A Distributed Self Spreading Algorithm for Mobile Wireless Sensor Networks

Transcription

1 A Distributed Self Spreadig Algorithm for Mobile Wireless Sesor Networks jeog Heo ad Pramod K. Varshey Departmet of Electrical Egieerig ad Computer Sciece, Syracuse Uiversity, Syracuse, NY 344,U Abstract - Sesor deploymet is a importat problem i mobile wireless sesor etworks. This paper presets a distributed self deploymet algorithm for mobile sesors. Performace metrics to evaluate algorithm performace are coverage, uiformity, time ad distace traveled till the algorithm coverges. algorithm is compared with a simulated aealig based algorithm for deploymet ad is show to exhibit excellet performace.. Itroductio Desig ad deploymet of ifrastructured etworks such as a cellular etwork has matured over the last two decades. I such etworks, mobile users access the etwork via fixed base statios. Plaig ad deploymet of these etworks is carried out based o radio propagatio ad terrai models with the goal of maximizig radio coverage. More recetly, there has bee a great deal of iterest i ad hoc wireless etworks. These etworks employ fixed or mobile odes ad dyamically orgaize themselves ito a etwork without requirig a ifrastructure. I ad hoc etworks, each ode acts ot oly as a ed ode but also as a router. Oe importat aspect i the desig of these etworks is the iitializatio procedure ad establishmet of routig structure. Most research o ad hoc etworks has bee focused o issues such as the developmet of routig protocols ad quality of service ad ot o topology ad deploymet. Wireless sesor etworkig has become a area of itese research activity. This is due to techical advaces i sesors, wireless commuicatios ad etworkig, ad sigal processig. May applicatios are evisaged icludig eviromet moitorig, battlefield surveillace, ad urba search ad rescue especially i hazardous situatios. Wireless sesor etworks (WSN) operate uder limited radio coverage ad attempt to coserve badwidth ad battery power. Most of the research o WSN has focused o the developmet of collaborative sigal processig ad power aware algorithms []. Sesor odes are geerally assumed to be fixed ad radomly placed because of practical reasos. The umber of sesors is assumed to be quite large so that coverage of the surveillace area is ot a issue. t much attetio has bee paid to optimizatio i terms of umber of odes or their topology. Recetly, Meguerdichia et al. have cosidered the problem of locatio ad deploymet of sesors i a WSN from a coverage stadpoit []. They implicitly assumed fixed wireless sesor odes. They argued that coverage is a primary performace metric that provides a idicatio regardig quality-of-service. They combied computatioal geometry ad graph theoretic approaches to develop algorithms for coverage calculatios. We also focus o coverage i WSN ad discuss it later i the paper. Bulusu et al. s work [3] is somewhat similar to the deploymet problem that is cosidered here. They have ivestigated the problem of adaptive beaco placemet for localizatio i a WSN. They also poited out the lack of viability ad iadequacy of fixed ad dese beaco placemet i some situatios due to ode perturbatio durig deploymet, oisy eviromet, ad self-iterferece. By placig additioal beacos icremetally, they achieve empirical adaptatio to terrai coditios. Aother related problem is the Art Gallery Problem i computatioal geometry [4]. The art gallery problem tries to fid the miimal umber of positios for guards or cameras so that every poit i a gallery is observed by at least oe guard or camera. A determiistic solutio ca be foud for the art gallery problem ad it appears to be a possible solutio to a variety of sesor placemet problems. Eve though there are may solutios to the art gallery problem, all of them assume the availability of a good model of the eviromet a priori. However, it is virtually impossible to have complete iformatio of the eviromet i a WSN. Too much commuicatio over log rage to obtai global iformatio requires a huge amout of eergy. This is a uaffordable burde o a system with limited power supply. Thus, determiistic deploymet is impractical due to may reasos such as the harshess of deploymet regio that may be remote ad ihospitable ad the icreased cost ad latecy due to the huge umber of odes deployed [5]. I our work, we are iterested i the self-deploymet of mobile sesor odes. This is quite similar to problems cosidered i cooperative mobile robotics [6]. Mobile sesors are ofte desirable sice they ca patrol a wide area, ad they ca be re-positioed for better surveillace [7]. Some researchers cosidered the use of mobile robots i sesor etworks. Wifield [8] cosidered autoomous dispersio of mobile odes i a sceario where mobility is required to cover the etire regio due to a lack of wireless etwork coectivity. He used a radom diffusio method for ode deploymet while collectig data over a fixed surveillace regio. I the icremetal deploymet algorithm [9], odes are added oe at a time. The goal is to maximize etwork coverage uder the costrait that odes maitai lie-of-sight with each other. Loo et al. cosidered a system cosistig of a umber of cooperatig mobile odes that move toward a set of prioritized destiatios uder sesig ad commuicatio costraits []. They show how idividual agets kow whe cooperatio betwee agets improves the performace ad whe they should susped cooperatio.

