Minimum Cost Localization Problem in Three-Dimensional Ocean Sensor Networks

Transcription

1 Minimm Cost Loliztion Prolem in Three-Dimensionl Oen Sensor Netorks Cho Zhng Yingjin Li Zhongen Go Goong Sn Y Wng Deprtment of Compter Siene, Uniersity of North Crolin t Chrlotte, Chrlotte, NC 8, USA. Deprtment of Compter Siene n Tehnology, Oen Uniersity of Chin, Qingo 66, Chin. Shool of Informtion Siene n Tehnology, Beijing Forestry Uniersity, Beijing 8, Chin. Astrt Loliztion is one of the most fnmentl prolems in oen sensor netorks. Crrent loliztion lgorithms minly fos on ho to lolize s mny sensors s possile gien set of moile or stti nhor noes n istne mesrements. In this pper, e onsier the optimiztion prolem, minimm ost loliztion prolem in D oen sensor netork, hih ims to lolize ll nerter sensors sing the minimm nmer of nhor noes or the minimm trel istne of the ship hih eploys n mesres the nhors. Gien the hrness of D loliztion, e propose set of greey methos to pik the nhor set n its isiting seqene. Aiming to minimize the loliztion errors, e lso opt onfiene-se pproh for ll propose methos to el ith noisy rnging mesrements n possile flip migity. Or simltion reslts emonstrte the effiieny of ll propose methos. I. INTRODUCTION Loliztion is one of the most fnmentl tsks in esigning oen sensor netorks (OSNs) [], [] or generl ireless sensor netorks []. Lotion informtion n e se in mny tsks of OSNs sh s eent eteting, trget/eie trking, enironmentl monitoring, tgging r sensing t, n netork eployment. Moreoer, lotion informtion n lso e se y netorking protools (e.g., position-se roting or geometri topology ontrol) to enhne the performne of OSNs. Hoeer, it is more hllenging to lote noes in nerter enironments thn in terrestril enironments. First, GPS signl oes not propgte throgh ter n RF signl nnot e se sine it ill ttente ery rpily in ter. Ths, osti signl is slly the est hoie in nerter enironments. Seon, seerl lterntie oopertie positioning shemes re not pplile in prtie e to osti hnnel properties (sh s lo nith, high propgtion ely n high it error rte). Sine the eloity of osti signl n hnge ith slinity, pressre n tempertre, it is iffilt to get qite preise rnges eteen noes nerter. Lst, D eployment of OSN reqires more The ork of C. Zhng n Y. Wng s spporte in prt y the US Ntionl Siene Fontion ner Grnt No. CNS-995 n CNS The ork of Y. Li s prtilly spporte y the Ntionl Ntrl Siene Fontion of Chin (NSFC) ner Grnt No The ork of Z. Go s prtilly spporte y the NSFC ner Grnt No n The ork of G. Sn s prtilly spporte y the NSFC ner Grnt No. 68. This ork s prtilly one hen Y. Li isite Deprtment of Compter Siene, the Uniersity of North Crolin t Chrlotte, ith sholrship from the Chin Sholrship Conil. Y. Wng (y.ng@n.e) is the orresponing thor. nhor noes to lote noes in D oen spe. All these mke rte loliztion in oen hllenging tsk. A lrge nmer of loliztion tehniqes [4] [7] he een propose for OSNs to lolize nerter sensors y exhnging informtion ith nhor noes. Utilizing time of rril (ToA) [4], [5] or time ifferene of rril (TDoA) [6], [7], these methos n estimte istnes eteen noes n then ompte positions of noes se on these istnes. Uslly, ertin mont of nhor noes re eploye, hose positions re knon eforehn or n e mesre y se srfe oys or essels. Most of existing methos try to lolize sensors ithot the grntee of oering ll sensors. They slly rely on one hypothesis tht there re enogh nhor noes to hiee the gol. Reently, Hng et l. [8] introe ne loliztion prolem, lle minimm ost loliztion prolem (MCLP), for D sensor netorks hih ims to lolize ll noes in netork sing the minimm nmer of nhor noes. A set of greey lgorithms sing oth triltertion n lol seep opertions [8] n geneti lgorithm [9] he een propose to ress this prolem in D netorks. In this pper, e frther exten n enhne the optimiztion prolem to D OSNs (Setion II). Note tht the ost of mnlly onfigring n nhor noe in oen is expensie n GPS eie oes not ork ell nerter. Therefore, e rely on ship isit to eploy n fin the lotions of nhor noes. In ition, e lso inle ne rition of MLCP here the length of isite pth for the ship is onsiere s the optimiztion ojetie. For oth ersions of MCLP, e propose mltiple greey lgorithms to lolize ll nerter sensors ith the minimm nmer of nhors or the minimm trel istne of the ship (Setion III). In ition, to hnle loliztion errors se y oth rnging errors n flip migity, e introe onfiene-se pproh to ontrol errors ine in loliztion proess (Setion IV). Or simltion reslts (Setion V) emonstrte the effiieny of ll propose methos. II. MINIMUM COST D LOCALIZATION PROBLEM A. Moels n Assmptions The D OSN n e moelle s grph G = (V, E), in hih V is the set of noes representing nerter sensor noes n E is the set of links onneting sensor noes if they re ithin eh other s sensing rnge. Here, e ssme ll sensor noes re stti, ignoring the moement se y

