CONSTRUCTING MINIMAL ADJACENT DOMINATING SETS IN SEMIGRAPHS FOR CLUSTERING IN WIRELESS NETWORKS

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1 CONSTRUCTING MINIMAL ADJACENT DOMINATING SETS IN SEMIGRAPHS FOR CLUSTERING IN WIRELESS NETWORKS S. Srvnn 1, R. Poovzhk 2 nd N. R. Shnker 3 1 Deprtment of Mthemtcs, R.M.D. Engneerng College, Kvrpett, Tml Ndu, Ind 2 E.M.G. Ydv Women s College, Mdur, Tml Ndu, Ind 3 Deprtment of Electroncs nd Communcton Engneerng, Alm Muhmmed Slegh College of Engneerng, Avd, Chenn, Tml Ndu, Ind E-Ml: mthssrvnn77@gml.com ABSTRACT Reserchers propose Connected Domntng Set (CDS) of grphs n whch ech node n the cluster wreless network cover v domntng neghbors, defne mny domntng sets such s strongly connected domntng sets nd wekly connected domntng sets etc. In ths pper, we extend the domntve cpcty of nodes such tht ech node domntes not only tself nd ll ts djcent neghbors completely clled Adjcent Domntng Set (ADS) n semgrphs. Furthermore, n ADS constructon lgorthm to fnd mnml ADS n wreless networks s proposed for cluster hed selecton. The effcency nd performnce of the ADS constructon lgorthm confrm through theoretcl nlyss nd smultons. Ths pper ddresses the behvor of the protocols n dfferent network model n ADS bsed cluster network. Smulton result shows tht DSR nd DSDV perform better n grph nd semgrph structure, wheres AODV s more dptble n the rndomly chosen network. Keywords: connected domntng set (CDS), semgrph, djcent domntng set (ADS), wreless sensor network (WSN), energy effcency. 1. INTRODUCTION Wreless sensor network (WSN) s n ctve reserch re, gn the ttenton of mny resercher n montorng Envronmentl prmeter, such s Helth cre montorng, Are montorng, Industrl montorng etc. Furthermore, the mjor problem n WSN s energy conservton, relblty, moblty, sclblty, nd ese of use. In ddton, clusterng the node n WSN provde beneft n terms of energy-effcency, reducng routng dely nd sclblty. However, wreless sensor network (WSN) consst of mny number of sensor node. In ddton, clusterng node s chllengng tsk n WSN for resercher. Among the node n the cluster, set of nodes ct s cluster-hed (CH). In lst decde, clusterng n WSN plys vtl role n the energy conservton for ndvdul node. However, cluster formton, CH selecton, nd dt trnsmsson re three phses of operton n clusterng protocol. The mjor work of clusterng strts wth the CH selecton lgorthm tht gves the energy effcency of ny network. In ddton to the energy effcency, CH covers mjor node nd fult-tolernce. 2. RELATED WORK The clusterng lgorthm wth convergence tme perform nd energy effcent. The convergence tme lgorthm executes wth constnt nd vrble convergent tme [1]. However, probblty theory plys n mportnt role n clusterng. The most common clusterng lgorthms re LEECH, HEED, nd EEHC. The probblty cluster clssfes the node ether n probblty or non-probblty method. In non-probblty scheme, the lgorthm dvdes nto node proxmty, grph, weght, bologcl nspred protocol, nd clusterng protocol. A survey n cluster routng protocol of 16 dfferent methods [2] show the QoSn WSN. However, the chllenges n cluster formton become dffcult when the sze of network ncrese. A study on clusterng nd ther chllenges such s selectng the optml frequency of CH rotton, ensurng connectvty, computng the optml cluster szes, nd clusterng the network n the presence of node duty cycle, presented n Youns et l.