2 Much research has bee doe about various issues related to the deploymet problem. Most approaches use either a cetralized solutio as i the circle coverig problem [,] ad the geometric problem [3], or are restricted to a certai topology as i the codig theoretic approach [4]. Sice some approaches adopt radom deploymet as i the set coverig problem [5] ad the topology discovery problem [5], iitial distributio determies the utilizatio of etworks. work is differet from prior work o the deploymet problem. deploymet algorithm s mai objective is topology improvemet for loger system lifetime by utilizig mobility of robots. We provide a decisio ad cotrol mechaism at each robot to be used durig deploymet rather tha radom diffusio, which is used i Wifield s work. I cotrast to Howard et al. who use a icremetal approach, the odes i our algorithm are deployed at the same time ad they orgaize themselves i a adaptive maer. Ulike Loo et al., our algorithm does ot require prespecified destiatios to form a eergy efficiet topology. self deploymet algorithm will be more useful i situatios where it is hard to esure precise iitial deploymet due to the fact that the deploymet area is too dagerous or iaccessible to humas. Radomly scattered sesors over a battlefield or a hazardous site are ot likely to form a uiform distributio ad provide desired coverage. Modificatio of WSN topology i a autoomous ad distributed maer usig our algorithm ca help i improvig sesor coverage ad also to prolog expected system lifetime. This is essetial i time-critical applicatios. For example, if some area is cotamiated by a hazardous material, a properly deployed sesor etwork ca quickly sese ad measure the amout of hazardous material such as poisoous gas or uclear leakage. By fully coverig the etire area of iterest, the overall coditio ca be assessed quickly ad this iformatio ca be used for search ad rescue missios as well as for evacuatio route plaig. The mai goals of this paper are ) discussio of the issue of coverage i WSN i detail ) developmet of a distributed algorithm for self-deploymet I the ext sectio, we discuss performace metrics for a mobile WSN. I Sectio 3, we formulate the sesor deploymet problem. algorithm is preseted i Sectio 4 followed by simulatio results i Sectio 5. Some cocludig remarks are provided i Sectio 6.. Performace Metrics i Mobile WSN Selectio of suitable measures to compare performaces of differet approaches is a importat issue i a mobile WSN. Coverage, uiformity, time, ad distace are cosidered as performace metrics i mobile wireless sesor etworks here. Coverage ad uiformity are related to the performace of sesor etworks after the deploymet of sesors is complete. Time ad distace are directly related to the performace of the deploymet scheme itself. Coverage Geerally, coverage ca be cosidered as the measure of quality of service of a sesor etwork. The cocept of coverage as a paradigm for the system level fuctioality of multi-robot systems was itroduced by Gage [6]. Gage described three types of coverage behavior for problems such as detectio of targets over a surveillace area. These behaviors are: blaket coverage, where the objective is to obtai the maximum detectio rate over a give area; barrier coverage, where the objective is to miimize the probability of miss through the barrier; ad sweep coverage, which is roughly equivalet to a movig barrier. self deploymet algorithm ca be classified as a blaket coverage problem accordig to Gage s taxoomy. Coverage ca be classified as either determiistic or stochastic by the maer i which odes are deployed. To obtai determiistic coverage, sesor odes should be placed at predefied locatios. The art gallery problem is a example of determiistic coverage. By dispersig sesors radomly i the eviromet, stochastic coverage ca be achieved. I this paper, coverage is defied by the ratio of the uio of covered areas of each ode ad the complete area of iterest. Here the covered area of each ode is defied as the area withi sesig radius R. Perfect detectio of all iterestig evets i the covered area is assumed. U Ai C = i=,..., N where A A i is the area covered by the ith ode, N is the total umber of odes, A stads for the area of the regio of iterest(roi). If a ode is located well iside the regio of iterest, the complete circle will lie withi the ROI. I this case, the full area of that circle, i.e., π R, is couted as the covered regio. If a ode is located ear the boudary of the ROI, the oly the partial area of the ROI covered by that ode is icluded i the computatio. Because