2 Z P nhor sensor X Y Ship sensing epth Fig.. Illstrtion of loliztion senrio in oen sensor netorks: lk noes re selete nhors from the shllo sensors, hile other green noes re lolize i D loliztion metho. P represents the trelling pth of the ship to isit ll nhors. ssrfe rrent. We lso ssme tht the sensing rnge of ll sensors re sme. A sset of sensor noes B V ill e efine s nhor noes hose positions nee to e knon t the eginning of loliztion (i.e., eploye n mesre y ship isit). Note tht GPS eies o not ork ner ter. The remining iely istrite nerter sensor noes ill rely on istne mesrements in E n the positions of nhor noes B to etermine their lotions ring the loliztion proere. Unerter sensor noes mye istrite in ifferent epths, t only those shllo sensor noes ithin ship s sensing rnge n e ptre. Ths, e ssme tht only the shllo sensors n e onsiere to eome nhor noes. We se V V to enote the set of shllo sensor noes. See Fig. for illstrtion. Note tht it is possile tht some nerter sensor noes nnot e lolize een thogh ll shllo sensor noes eome nhors. Therefore, e ssme tht V only ontins sensors ho re lolizle hen ll shllo sensors re set s nhors. B. The MCLP Prolems The min prpose of the minimm ost loliztion prolems (MCLPs) in D OSN is to lolize ll nerter noes sing the minimm ost for nhor noes. The ost inles the eqipment ost n ny osts ring eployment n mesrement. Similr to [8], y ssming nit ost per nhor noe, e n efine the folloing D MCLP prolems: Definition : Minimm Cost Loliztion Prolem (MCLP-): Gien D oen sensor netork G, fin sset B of shllo sensor noes to e nhor noes sh tht () ll sensor noes in V n e lolize gien the grph, the lengths of ll links, n positions of ll nhor noes; n () the totl nmer of nhor noes B is minimize. In ition to the ost per nhor noe, the ost of eployment n mesrement for ll nhors my epen on the length of the rote P trelle y the ship (shon in Fig. ). In tht se, the MLCP prolem n e efine s follos: Definition : Minimm Cost Loliztion Prolem (MCLP-): Gien D oen sensor netork G, fin sset B of shllo sensor noes to e nhor noes sh tht () ll sensor noes in V n e lolize gien the grph, the lengths of ll links, n positions of ll nhor noes; n () the totl length of the rote P tken y the ship to isit ll nhors is minimize. It is ler tht oth MCLPs lys he fesile soltion, sine se on or ssmption in the orst se eery shllo sensors re selete s nhors (i.e., B = V ) n ll sensors in V ill e lolize. Hoeer, fining the optiml soltion of sh prolems is still ery hllenging. Sine the D MCLP [8] is speil se of MCLP- n it is NP-hr, MCLP- is lso NP-hr. In MCLP-, ithot onsiering the loliztion prt, fining the minimm length rote to isit ll nhors lone is the ell-knon trel slesmn prolem (TSP), hih is NP-hr. Therefore, MCLP- is lso NP-hr. III. MIN-COST D LOCALIZATION ALGORITHMS In this setion, e introe ifferent greey lgorithms to pproximtely sole MCLP prolems y fining the nhor set. All propose greey lgorithms se simple olor oing, here eh sensor noe ill e mrke ith ifferent olors. We se s() to enote s olor n stts. A hite sensor noe represents the noe hih hs not een lolize yet; lk sensor noe enotes the selete nhor noe hih n e iretly ontte y the ship to get its position; green sensor noe is non-nhor noe hose lotion n e otine i loliztion. Initilly, ll sensor noes re hite. The prpose of or lgorithms is to fin the smllest set of nhors (lk noes) or the nhor set ith the minimm trelling istne to get the positions of ll noes (oloring ll noes in either lk or green) s shon in Fig.. A. Algorithms for MCLP- For soling the D MCLP-, e simply exten to propose methos in [8] for D MCLP (one ses triltertion n the other ses lol seep for loliztion) to D netorks. Both methos se generl greey frmeork s shon in Algorithm. The si ie is s follos. Initilly ll sensor noes ill e olore s hite n ith its rnk r() = (Line -). Here, noe rnk r() of noe inites the totl ont of lolize neighors (ith olor of lk or green). In ition, for ny sensor