Efficient Processing of Continuous Reverse k Nearest Neighbor on Moving Objects in Road Networks

Transcription

1 Iteratial Jural f Ge-Ifrmati Article Efficiet Prcessig f Ctiuus Reverse k Nearest Neighbr Mvig Objects i Rad Netwrks Muhammad Attique, Hyug-Ju Ch, Rize Ji ad Tae-Su Chug, * Departmet f Cmputer Egieerig, Aju Uiversity, Suw 699, Krea; attique@aju.ac.kr (M.A.); rizeji@aju.ac.kr (R.J.) Departmet f Sftware, Kyugpk Natial Uiversity, Sagju-si 7, Krea; hyugju@ku.ac.kr * Crrespdece: tschug@aju.ac.kr; Tel.: Academic Editrs: Nrbert Bartelme ad Wlfgag Kaiz Received: 8 August 06; Accepted: Nvember 06; Published: 0 December 06 Abstract: A reverse k earest eighbr (RkNN) query retrieves all the data pits that have q as e f their k clsest pits. I recet years, csiderable research has bee cducted it mitrig reverse k earest eighbr queries. I this paper, we study the prblem f ctiuus reverse earest eighbr queries where bth the query bject q ad data bjects are mvig. Existig state-f-the-art techiques are sesitive twards the mvemet f data bjects, e.g., a cadidate bject must be verified wheever it chages its lcati. Further, isufficiet atteti has bee give t the mitrig f RNN queries i dyamic rad etwrks where the etwrk weight chages depedig the traffic cditis. I this paper, we address these prblems by prpsig a ew safe exit-based algrithm called CORE-X fr efficietly cmputig the safe exit pits f bth query ad data bjects. The safe exit pit f a bject idicates the pit at which its safe regi ad -safe regi meet, thus a set f safe exit pits represets the brder f the safe regi. Withi the safe regi, the query result remais uchaged prvided the query ad data bjects remai iside their respective safe regis. The results f extesive experimets cducted usig real rad maps idicate that the prpsed algrithm sigificatly reduces cmmuicati ad cmputati csts cmpared t the state-f-the-art algrithm. Keywrds: ctiuus mitrig; lcati-based applicatis; reverse earest eighbr query; safe exit algrithm; mbile cmputig; rad etwrk. Itrducti The rapid techlgical advaces i wireless etwrks ad develpmet f hadheld devices equipped with lcati sesig techlgy (e.g., smart phes ad tablets) have ppularized lcati-based services i the past decades. These systems eable real-wrld applicatis such as mixed reality games, army strategy plaig, bject surveillace i wireless sesr etwrks, ad ehaced emergecy services [,]. The ctiuus mvemet f data bjects demads ew query prcessig techiques permittig frequet lcati updates. Althugh csiderable effrt has bee devted t mvig-query prcessig [ 7], these studies have fcused rage queries ad earest eighbr (NN) queries; there ctiues t be a deficiecy f research addressig ctiuus reverse earest eighbr (RNN) queries. Further, the majrity f the research has bee cducted i the Euclidea space, t a rad etwrk. Csider a query bject q ad a set f data bjects O (e.g., places f accmmdati, restaurats, ad gas statis). We use dist(, q) ad dist(, ) t represet the shrtest distace frm bject t query q ad ather data bject, respectively. A reverse k earest eighbr (RkNN) query retrieves all f the data bjects O fr which q is e f their k clsest bjects. ISPRS It. J. Ge-If. 06,, 7; di:0.90/ijgi07