of areas covered by odes that fall out of the ROI ad the fact that overlap of covered areas betwee odes should ot be icluded while computig coverage, we eed to use more odes tha simply the ratio of A ad area sesed by a sigle ode. We distiguish sesig rage ad commuicatio rage of a ode. I geeral, they will be differet ad accordigly sesig coverage ad commuicatio coverage will be differet. Sesig coverage ca be accrued whe sesor odes are coected via wireless liks. Uiformity Uiformly distributed sesor odes sped eergy more evely through the WSN tha a irregular topology does. So, uiformity of etwork topology ca be a good estimator for the expected system lifetime. Also, fewer odes are required to cover a ROI whe odes are more evely distributed. Uiformity ca be defied by the average local stadard deviatio of the distaces betwee odes. N U = U i N i= K i ( ( Di, j M i ) ) Ui = K where i j= N is the total umber of odes K i is the umber of eighbors of the ith ode, D i,j is the distace betwee ith ad jth odes, M i is the mea of iterodal distaces betwee the ith ode ad its eighbors. I the calculatio of the local uiformity U i at the ith ode, oly eighborig odes that reside withi its commuicatio rage are cosidered. The uiformity measure is a local measure ad is computed locally because each ode has access to local iformatio oly. A smaller value of U meas that odes are more uiformly distributed i the ROI. I uiformly distributed etworks, iterodal distaces are almost the same; the expected eergy cosumptio per commuicatio as well as the expected lifetime of each ode is almost the same. Therefore, we ca expect full eergy utilizatio at each ode ad loger system lifetime for uiformly distributed etworks.

3 Time The time spet for deploymet is also importat i may timecritical applicatios such as search ad rescue ad disaster recovery operatios. Mostly, the required time depeds o the complexity of the reasoig algorithm ad physical time for the movemet of odes. The total time elapsed is defied here as the time elapsed util all the odes reach their fial locatios. We focus here o the time spet for deploymet itself ad ot o data trasmissio delays from a source ode to a destiatio ode that is commoly used for etwork performace evaluatio ad its quality of service. Distace The average distace traveled by each ode is related to the required eergy for its movemet. So, the expected distace traveled is importat for the estimatio of required eergy (fuel) whe each ode has a limited eergy supply. The variace of traveled distace is also importat to determie the fairess of the deploymet algorithm ad for system eergy utilizatio. 3. Mobile de Placemet/Deploymet Problem 3.. Assumptios ad restrictios We assume that all sesor odes have capabilities for sesig, commuicatio, computatio, ad mobility. Sesig coverage ad commuicatio coverage of each ode is assumed to be ideal, which meas that both coverage areas have a circular shape without ay irregularity. Computatio capability is required at each ode to support a distributed algorithm that icludes a reasoig process for deploymet ad routig. We assume that the iitial deploymet is radom ad a distributed deploymet algorithm will be executed startig from the iitial radom topology usig each ode s mobility. Aother assumptio is that every ode has the ability to kow its ow locatio by some method such as GPS or iterative multilateratio [7]. This locatioig ability is eeded by each ode while makig a decisio regardig its ext movemet i the deploymet process. Also, we assume that there are o errors durig trasmissios of data ad i the calculatio of locatios. The restrictio is that each ode has oly local iformatio from the eighborig odes withi its direct commuicatio rage. The commuicatio rage of each ode is defied by the maximum distace at which the sigal to oise ratio is above the threshold required for achievig the desig goal i terms of power coservatio. 3.. Problem formulatio Coverig ad placemet problem ca occur i may applicatios from eviromet moitorig to battlefield surveillace. Without loss of geerality, we ca cosider the coverig problem i a rectagular ROI with a certai umber of odes that form a ad hoc wireless etwork. The goal is to fid positios ad movemets of odes to achieve maximum coverage ad to form a uiformly distributed wireless etwork i miimum time ad with miimum eergy cosumptio. We develop a heuristic algorithm ad evaluate its performace i terms of these performace metrics. 