[2], lso clssfes the clusterng bsed on the prmeters of the CH selecton nd the executon nture of clusterng lgorthm (probblstc or tertve). Furthermore, the vrous routng protocols studed by Henzelmn et l. [3], Youns et l [2]. Bsgn, Xu et l. [4], nd Chen et l. [5], respectvely. In [6], descrbes PEGASIS protocol s ner optml for dt gtherng problem n sensor network. However, comprtve study on herrchcl routng protocol performed by Xu nd Go. The mportnt herrchcl routng protocol, such s LEACH, TEEN, APTEEN (Mnjeshwr et l.[7,8]) nd PEGASIS (Lndsey et l.[6]), etc. exmned. In our dly lfe, the usge of sensor network ncreses consderbly. Wreless sensor network (WSN) trnsfer the dt between node n pproprte tme nd effcent routng. The effcent routng nd tme cn obtn from vrous routng lgorthm such s herrchcl, clusterng, grph theory, color theory, nd hypergrph theory. Herrchcl routng protocol (HRP) [9,10,11] show consderble conservton n totl energy consumpton of WSN. The HRP crete the cluster wth cluster hed for ech cluster. Moreover, the selecton for best Cluster heds nd Clusters mke dffcult n ny cluster routng of the network. Durng the selecton of cluster heds n ny cluster consume more energy nd tme. The Color-theory-bsed Dynmc Loclzton (CDL) lgorthm [12] dentfes ech sensor node locton for effcent energy routng. 1063

2 Fgure-1. The method to connect domntng set (CDS) s vrtul bckbone for routng ws frst proposed by Ephermdes et l.[13] n Furthermore, Clrk et l.[14] extended to the reltve complexty of problems under the restrcton from Unt Dsk Grphs (UDG) to grd grphs. The vertces dstnce n the crcle must below or equl to unt vlue. In ddton, Guh nd Kuller[15] ntroduce the pproxmton lgorthms to clculte mnmum CDS s vrtul bckbone n UDG n grph theory. In grph model, s shown n Fgure-1 ech edge s connected wth two vertces. However, the vrtul bckbone constructon plys n mportnt role n clusterng the network. The Fgure-1, need t lest 3 cluster heds (nodes) to form connected domntng set (CDS) sy, 1, 2, nd 4. Furthermore, n hypergrph [16] nsted of connectng two vertces, set of vertces connects wth n edge. However n semgrph theory, the number of vertces n n edge s rrnged n n order. Hence we consder semgrph [18,19] model n our work. 3. SEMIGRAPH MODEL In WSN, semgrph S ( U, X ) consder s network where U s the set of sensor nodes, nd X s the set of lnks between the nodes. A semgrph S ( U, X ) s nturl extenson of grph obtned by removng the constrnt on the crdnlty of n edge: Any nonempty subset of U cn be n element of the edge set X. Defnton 1 (Domntng set). [17] A domntng set (DS) for grph G (V, E) s subset D V such tht ech node n V D djcent to t lest one node n D. Defnton 2.(Connected domntng set). [17] A connected domntng set (CDS) n connected grphg s domntng set n G whose vertces nduce connected subgrph. Defnton 3.[18]. A semgrph s pr S (V, E) nd V s nonempty set of vertces of S nd E s set of n-tuples, clled edges of S of dstnct vertces for n 2 stsfyng the followng condtons: (SG1) Any two edges hve t most one vertex n common. (SG2) Two edges (u 1, u 2,., u n ) nd (v 1, v 2,., v m ) re consdered to be equl f nd only f () m = n nd () u = v for 1 nor u = v n-+1 for 1 n. Thus the edge (u 1, u 2,., u n ) s the sme s (u n, u n-1,., u 1 ). For the edge e 1 = (u 1, u 2,, u n ), u 1 nd u n re to be the end vertces of e 1 nd u 2, u 3, u n-1 re the