noes ith less thn for neighors ill e mrke s lk (Line -5), sine they nnot e lolize y others. Note tht se on or ssmption, ll these noes re shllo sensor noes.then in eh step (Line 6-) the lgorithm greeily piks one hite sensor noe hih n enefit the loliztion proere most in next step if it is mrke s lk, n olors it s lk. Here e efine the enefit of mrking noe lk s g() hih is the nmer of nely mrke green noes if is mrke s lk. This proere termintes ntil there is no hite noe n ll lk noes re the selete nhor noes y the lgorithm. In Algorithm, hen noe is mrke s lk or green, fntion MARK is lle, hih is rersie proess of loliztion se on the nerline loliztion metho (triltertion or lol seep).

3 Algorithm Greey Algorithm for MCLP- : for eh V o : s() = hite n r() =. : for eh V o 4: if egree of () then 5: MARK(, lk). 6: hile hose s()=hite o 7: for eh hose s()=hite o 8: Bkp rrent stts of ll noes. 9: MARK(, green). : g() = totl nmer of nely e green noes. : Restore stts of ll noes. : Let mx is the hite noe ith the mximm g(). : MARK( mx, lk). 4: retrn ll lk noes s the selete nhors. Algorithm MARK(, olor) se on Triltertion : s()=olor. : for eh of s hite neighor o : r() : for eh of s hite neighor ith r() 4 o 5: MARK(, green). Algorithm MARK(, olor) se on Lol Seep : s()=olor. : for eh of s hite neighor o : r() + +. {Phse : triltertion} 4: for eh of s hite neighor ith r() 4 o 5: MARK(, green). {Phse : lol seep ithin to-hop neighorhoo} 6: if to hite neighors of (sy n ) oth he rnks of n re neighor to eh other then 7: if oth n he niqe positions hih re onsistent ith istne mesrement then 8: MARK(, green) n MARK(, green). {Phse : lol seep ithin three-hop neighorhoo} 9: if hite neighor of n nother hite neighor of oth he rnks of then : if oth n he niqe positions hih re onsistent ith istne mesrement then : MARK(, green) n MARK(, green). ) MARK se on Triltertion: Algorithm shos the mrk fntion for mltiltertion (itertie triltertion). When noe is mrke s lk or green, ll its hite neighors rnk stts nee to e inrese one. If its hite neighor s rnk rehes 4, this neighor n e mrke s green too. ) MARK se on Lol Seep: Triltertion hs its on limittion s issse in []. Similr to D methos propose in [8] n the loliztion metho y [], e n se seep opertions to improe the loliztion y heking the onsisteny of possile positions of noes in lol neighorhoo n lolizing more noes if possile. When to () niqe mth () no niqe mth Fig.. Lol seep ithin to-hop: () there is niqe mth; () there is no niqe mth. Here oth n re s neighors ith rnk of. () niqe mth e () no niqe mth Fig.. Lol seep ithin three-hop: () niqe mth; () no niqe mth. is s neighor n is s neighor, n oth he rnk of. Algorithm 4 Greey Algorithm for MCLP- -9: sme s Algorithm. : g() = -4: sme s Algorithm. totl nmer of nely e green noes istne from to the lst selete lk noe. neighoring noes hose rnks re oth (i.e., the possile positions of eh of them re limite to to lotions), the istne eteen these to noes n e se to eliminte the ogs positions. Note tht it is possile tht niqe mth nnot e fon, then these to noes nnot e relize y the lol seep. To ree the oerhe, e limit the seeps ithin to-hop or three-hop rnges. Fig. n Fig. illstrte exmples for ll ses. The etile MARK proere is gien in Algorithm. B. Algorithms for MCLP- In MCLP-, the ltimte gol is to minimize totl nmer of nhors in D OSNs. Hoeer, the length of the rote P trelle y the ship hih eploys n mesres ll nhors n lso le to time n finnil ost. Therefore, in MCLP-, e fos on minimizing the trelling istne of the ship inste. Oiosly, e n still se the greey lgorithms propose for MCLP- n let the seletion orer of nhors e the isiting seqene of nhors y the ship. Ntrlly less selete nhors reslts in shorter trelling istne of the ship. Therefore, the greey metho se on lol seep my gie etter performne thn the one se on triltertion. Sine ll of preios greey lgorithms o not onsier the trel istnes mong selete nhor noes, one possile improement ol e inling the istne metri in the seletion riteri of or greey lgorithm. Bsilly in eh step, e n selet the next hite noe, hih yiels the est rtio eteen the gin of nmer of green noes n the nely e