4. Distributed Self-Spreadig Algorithm The algorithm is ispired by the equilibrium of molecules, which miimizes molecular electroic eergy ad iter-uclear repulsio. Each particle obtais its ow lowest eergy poit i a distributed maer ad its spacig from the other particles is almost the same. While deployig a wireless sesor etwork usig mobile odes, we observe that a similar pheomeo is eeded. We propose a algorithm that ca cover the regio of iterest without ay itervetio from a cetral cotroller that acts remotely. We call this algorithm Distributed Self-Spreadig Algorithm(DS) ad discuss it i detail. To begi with, a specified umber of odes are pre-deployed radomly i a give regio, for istace, iside a rectagle. The sesig rage (sr) ad the commuicatio rage (cr) are assumed to be give. Each ode ca sese or detect a evet withi its sesig rage ad ay pair of odes withi their commuicatio rage ca commuicate with each other. This commuicatio is eeded for fidig eighborhood, obtaiig locatios of odes i the eighborhood, ad trasmittig ad forwardig sesed data. Neighborhood is defied here as odes withi the commuicatio rage. The flow chart of the algorithm is give i Figure. This distributed algorithm is executed at each ode i. Start Iitializatio p, sr, cr Calculatio of desity µ,d, D=D Calculatio of partial force f i,j (µ,d,cr,p ) Temporary positio p i + ' = pi + sum(f i,j ) Oscillatio? p i - -pi + ' < e p i + =pi + ' Update D Stable? p i + -pi < e Cout stable status S cout = S cout + S cout < S lim p i + = pi Cout oscillatio O cout = O cout + O cout < O lim p i + =/(pi +pi + ') Update D Stop Fig. Distributed Self-Spreadig Algorithm (DS) The algorithm begis with the specificatio of cr, sr, ad the iitial ode locatios (p ). I our algorithm, we require the quatity called expected desity. This ca be calculated by usig N π cr µ ( cr) =, where N is the umber of odes ad cr is the A commuicatio rage of each ode, ad A is the ROI. Thus, expected desity is the average umber of odes required to cover the etire area whe these odes are deployed uiformly. Iitial local desity D of a ode is equal to the umber of odes withi its commuicatio rage. These desities will be used whe decisios regardig positios of odes are made. We itroduce the cocept of force to defie the movemet of odes durig the deploymet process. The force is depedet o the distace betwee odes ad the curret local desity. The force from a ode that is closer is greater tha that from a ode that is farther just like the particles i Physics that follow Coulomb s law.

4 We defie a force fuctio that satisfies the followig coditios. (i) Iverse relatio: f(d ) f(d ), whe d d, where d ad d are ode separatios from the origi. (ii) Upper limit: f( + ) = f max. (iii) Lower limit: f(d) =, where d > d th, d is the ode separatio ad d th is the threshold to defie the local eighborhood. Coditio (i) is the same as i Physics, but coditios (ii) ad (iii) are icluded to modify the model to icorporate the otio of locality. I other words, a limitig fuctio is applied via coditios (ii) ad (iii). The partial force at time step o the ith ode from the jth ode that is i the eighborhood of the ith ode is calculated i this paper as j i i, j D i j p p f = ( cr p ) p () µ j i p p where cr stads for commuicatio rage p stads for the locatio of ith ode at time step i 5. Simulatio Results We cosider 3 radomly placed odes i a regio of size to ru the Distributed Self-Spreadig Algorithm. We assume sr= ad cr=4. Fig. shows the locatios ad coverage of the iitial deploymet before ruig the Distributed Self-Spreadig Algorithm. Tiy circles represet the positios of odes ad small (shaded) ad large circles are used to show the sesig rage ad the commuicatio rage respectively. Sesor iformatio may be collected withi the sesig rage ad commuicatios betwee odes are possible withi the commuicatio rage. Commuicatios are possible betwee odes that are coected by a lie i the figure. As see i Fig., some parts of the regio caot be covered by the odes that are radomly dispersed, eve though there are eough umbers of them i the give ROI. I this particular example, the etwork is ot fully coected, so the actual coverage is much smaller tha just addig the etire covered regio. The calculated coverage is more tha 9%, but the actual coverage is well below 5% because the etwork is separated. This situatio is exactly the case where topology improvemet is required. The desity factor, which is defied as the ratio of the local desity (D) ad the expected desity (µ) at each ode, is small i sparse regios ad is large i dese regios. Also iterodal distace affects the partial force iversely. Closely located odes impose larger partial forces ad odes that are far apart iduce smaller partial forces. 