2 ISPRS It. J. Ge-If. 06,, 7 f 6 RkNN queries are geerally categrized it tw types: mchrmatic reverse knn (MRkNN) queries ad bichrmatic reverse knn (BRkNN) queries. I MRkNN, all mvig data ad query bjects are f the same type. Applicatis f the ctiuus MRkNN iclude mixed reality games where the gal f each player is t sht the earest player. Each player must ctiuusly mitr their w RNN t avid beig sht by ther players. Give a bject set O ad a query q, the MRkNN query is frmally defied as MRkNN(q) = { O q knn()}, where knn() is the knn set f. Ulike, MRkNN queries, i BRkNN, query bjects ad data bjects belg t tw differet types f bjects. Applicatis f the ctiuus BRkNN query iclude decisi supprt system. Fr example, csider a applicati where the gal is t determie the best lcati fr peig a Chiese restaurat. The mai decidig factr ca be Hw may users csider this ptetial lcati t be the earest Chiese fd restaurat? Each cadidate lcati fr the Chiese fd restaurat geerates a RNN query; the results ca the be cmpared t chse the best lcati. The abve example emplys BRkNN queries because restaurats ad users belg t differet types f bjects. Give tw differet bject sets O ad S ad a query q S, a BRkNN query is frmally defied as BRkNN(q) = { O q knn(, S)}, where knn(, S) detes the knn f amg all bjects i dataset S. May real-life applicatis exist t illustrate the usefuless f mvig RkNN queries, particularly i applicatis which require ctiuus mitrig f reverse earest mvig bjects. Fr example, fr ehaced 9 service, wheever a service ceter receives ay emergecy call, the team that is clsest t the emergecy call lcati is tified. Other examples culd be taxi cmpay dispatchig taxis t the passegers, r army backup uits iterested i mitrig their clsest uits which may require ay assistace. The mai challege fr ctiuus mitrig algrithms is maitaiig the freshess f the query results whe the query ad data bjects mve freely ad arbitrarily. A simple apprach is t icrease the frequecy f updates; query q peridically seds requests t reevaluate the query results. Hwever, this apprach des t guaratee that results are fresh because the query results may becme stale betwee each call t the server. Mrever, a excessive cmputatial burde may be impsed the server side as a high cmmuicati frequecy icreases the cmmuicati cst. The existig state-f-the-art algrithm kw as SAC [8] attempts t address the abvemetied prblem ad mitrs the RkNN queries i tw majr steps, i.e., filterig ad verificati. I the filterig phase, usig pruig rules, the part f the etwrk that cat ctai ay RkNN f q is prued. The bjects that are i the uprued etwrk are called the cadidate bjects cad. I the verificati phase, the server mitrs whether the bject cad cadidate bject list is the RkNN f q. Mre specifically, a cad is reprted as a RkNN if ad ly if q is the clsest bject f cad. Hwever, at each timestamp, the algrithm must verify the cadidate bject wheever it updates its lcati. The verificati is csiderably expesive because (a) verifyig a cad requires checkig whether q is e f the k clsest bjects f cad ad (b) verificati is required wheever a cad chages its lcati. T address the afremetied limitati, we preset a safe exit-based apprach fr ctiuus mitrig f reverse k earest queries i a rad etwrk where bth query bjects ad data bjects mve arbitrarily. The prpsed algrithm cmputes safe exit pits fr bth query ad data bjects. The query result remais uchaged prvidig query ad data bjects reside withi their respective safe regis. The safe exit techique reduces the tw-way cmmuicati betwee cliet ad server resultig i a reducti f the cmmuicati ad cmputati cst. We first csider the ctiuus mitrig f reverse k earest queries i static rad etwrks where the etwrk distace des t chage ver time. The, we exted the prpsed apprach t dyamic rad etwrks where the etwrk weight such as the travel time chages depedig traffic cditis icludig traffic cgesti r reversible laes. Our key ctributis are summarized as fllws: We preset a framewrk fr ctiuus mitrig f RkNN queries where bth query ad data bjects are mvig a rad etwrk. Our algrithm prcesses bth MRkNN ad BRkNN queries fr arbitrary values f k.

