [9]. Currently theNational Science Foundation (NSF) has joined the ride indevelopment. Last but not the least, the group ofresearchers and developers in the community areconstantly working to keep NS2 strong and versatile [7].5.1.1 Basic Architecture OF NS-2

Figure 2.1 shows the basic architecture of NS2. NS2provides users with an executable command ns whichtakes on input argument, the name of a Tcl simulationscripting file. Users are feeding the name of a Tclsimulation script (which sets up a simulation) as an inputargument of an NS2 executable command ns. In mostcases, a simulation trace file is created, and is used to plotgraph and/or to create animation[7].










BASED ON THE PDS {0, 1, 4, 14, 16}




















BASED ON THE PDS {0, 1, 4, 14, 16}













Fig.5. Basic Architecture OF NS

5.1.2 Basics of Using Two LanguagesNS2 consists of two key languages: C++ and Object-

oriented Tool Command Language (OTcl). While the C++defines the internal mechanism (i.e., a backend) of thesimulation objects, the OTcl sets up simulation byassembling and configuring the objects as well asscheduling discrete events (i.e., afrontend). The C++ andthe OTcl are linked together using TclCL. Mapped to aC++ object, variables in the OTcl domains are sometimesreferred to as handles. Conceptually, a handle (e.g., n as aNode handle) is just a string (e.g.,o10) in the OTcldomain, and does not contain any functionality. Instead,the functionality (e.g., receiving a packet) is defined in themapped C++ object (e.g., of class Connector). In the OTcldomain, a handle acts as a frontend which interacts withusers and other OTcl objects. It may defines its ownprocedures and variables to facilitate the interaction. Notethat the member procedures and variables in the OTcldomain are called instance procedures (instprocs) andinstance variables (instvars), respectively.

NS2 provides a large number of built-in C++ objects. Itis advisable to use these C++ objects to set up a simulationusing a Tcl simulation script. However, advance users mayfind these objects insufficient. They need to develop theirown C++ objects, and use a OTcl configuration interfaceto put together these objects. After simulation, NS2outputs either text-based or animation-based simulationresults. To interpret these results graphically andinteractively, tools such as NAM (Network AniMator) andXGraph are used. To analyze a particular behaviour of thenetwork, users can extract a relevant subset of text-baseddata and transform it to a more conceivable presentation[7].The figure below shows an example of ns-2implementation [6].

Fig.6. Example of ns-2 implementation [9]

This is a data plotter with interactive buttons forpanning, zooming, printing, and selecting display options[9].• Create your own output files [8].• Collect statistical data synchronized.• Run xgraph:

– $xgraph out0.tr, out1.tr –geometry 800x400– In ns-2 script:Proc finish{} {……

exec xgraph out0.tr, out1.tr out2.tr –geometry 800x400&

exit}

Fig.7. a simple view of x-graph utility

1) First we save the above tcl script by .tcl extension, say“node_7.tcl”.

2) Now save the above program on desktop and open theterminal and execute by using the followingcommands:

Cd Desktop/ ii Ns node_7.tcl1. Then we will get the NAM editor where we can see thewhole simulation steps as shown in the screenshots in thetesting stage.First we select the source and destinationnode in the topology. Say node-0 and node-5.

2. Now after clicking the play button we will see the dataflow from n-0 to n-5 when there is no traffic. By clickingon the link we can able to see the bandwidth and latencyof the topology as shown in below.











exit}
















exit}








3. By clicking on the graph option we can see the graphbandwidth as well as the graph loss as shown below:4. When there is traffic in the network and we want tosend data packets from node-0 to node-5 then the flow ofpacket will be like as shown in below:

5. If there is any link failure, say link 1-5 become fail thenit is indicated by the red line and the packets will flowthrough another path according to the routing table whichwill show the shortest path from node-0 to node-5 asshown below:

6. In all the above screenshots it has been seen that thepackets were going through only one hope in the PDNnetwork. Now the screenshot given below shows that inPDN the packets may follow at most two hop to send datafrom source to destination here the packets are flow fromnode-0 to node-5 and from node-1 to node-6 by atmost two hops.

7. We can monitor the detailed about the data packets,CBR traffic the time step at which the packets are sendingfrom one node to another by just right clicking the flowarrow and monitor will be displayed as shown below:

IV. CONCLUSION

We have introduced perfect difference networks and themathematical underpinnings that make them desirable asrobust, high-performance interconnection networks forparallel and distributed computation. Although otherinterconnection networks with topological andperformance parameters similar to PDNs exist, we viewthese networks as worthy additions to the repertoire ofcomputer system designers. Alternative networktopologies offer additional design points that can beexploited to accommodate the needs of current andemerging technologies. Further study is needed to resolvesome open questions and to derive cost/performancecomparisons for PDNs and their derivatives.

Perfect difference networks constitute a class of robust,high-performance interconnection networks for paralleland distributed computation. Whereas PDNs may not bedesirable for large networks with wired connectivity, theydo offer attractive alternatives for wireless and opticalinterconnections and as smaller component networks inhierarchical architectures.







IV. CONCLUSION









IV. CONCLUSION





REFERENCES

[1] Parhami, B. and D.-M. Kwai, “Periodically Regular ChordalRings,” IEEE Trans. Parallel and Distributed Systems,Vol. 10, No. 6, pp. 658-672, June 1999. (Printer errorscorrected in Vol. 10, No. 7, pp. 767-768, July 1999.)

[2] Parhami, B., “Swapped Interconnection Networks: Topological,Performance, and Robustness Attributes,” J. Parallel andDistributed Computing.

[3] Singer, J., “A Theorem in Finite Projective Geometry and SomeApplications to Number Theory,” Trans. AmericanMathematical Society, Vol. 43, pp. 377-385, 1938

[4] Parhami, B., Introduction to Parallel Processing: Algorithms andArchitectures, Plenu Press, 1999.

[5] Parhami, B., “Swapped Interconnection Networks: Topological,Performance, and Robustness Attributes,” J. Parallel andDistributed Computing, to appear.

[6] Network Simulator Tutorial, Vacha Dave, University of Texas atAustin.

[7] Introduction to Network Simulator NS2, Teerawat Issariyakul •Ekram Hossain.

[8] Ns-2 Tutorial (1)Multimedia Networking Group,TheDepartment of Computer Science, UVA Jianping Wang.

[9] Funded by DARPA, the VINT project aimed at creating anetwork simulator that will initiate the study of differentprotocols for communication networking.

AUTHOR’S PROFILE

Madhuri M.PalM.E. (ESC)Suresh Deshmukh College of EngineeringEmail ID : [email protected]



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AUTHOR’S PROFILE




REFERENCES










AUTHOR’S PROFILE





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