8 6 Iitial positio of 3 sesors, C = After addig all the partial forces at the curret locatio, each ode ca decide its ext movemet. The local iformatio is collected from the odes that are withi the commuicatio rage ad that iformatio is used for the calculatio of the local desity at each ode. Each ode s movemet is decided by the combied force at that ode due to odes i its eighborhood. Whe should a ode stop its movemet is a importat issue. Two stoppig criteria are itroduced i the Distributed Self- Spreadig Algorithm. If a ode moves less tha e for the time duratio S lim, this ode ca be cosidered to be i the stable status ad that ode stops its movemet. This stoppig criterio is for statioary odes because of either empty fuel or broke mobile uits ad also for the odes that have reached the stable status. If a ode moves back ad forth betwee almost the same locatios may times, this ode is regarded i the oscillatio status. By comparig with the history of its movemet, each ode ca kow if oscillatio is goig o. The oe couts the umber of oscillatios ad if this oscillatio cout (O cout ) is over the oscillatio limit (O lim ), we stop that ode s movemet at the ceter of gravity of the oscillatig poits. Whe sesor odes are deployed i a remote ad hostile regio, some odes ca be affected durig ad after deploymet period. Some odes ca lose their mobility ad other odes ca lose their commuicatio fuctioality. So we eed a robust deploymet method that ca hadle these difficulties. algorithm exhibits this kid of robustess. First, whe the sesor ode loses its mobility, that sesor ode does ot move ad is cosidered to be a early stopped ode. Neighborig odes, if they ca move, may still improve the irregular topology. Secod, whe a sesor ode loses its commuicatio capability, that sesor ode is of o use i a sesor etwork. Because each ode s movemet is oly affected by the curret status of eighborig odes, each ode is adaptive to eviromet chages such as ode failures, various terrai shapes, etc Fig. Iitial distributio of sesor odes Fig. 3 shows odes locatio ad coverage after ruig the Distributed Self-Spreadig Algorithm. The rectagle is fully covered after ruig the algorithm. The parameter values used i this simulatio ru are: stable status limit(s lim ) =, oscillatio limit (O lim ) =, ad threshold (e) for oscillatio ad stable status = w the etwork is fully coected ad also ca cover the etire ROI. te the spatial ode distributio is more uiform tha the iitial radom distributio i Fig.. Iteratio # : 36, Covered Area:, Slim:, Olim:, e: Fig. 3 Fial distributio after ruig DS

5 For the purpose of compariso, Simulated Aealig was also used for self-spreadig. Simulated Aealig algorithm is kow as a good solutio of may combiatorial optimizatio problems. To implemet a Simulated Aealig algorithm, 4 mai desig issues should be cosidered. These are: the defiitio of the eighborhood, move operator, local eergy calculatio, ad aealig schedule. The defiitio of eighborhood is the same as for DS, i.e., it is set equal to the commuicatio rage. This cocept of eighborhood is reasoable because each ode ca oly reach the eighborig ode i a sigle hop commuicatio. The move operator is chose to be a radom movemet withi the eighborhood. The local eergy calculatio is doe by addig up sub-forces i the eighborhood just like our algorithm. The expoetial coolig schedule is used as the aealig schedule for efficiecy as i [8]. The parameters used are iitial temperature T = ad as the stoppig criterio 3 cosecutive failure of achievig the desired acceptace ratio is used. The result after applyig the simulated aealig algorithm is show i Fig. 4. Fig. 4 shows that Simulated Aealig also works well for this iitial distributio. The etire area is covered by 3 sesor odes ad these odes are well spread over the regio. Iteratio # : 9, Covered Area:, T:, failureidexlimit: IitstdSeparatio, stdseparatio Iitial # of des Fig. 6 Uiformity versus etwork size Fig. 5 shows the improvemet i coverage area from the iitial radom deploymet for both algorithms; simulated aealig ad ours. Though both algorithms have quite similar performace, our algorithm has better coverage for smaller umber of odes. As the umber of odes icreases, the improvemet i coverage dimiishes. Stadard deviatio of iterodal distaces ca be cosidered as the metric of uiformity of the etworks. Fig. 6 shows the reductio i the stadard deviatio from the iitial case. Both algorithms obtai better uiformity tha the iitial oe ad ours outperforms simulated aealig. The improvemet i uiformity is ot that differet with varyig etwork desity stoppednum Fig. 4 Fial distributio after ruig The performace of our algorithm is evaluated i terms of the metrics preseted i Sectio. Results are preseted i Fig. 5 ~ 8. These results are obtaied for differet umber of odes dispersed over a fixed ROI of size, i.e., for differet ode desities. Number of odes varies from to 5 ad results are averaged over rus..95 IitArea & Covered Area # of des 8 7 Fig. 7 Termiatio times versus etwork size meatotaldistace Iitial Area # of des Fig. 5 Coverage versus etwork size # of des Fig. 8 Distace traveled versus etwork size