4 e trel istne (from this noe to the preios nhor), s the next nhor. See Algorithm 4 for the etile lgorithm. Moreoer, one ll nhors (lk noes) re selete y oe methos, frther optimiztion on the trel istne n e performe. Rell tht trelling slesmn prolem (TSP) ims to minimize the trel istne hile isiting eh noe extly one. TSP is ell-knon NP-omplete prolem n nnot e esily sole to fin the optiml soltion, hile there re mny existing pproximtion lgorithms or heristis hih ork ell in prtie. We n se ll nhors fon y or greey lgorithms s the inpt of TSP lgorithm to fin the isiting seqene of the ship. This n frther ree the finl length of its isiting pth. IV. MIN-COST D LOCALIZATION WITH ERRORS So fr, e ssme tht there is no error in istne mesrements n the llte positions i triltertion or other loliztion methos re rte one the noe is lolizle. Hoeer, this is not the se in rel orl, espeilly for OSNs. First, e to high ely n error rte of osti hnnel n rios eloities of osti signls ith hnging slinity, pressre n tempertre, it is iffilt to get preise istne mesrements eteen nerter noes in the oen. Seon, een hen the istne mesrement error is minimize, flip migity n still reslt in totlly ontrry reslts s emonstrte y []. Lst, sine most of the nerter noes nee to e lolize i other noes sing itertie loliztion methos, ny lol errors n propgte to lrge region, mlte throgh the loliztion proess, n ine hge eitions from the orret positions. This is espeilly tre for lrge-sle netorks. Therefore, e nee to onsier ho to hnle the loliztion errors se y oth rnging errors n flip migity. Similr to [] n [], e opt onfiene-se pproh to ontrol errors ine in loliztion proess. For eh sensor noe, e efine noe onfiene (), hih inites the proility of its rte loliztion. The onfiene () epens on the onfienes of referene noes of n the lyot of the referene noes. Here, e se referene noes to enote the lolize noes (those mrke s green or lk) se on hih e n lote other noes y triltertion. For exmple, in Fig. 4(), hite noe n e lolize i for referene noes (,, 4, ll mrke s green or lk noes). Similr to [], e se the ith of the referene noe set to estimte the onfiene of triltertion, sine nrro ith of referene noes my le to higher proility of migity n errors. Here, the ith of for referene noes n e efine s the miniml height of tetrheron 4 (mrke s le in the figre) hih epens on the lyot of referene noes. To normlize the onfiene of triltertion into rnge [, ], e iie the ith of referene noes to the height of reglr tetrheron ith its size length eql to the sensing rnge. Let t (,,,, 4 ) represent the onfiene of triltertion t y sing,, 4 s the referene noes. Then the onfiene of is gien y () = min{( ), ( ), ( ), ( 4 ), t (,,,, 4 )}. In 4 () triltertion () lol seep Fig. 4. Clltion of Noe Confiene: () s onfiene epens on its for referenes onfienes n the onfiene of triltertion (from le tetrheron); () onfienes of n he the sme le n epen on onfienes of six referenes n the onfienes of triltertion (from oth re n le tetrherons). other ors, the onfiene of noe is the minimm le mong the onfienes of its referene noes n the onfiene of triltertion. By this efinition, the seletion of its referene noes inflenes the onfiene of noe. We lys ssme tht the onfiene of n nhor noe is. Dring lol seep, to noes re lolize i fie to six referene noes. Fig. 4() shos n exmple for lol seep ithin three-hop. In this exmple oth n n e lolize i six referenes,, 6. Sine the positions of n epen on eh other, ths ( ) = ( ). The onfiene of triltertion t (,,,,, 4, 5, 6 ) = min{ t (,,,, ), t (,, 4, 5, 6 )}, i.e., the minimm one of the onfienes of to tetrherons ( in re n in le). Then the noe onfienes of n re gien y ( ) = ( ) = min{( ),, ( 6 ), t (,,,, 6 )}. By hing the noe onfiene, e n moify ll propose methos y reqiring minimm onfiene threshol α. A noe n e lolize i either triltertion (Line 4 in Algorithm n Line 4 in Algorithm ) or lol seep (Line 6 n Line 9 in Algorithm ) if n only if () α. A onfiene lltion n hek proere is introe into those lines efore the noe(s) n e mrke s lolize. If () < α, this noe nees to it for more referene noes (green noes) in its neighors. If noe is selete s n nhor (mrke s lk), its onfiene is set to e. V. SIMULATIONS In orer to elte the performne of propose methos, extensie simltions re onte on rnom generte D OSNs. In ll simltions, 5 sensor noes re rnomly eploye ithin (m) i region. The sensing rnge of eh noe is set to e m, i.e., if the istne eteen to noes is less thn or eql to m, e ssme tht there is istne mesrement eteen them. The sensing rnge of the ship is set s 9m, i.e., ll sensor In lol seep ith to-hop, fie referene noes re se. It jst like the se hen n 6 re the sme noe.