mddle vertces of edge e 1. Defnton 4 ([19]). Let S(U, E) be semgrph. For ny vertex u U, the djcent neghborhood of u s N (u)={ x U /x, djcent to u} nd N [ u] N ( u) { u}. Defnton 5 [18].The hghest djcent degree of ll the nodes of semgrph (network) S s denoted by ( S ). Defnton 6.Adjcent Domntng Set[18] A set D U s clled djcent domntng set [7] (d-set) f for every u U D there exsts v D such tht u s djcent to u n S. The djcency domnton number ( ) s the mnmum crdnlty of n d-set of S. S Fgure-2. In the semgrph model shown n Fg 2, the vertex set D { v1, v4} s n ADS. Snce every node not n D hs t lest one djcent neghbor n D. A survey of well-known results relted to domnton prmeters contned n the book [17]. 4. MOTIVATION AND CONTRIBUTION Reserchers consder UDG s network model. However, Mohnty et l. [20] proposed Connected Pseudo Domntng Set usng 2 Hop Informton 1064

3 (CPDS2HI). The lgorthm constructs Pseudo Domntng Set (PDS) n the frst phse, nd then mproves the Stener tree constructon method, the PDS nodes re connected nd fnlly, the redundnt domntors re omtted from the CDS. The performnce rto of CPDS2HI s (4.8+ln5) opt +1.2, wheres opt s the sze of ny optml CDS. Also, the tme nd messge complextes of On ( ) nd O(D), wheres D s the network dmeter nd s the mxmum degree of nodes of the network. Furthermore, the mjor contrbuton of the proposed lgorthm mproves the energy effcency nd performnce of the network. However, the ADS constructon method 2 hs very low tme complexty O( p ). Also the method hs the best performnce rto of (1 l n ) opt, where opt to the optmum sze of the network. Fnlly, we nlyze the three dfferent network performnces n three routng protocols AODV, DSR, nd DSDV n the bckbone of ADS usng network smultor (NS2). 5. ADS CONSTRUCTION-ALGORITHM In the frst phse of ths secton, we obtn djcent neghborhood set. In the lst phse, we ntroduce the proposed lgorthm to buld ADS of semgrph S. Algorthm 1. Determnton of djcent neghborhood sets of semgrph. Input: A connected semgrph S(U, E) wth U = p nd E = q. Output: Adjcent neghborhood sets N ( u ) of ech node n u U nd N ( u ) U. 1: N u U u1 u2 u p ( ), {,,... }, E { E1, E2,... E q } 2: Let E j {( u1, u2,... u.. uk ) u U, k 2} 3: for 1to p do 4: for ll postve nteger k <p nd k do 5: f u stsfes t lest one of the followng condtons: () uk belongs to n edge contnng u () uk djcent to the node u then 6: Add k 7: end f 8: end for 9: end for u to N ( u ) 10: return N ( u ) The next lgorthm strts wth sortng the nodes nto the scendng order of ts djcent degrees nd end wth ADS of connected semgrph. Algorthm 2. Constructon of mnml djcent domntng set (ADS) of semgrph. Input: A connected semgrph S(U, E) wth U = p nd E = q. Output: Mnml ADS Dof the semgrph S(U,E), D U. 1:ADS D, U { u1, u2,... u p }, E { E1, E2,... E q } 2: for 1to p do 3: Sort the nodes n the scendng order of N ( u ) 4: end for p 5: Add lst 1 6: for 1to p do 7: for every node u U D 8: f N ( u ) D 0 then 9:dd u to D 10: end f 11: end for 12: end for 13: return D nodes to D In lgorthm 2, whle constructng ADS n the hghest degree of the lst p 1 node dded to D. Snce the bove number s lower bound for [18]. Hence the djcent domntng set D obtned from lgorthm 2 s the mnml ADS. 6. OPTIMALITY ANALYSIS OF ADS CONSTRUCTION ALGORITHM Theorem 1. D s n ADS. Proof. Let D be the set of end node obtned from lne 13 of Algorthm 2, nd there s no