6 Fig. 7 shows that our algorithm leads to faster deploymet tha simulated aealig. Also termiatio times of our algorithm are similar over a wide rage of umber of odes. This meas that our algorithm is more isesitive to the umber of odes, i.e., etwork desity. Fig. 8 shows mea distace traveled to reach the fial locatios for deploymet. algorithm requires less distace traveled tha the simulated aealig algorithm. This distace is related to the required eergy (fuel) for deploymet. Also, we observe that the required eergy as well as the distace traveled at differet ode desities is almost costat. Thus, the required eergy (fuel) is quite isesitive to etwork desity. As see i Fig.5~Fig.8, 5~4 odes are required to attai acceptable performace. Whe too few odes are used, we caot obtai full coverage over the regio of iterest. Whe too may odes are used, we do ot gai that much coverage improvemet because of the dimiishig margial gai i terms of coverage, though we ca still obtai more uiform distributio. With the umber of odes i this rage, the required time ad distace of our algorithm is much smaller tha that of the Simulated Aealig algorithm. Because the variatio of required time ad traveled distace is small over this rage of ode desities, it is easy to estimate the required eergy for deploymet. 6. Coclusios We cosidered the sesor coverage problem for the deploymet of wireless sesor etworks here. A regio of iterest eeds to be covered by a give umber of odes with limited sesig ad commuicatio rage. We start with a radom distributio of the odes over the regio of iterest. Though may scearios adopt radom deploymet because of practical reasos such as deploymet cost ad time, radom deploymet may ot provide a uiform distributio which is desirable for a loger system lifetime over the regio of iterest. I this paper, we propose a distributed algorithm for the deploymet of mobile odes to improve a irregular iitial deploymet of odes. After goig through the algorithm, the area of iterest is covered by uiformly distributed odes. While developig the algorithm, oe should cosider factors such as desity of odes, memory costraits, localizatio errors, ad scalability of mobile odes. Through the requiremet of mobility ad locatioig ability of odes, this algorithm provides a way to avoid expesive redeploymet process. This postdeploymet idea will be more useful especially whe a large fractio of odes are destroyed or broke durig deploymet or i a hostile situatio, where iitial distributio is quite ueve ad whe redeploymet is too costly or too risky. The performace of the algorithm is determied by the percetage of regio covered, by computatioal/deploymet time, by the mea distace that is required for the deploymet, ad by the uiformity of the etworks. Simulatio results show that our algorithm successfully obtais a uiform distributio from iitial ueve distributio. The performace of our algorithm is compared with the Simulated Aealig algorithm ad exhibits excellet performace. Ackowledgmets This work was supported i part by the Natioal Sciece Foudatio uder grat umber ECS-9936 ad by NA uder grat umber NAG5-7. Refereces [] Kumar, S.; Feg Zhao; Shepherd, D., Collaborative sigal ad iformatio processig i microsesor etworks, IEEE Sigal Processig Magazie, vol.9, o., pp. 3-4, Mar.. [] Seapah Meguerdichia, Fariaz Koushafar, Miodrag Potkojak, ad Mai Srivastava, Coverage Problems i Wireless Ad-hoc Sesor Networks, IEEE Ifocom, pp ,. [3] Nirupama Bulusu, Joh Heidema, ad Deborah Estri, Adaptive Beaco Placemet, i Proceedigs of the st Iteratioal Coferece o Distributed Computig Systems, pp , Phoeix, AZ, Apr.. [4] O'Rourke, J., Art Gallery Theorem ad Algorithms. New York: Oxford Uiversity Press, 987. [5] S. Slijepcevic ad M. Potkojak, Power efficiet orgaizatio of wireless sesor etworks, Proceedigs of IEEE Iteratioal Coferece o Commuicatios,. [6] Cao, Y. 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