5 # of nhors portion of nhors trelling istne 4 x trelling istne 7 x () nmer of nhors () portion of nhors () trel istne efore TSP () trel istne fter TSP Fig. 5. Performnes of ifferent methos for MCLP- n MCLP-. # of nhors # of nhors loliztion error loliztion error () # of nhors/rnom () # of nhors/triltertion () loliztion error/rnom () loliztion error/triltertion Fig. 6. Performnes of Greey-Rnom n Greey-Triltertion ith ifferent onfiene threshols α. noes ithin 9m epth re the shllo sensor noes. A ertin nmer of sensor noes oorintes re generte ithin the i region. Then sing ll the shllo noes s nhors n triltertion, ll sensor noes hih nnot e lolize ill e elete. We only onsier ll sensor noes ho re lolizle hen ll shllo sensor noes re set s nhors. For eh set of simltions, e perform the simltions for times (i.e. oer rnom netorks), n report the erge reslts. We implement the three propose methos: Greey-Triltertion (Algorithms n ), Greey-LolSeep (Algorithms n ), Greey-TrelDist (Algorithms 4 n ). In ition, e lso test rnom metho Greey-Rnom. In Greey-Rnom, triltertion is rersiely se to lolize s mny noes s possile, n hen there is no more noes n e lolize, rnom noe is pike to e the next nhor. This proere is repete ntil eeryone re lolize. In the folloing setions, e ill test these for methos in ifferent experiments. A. Experiments for MCLP- In MCLP-, or ojetie is to minimize the nmer of nhors. Fig.5() shos the totl nmer of nhors selete y eh lgorithm. It is oios tht the less nhor noes selete the etter. Among ll methos, the rnom metho nees the most nmer of nhors. Greey-LolSeep yiels the est performne, sine it n lolize more noes t eh step. All the greey lgorithms ill onerge fter the sensor noes eome enser. Oerll, the totl nmer of nhors first inreses ith the noe nmer n then rops on shrply ron 4. Fig.5() shos the perentge of nhors to the totl nmer of shllo sensor noes. Clerly, the perentge of nhors rops hile the nmer of sensors inreses. When the sensor netork is sprse, most of shllo sensors he to e nhors. When the sensor netork is ery ense, only ery smll perentge of noes nees to e nhors. This shos tht or propose lgorithms n hiee etter performnes for lrge-sle D sensor netorks. B. Experiments for MCLP- In MCLP-, e im to minimize the trelling istne of the ship e to high ost of trelling in the se. Fig.5() shos the trelling istne of eh lgorithm se on the isiting seqene of the greey lgorithms (i.e. the orering of selete nhor noes). As expete, feer nhor noes reslts in shorter trelling istne. Hoeer, the rnom metho hs mh longer trelling istne thn other greey methos. Moreoer, Greey-TrelDist hs similr performnes s Greey-Triltertion. It mens tht the nhor nmer plys more importnt role in the rnomly eploye netork. This mye e to the similr istne mong sensor noes in niformly rnom eployment. As issse in Setion III-B, e n lso se existing TSP lgorithm to optimize the trelling istne of eh lgorithm. We fee the positions of ll selete nhors in geneti TSP lgorithm, n se the opt pth s the orse of the ship. Fig.5() shos the reslts. Roghly the totl trelling istnes re shorten to hlf of originl ones. Moreoer, the ifferene eteen the rnom metho n other greey methos eome smller n Greey-LolSeep still yiels the est reslt in term of trelling istne. C. Experiments for MCLP- ith Errors Lst, e onsier the MCLP prolem ner mesrement errors. Similr s [], e introe sle zero men Gssin noise β N(, σ ) to the istne mesrements. In or simltions, e let σ = n s the eflt le of