mddle node n D. By the constructon of D, every mddle node u D A should be djcent to t lest one end node n D. Hence D s n djcent domntng set (ADS). In ths secton, the performnce rto of the proposed lgorthm nlyze wth messge nd tme complexty. For lgorthm, consder ADS, the sze of ADS s the prmeter. Theorem 2. The proposed ADS lgorthm determnes the correspondng ADS n fnte tme. Proof. Algorthm 1 shown n secton 5 construct djcent neghborhood set N ( u ) for ech u U. Snce there re p node n semgrph S, the number of terton s p bounded. In Algorthm 2, we dd the lst node, 1 fter rrngng the node nto the scendng order of djcent degree. Then set of node dded to ADS D, n such wy tht ech node not n D djcent to t lest one node n D. So the number of round s fnte. Hence the fnte number of node n ADS, wll tke fnte tme. Theorem 3. The sze of ADS obtned by ADS constructon lgorthm s t most (1 ln ) opt, 1065

4 wheres opt the sze of ny optml ADS, nd the mxmum djcent degree of the network. Proof. Let ADS be the sze of ADS of the network (Semgrph) S. The frst phse of constructon lgorthm determnes djcent neghborhood set for ech node of S. In the second phse, mnml djcent domntng set (ADS) s determned. Then ADS (ln 1) opt [21]. Hence the performnce rto of ADS constructon lgorthm s (ln 1) opt. Theorem 4. The messge nd tme the complextes of ADS lgorthm re O(p 2 ) nd O[ p( p log p)] respectvely. Proof. The tme nd messge complextes n phse 1 re O(p 2 ) nd O(p 2 ) respectvely. In phse 2, ech node u U D hs t lest one djcent node n D[22]. The lgorthm need O(p 2 ) tme nd O(plogp+ p 2 ) messge [24]. Therefore, the overll tme complexty of the proposed lgorthm s O(p 2 ) nd totl messge complexty s O p p p O p p p 2 ( log ) [ ( log )]. Snce the smller vrtul bckbone gves the better performnce [25]. As shown n Fgure-1 nd Fgure- 2, sze of ADS n semgrph s smller thn the sze of CDS n grph. Snce ech edge of semgrph not only connected wth end nodes but lso connected wth mddle nodes. Furthermore, smulton of proposed ADS constructon lgorthm pples wth three routng protocols AODV, DSDV, nd DSR nd compre wth performnce metrcs such s throughput, pcket loss rto, resdul energy, nd end-to-end dely. Furthermore, ll the bove metrc s dscussed n further secton. 7. PERFORMANCE METRICS OF ADS The performnce metrcs n the experments re throughput, the pcket loss rto, the resdul energy, nd end-to-end dely. Furthermore, Smultor (NS-2) softwre used to smulte the proposed novel ADS constructon lgorthm. In ddton, n vrtul bckbone smulton, the deployment re for ech protocol s of dmenson 2000m 1000msqureunts. Also, n the smulton work, we consder the followng models connected grph, connected semgrph nd connected rndom network. However, the result of the smulton s reported n the subsequent Fgures. 5 Are of smulton 2000m 1000m 6 Trffc ntervl 0.55 sec 7 Pckets generted by ech source Totl pckets generted n N/W = Sze of ech pcket 100 bytes 10 Model Energy model 11 Intl energy 1000J 12 Trnsmttng power 36.00mw 13 Recevng power 14.4mw 14 Protocol AODV/DSR/ DSDV 15 MAC lyer Throughput s the mesure of number of pckets or dt successfully trnsmtted to ther fnl destnton v communcton lnk per unt tme [24]. Furthermore, t s mesured n bts per second (bt/s or bps). End to End Dely defnes s the verge tme between pckets sent nd receved. It s gven by A p 1 ( r s ), N s the number of successfully N 1 receved pckets, s unque pcket dentfer, r s tme n whch pcket wth unque d s receved, s s tme t whch pcket wth unque d s sent nd A mesure n ms. [24]. Pcket Loss rto s clculted by subtrct to the number of dt pckets sent to source nd number of dt pckets receved t destnton through the number of pcket orgnted by the pplcton lyer of the source. 