3 ISPRS It. J. Ge-If. 06,, 7 f 6 We preset vel pruig rules that ptimize the cmputati f safe exit pits by miimizig the size f the uprued etwrk ad umber f bjects. We demstrate the extesi f the prpsed algrithm t a directed rad etwrk where each rad segmet has a particular rietati ad t a dyamic rad etwrk where the etwrk weight chages depedig the traffic cditis. We cfirm thrugh a extesive experimetal evaluati that the prpsed algrithm utperfrms the state-f-the-art algrithm i terms f bth cmmuicati ad cmputati csts fr the majrity f the settigs. The remaider f this paper is structured as fllws. Secti reviews the existig wrk the ctiuus mitrig f RkNN queries i Euclidea space ad rad etwrks. Secti prvides termilgy defiitis ad describes the prblem. Secti elabrates the prpsed safe exit algrithm (CORE-X) fr cmputig safe exit pits f mvig RkNN queries i rad etwrks. Secti presets a perfrmace aalysis f the prpsed techique. Secti 6 ccludes this paper.. Related Wrk A RNN query fr mvig bjects searches fr thse bjects that select bject q as their earest eighbr. The prcessig f RNN queries has becme a recet emergig area f research. May algrithms have bee prpsed fr the mitrig f reverse earest eighbrs, especially i Euclidea space. Hwever, there remais a lack f efficiet algrithms fr rad etwrks. Our related wrk is divided it tw sectis: Secti. addresses ctiuus RNN query prcessig i Euclidia space ad Secti. reviews ctiuus RNN query prcessig i rad etwrks... Algrithms fr Ctiuus Reverse Nearest Neighbr Query Prcessig i Euclidia Space Kr et al. [9] were the first t itrduce the ccept f RNN queries. They used a precmputig techique t search fr RNNs. The mai drawback f the preprcessig apprach is that it is limited t supprtig RkNN queries fr a fixed umber f k; mrever, it is iefficiet whe prcessig bject mvemets. T address the shrtcmigs f the preprcessig techique, a ew categry f algrithms has bee itrduced, kw as sapsht RNN algrithms. These sapsht algrithms have tw mai phases. The first phase is a filterig phase that fcuses the pruig f uecessary bjects; the secd phase is refiig, which verifies whether the remaiig bjects are valid query results. The tw mai filterig methds that have bee develped fr RkNN queries are 60-degree-pruig [0] ad TPL-pruig [] by Stai et al. ad Ta et al., respectively. Ather categry is ctiuus RNN query algrithms, which ca prduce icremetal RNN results. A umber f algrithms have als bee prpsed fr efficiet mitrig f NN queries, ctiuus rage queries, ad RNNs [,8, 7]. The existig ctiuus query prcessig methds fcus defiig the mitrig regi ad updatig the query results based the mvig bject s lcati. Beetis et al. [8] were the first t study ctiuus RNN mitrig; hwever, their prpsed scheme assumes that the velcities f the bjects are kw. Xia et al. [9] itrduced a icremetal ad scalable algrithm fr mchrmatic RNN queries based the 60-degree-pruig techique. I their apprach, the mitrig regi f a ctiuus RNN query is defied by six pie regis (determied by the query pit ad the six cadidates) ad six arc regis (determied by the six cadidates ad their earest eighbrs). The efficiecy f this algrithm is superir t that f cvetial methds because it idetifies ad prcesses the updates that fall it the mitrig regi. Kag et al. [0] prpsed a vel algrithm fr mitrig ctiuus RNNs called IGERN. It is mre efficiet tha 60-degree-pruig-based slutis because it mitrs a small umber f cadidates rather tha the etire space. Further, this methd is applicable t bth MRkNN ad BRkNN queries. The tradeff f this scheme is that it cat be easily exteded t ctiuus RkNN queries fr k >. Therefre, the defied mitrig regi applies ly t ctiuus RNNs fr k =.

4 ISPRS It. J. Ge-If. 06,, 7 f 6 Wu et al. [] prpsed a techique t mitr RkNNs that ivlves ctiuus filterig ad ctiuus refiig. They prpsed a ew refiig framewrk called CRage-k that verifies the cadidate bjects by issuig knn queries i each regi rather tha sigle earest eighbr queries. Users that are clser tha the kth NN i each regi are the cadidate bjects; these cadidate bjects are verified if q is e f the k clsest facilities. T mitr the results, fr each cadidate bject, the methd ctiuusly mitrs the circular regi arud it that ctais the k earest facilities. Cheema et al. [8,] prpsed several schemes fr the mitrig f ctiuus RNNs. I [], they develped a algrithm fr ctiuus BRkNN queries. They used the ccept f the ifluece ze where the query bject is static ad the data bjects are mvig. I [8], they prpsed a ew framewrk based safe regis fr bth Euclidea ad rad etwrks where bth query ad data bjects are mvig. This scheme sigificatly imprves the cmputati cst as it assigs each