6 # of nhors () nmer of nhors loliztion error () loliztion error Fig. 7. Performnes of ifferent methos ith α =.5. the sle ftor β. All propose lgorithms no mst onsier the noes onfienes ring the loliztion proess. As expline in Setion IV, noe is lolize only hen its onfiene () α here α is the onfiene threshol. The lotion of the lolize noe is llte sing the lest sqre estimtion, sine it nnot e soltely sole e to mesrement errors. In the folloing experiments, e lso mesre the lotion errors of eh lgorithm in ition to the nmer of selete nhors. Here, e se the rtio eteen the error of estimte istne n the tl istne se on rte positions to represent the lotion error. ) The impt of onfiene threshol: We first hnge the onfiene threshol α from to.. Note tht hen α =, ll methos o not onsier the noe onfiene t ll ring the loliztion proess. De to spe limit, e only sho the reslts for Greey-Rnom n Greey-Triltertion in Fig. 6. As expete, more nhors re neee hen onfiene threshol inreses ese higher threshol mens striter seletion rles for nhor noes. De to striter seletion rles of nhor noes ith higher onfiene threshols, the loliztion ry inreses ith onfiene threshol s shon in Fig. 6() n (). Note tht the seletion of onfiene threshol my limit the loliztion pity. For instne, in Fig. 6() if α is set to e.5, the mximm size of lolize sensor set is ron 4. For netork ith more thn 4 sensors, the loliztion error inreses rmtilly n the loliztion reslts re not relile. Hoeer if the le of α is lrge enogh (sh s.75), the portion of lolizle noes n inrese ith little error inreses. ) Performnes of ifferent methos: We then sty the performnes of ifferent propose greey lgorithms. In this set of simltions, e set α =.5. In this se, the nhor seletion strtegies ill ply ritil role in minimizing totl nmer of nhor noes. In Fig. 7(), Greey-LolSeep n Greey-Triltertion n hiee the etter performnes. Moreoer, similr s experiments for MCLP-, initilly hen sensor netork is sprse, most of the shllo noes he to e nhors. Grlly, s sensor netork eome enser, only smll portion of shllo noes nee to e nhors. Hoeer loliztion errors ine y ifferent methos re similr. This onfirms tht the onfiene threshol plys more ritil role in ontrolling loliztion errors s shon in Fig. 7(). ) The impt of mesrement errors: At lst, e inestigte the inflenes of mesrement errors in or loliztion prolem. Here e fix the totl s n α =.5. Different methos re ompre for.9 ifferent leels of mesrement errors (gener te y rios les of.5 β) s shon in Fig.8..4 Generlly, lrger le. of β les to lrger.. mesrement errors. As expete, erge loliztion errors inrese β steily ith lrger mesrement errors. Hoels of errors (rios les of β ). Fig. 8. Performnes ner ifferent leeer the inrese rte hnges rmtilly hen β = 5. The tren ill inrese n eome ot of ontrol if the mesrement errors keep inresing fter ertin threshol le. loliztion error VI. CONCLUSION In this pper, e exten the minimm ost loliztion prolem (MCLP) to D OSNs in orer to fin the optiml nhor set to lolize ll sensor noes in the netork. To ersions of MCLP re introe: one tries to minimize the nmer of nhors, hile the other foses on optimizing the trelling istne of ship to isit ll selete nhors. Both prolems re ompttionlly hllenging. We propose three ifferent greey lgorithms to fin the nhor set for gien D OSN. We lso onsier ho to hnle mesrement errors n flip migity y opting onfiene-se pproh. Extensie simltions re onte n emonstrte the effiieny of these lgorithms. We lee the prolem to hnle the noe moility s or ftre ork. REFERENCES [] M. Erol-Kntri, H. Mofth, n S. Oktg, A srey of rhitetres n loliztion tehniqes for nerter osti sensor netorks, IEEE Commnitions Sreys Ttorils, ():487 5,. [] Y. Wng, Y. Li, n Z. Go, Three-imensionl oen sensor netorks: A srey, J. of Oen Uniersity of Chin, (4):46 45,. [] Y. Wng n L. 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