8. SIMULATION RESULTS Throughput In the frst set of smulton, we compre throughput for the DSR, DSDV nd AODV protocols s shown n Fgure-3, Fgure-4, nd Fgure-5. However, the semgrph model obtns the best throughput of 12kb/sec over tme n DSR. Stll t cheves smlr results for AODV nd DSDV wth ll three network models s shown n Fgure-3, Fgure-4, nd Fgure-5. Tble-1.The tble shows the prmeters of smultonn NS2. S.No. Prmeters Detls 1 Node plcement semgrph 2 Number of nodes 20 3 Number of snk(destnton) One(Node 0) 4 Number of sources

5 Resdul energy Semgrph model (ADS) consume low energy compred to grph model nd rndom network models s shown n Fgure-6, Fgure-7, nd Fgure-8. Fgure-3. Fgure-6. Fgure-4. Fgure-7. Fgure

6 Fgure-10. Fgure-8. Pcket loss rto Refer to the grph of Fgure-9, Fgure-10, nd Fgure-11 the pcket loss rto of ADS s lower thn the other two networks n ll the three protocol. Furthermore, for AODV, the pcket loss s hgher thn DSDV nd DSR for ll the three network models. Fgure-11. End-to-end dely rto From the grphs of Fgure-12, Fgure-13, nd Fgure-14, the AODV hs hgh dely compre to DSDV nd DSR n ll the three dfferent network models. In ddton, the semgrph model (ADS) shows lower endto-end dely whle comprng the other two network models. Fgure

7 9. CONCLUSION In ths pper, we hve nvestgted the problem of clusterng nd cluster hed selecton by constructng mnml ADS for wreless network. The performnce rto of our ADS constructon lgorthm s (1 ln ) opt, wheres opt the sze of ny optml ADS, nd the mxmum djcent degree of the network. Furthermore, from the smulton result, we conclude the semgrph model (ADS) s the energy effcent model n ll the metrcs of three protocols AODV, DSDV, nd DSR. As future work, we cn buld new ADS constructon schemes consderng other ssues such s bndwdth, nd fult tolernce. REFERENCES Fgure-12. [1] Abbs A, Youns M A survey on clusterng lgorthms for wreless sensor networks. ComputCommun. 30: [2] Youns O, Krunz M, Rmsubrmnn S Node clusterng n wreless sensor networks: recent developments nd deployment chllenges. IEEE Netw [3] Henzelmn WB, Chndrksn AP, Blkrshnn H Energy-effcent communcton protocol for wreless mcrosensor networks. In: Hw nterntonl conference on system scences (HICSS), Wle Mu, Hw. pp Fgure-13. [4] Bsgn S Dstrbuted clusterng for d hoc networks.in: Interntonl symposum on prllel rchtectures, lgorthms, nd networks (I-SPAN), Fremntle, Austrl. pp [5] Chen B, Jmeson K, Blkrshnn H, Morrs R Spn: nenergy-effcent coordnton lgorthm for topology mntennce n d hoc wreless networks. WrelNetw. 8: [6] Lndsey S, Rghvendr C, Svlngm K Dt gtherng lgorthms n sensor networks usng energy metrcs. IEEE Trns Prllel Dstrb Syst. 13: [7] Mnjeshwr A, Agrwl D Teen: routng protocol for enhnced effcency n wreless sensor networks. In: IEEE prllel nd dstrbuted processng symposum (IPDPS), Sn Frncsco, USA.pp Fgure-14. [8] Mnjeshwr A, Agrwl D Apteen: hybrd protocol for effcent routng nd comprehensve nformton retrevl n wreless sensor networks. In: 1069

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