bject ad query a safe regi such that expesive recmputati is t required prvided the query ad bjects remai i their respective safe regis... Algrithms fr Ctiuus Reverse Nearest Neighbr Query Prcessig i Rad Netwrks I recet years, reverse eighbr query prcessig i rad etwrks has received atteti by the spatial database systems research cmmuity. Yiu et al. [7] first addressed the issue f RNN i rad etwrks (they represeted rad etwrks as graphs) ad prpsed a algrithm fr bth MRkNN ad BRkNN queries. Safar et al. [] preseted a framewrk fr RNN queries based etwrk Vri diagrams (NVDs) t efficietly prcess RNN queries i rad etwrks. Hwever, their scheme is t suitable fr ctiuus RNN queries because NVDs chage wheever a dataset chages its lcati, resultig i high cmputati csts. Su et al. [6] studied the ctiuus mitrig f bichrmatic RNN queries. They assciated a multiway tree with each query t defie the mitrig regi ad ly updates i the mitrig regi affect the results. Hwever, this methd is limited t bichrmatic queries ad des t apply whe k >. Mrever, their prpsed scheme assumes that the query bjects are static. Li et al. [] prpsed a vel algrithm fr ctiuus mitrig f RkNNs based a dual layer multiway tree (DLM tree) where they itrduced several lemmas t reduce the mitrig regi ad filter the cadidate bjects. Their ctiuus mitrig f RkNN methd cmprises tw phases: the iitial result geerati phase ad the icremetal maiteace phase. Cheema et al. [8] prpsed a safe regi apprach fr the mitrig f ctiuus MRkNN ad BRkNN queries i Euclidea ad rad etwrks as metied i Secti. ad devised pruig rules that reduce the mitrig regi. Gth et al. [] preseted a simple rutig methd t prcess BRkNN queries. Wag et al. [6] prpsed a ifluece ze-based algrithm fr mitrig ctiuus RkNN i a rad etwrk. Attique et al. [7] prpsed a safe exit-based scheme fr ctiuus mitrig f RNN queries. Hwever, this apprach is applied t BRkNN queries; mrever, it des t address the mvemet f data bjects. I this paper, we develp a scheme that is applicable t bth MRkNN ad BRkNN queries; it ca als address the mvemet f bth query bjects ad data bjects. This paper is a exteded versi f ur previus wrk the mitrig f RkNN queries fr mvig data bjects ad queries [8]. Hwever, we preset five relevat extesis that were t ivestigated i ur prelimiary wrk [8]. The first extesi mitrs the prued bjects (Secti..). The secd extesi studies adaptatis f the prpsed algrithm t address a directed rad etwrk (Secti..). I the third extesi, we exted CORE-X t a dyamic rad etwrk where the etwrk distace chages depedig the traffic cditis (Secti..). The furth extesi studies bichrmatic RkNN queries (Secti..). I the fifth secti, we cduct a extesive experimetal study fr bth static ad dyamic rad etwrks (Secti ). I additi t these majr extesis, we als preset a time cmplexity aalysis f the prpsed algrithm that was t preseted i ur previus wrk (Secti.).

5 ISPRS It. J. Ge-If. 06,, 7 f 6. Prelimiaries Secti. defies the terms ad tati used i this paper. Secti. prvides the prblem descripti based a example... Defiiti f Terms ad Ntatis Rad Netwrk: A rad etwrk is represeted by a weighted udirected graph G = (N, E, W), where N icludes the set f des N = {,,, N }, E is a set f edges that cects tw distict des E = {e, e,, e E }, ad W(e) detes the weight f a edge e. A edge betwee tw des is deted by e( s, e ), where s ad e are referred t as budary des. A budary de with a smaller de id is referred t as the base de f the sequece. This study assumes that s e. ISPRS It. J. Ge-If. 06,, 7 f 6 Therefre, s is the base de. Each edge cectig tw distict des has a psitive weight that represets.. thedefiiti cst f f travelig Terms ad the Ntatis legth f the edge, e.g., the time required t travel alg the edge. Segmet: Segmet s (p,p ) is a part f a edge betwee tw pits, p ad p, the edge. Rad Netwrk: A rad etwrk is represeted by a weighted udirected graph =(,, ), A edge csists where N ficludes e r mre the set segmets. f des A ={ edge,,, is als }, csidered is a set f tedges be a segmet that cects where tw the des are the eddistict pits des f the ={ edge., The,, weight }, ad W(e) f adetes segmet the weight s (p,p ) f isa deted edge e. A byedge W(s). betwee tw Figuredes presets is deted a by example (, ), fwhere a udirected ad are rad referred etwrk t as with budary seve des. des, A budary t 7. Several de with a smaller de id is referred t as the base de f the sequece. This study assumes that edges ad segmets are illustrated with their respective weights. Fr example, the edge e(, ). Therefre, csists f segmets s (, ) ad is the s base de. Each edge cectig tw distict des has a psitive weight that represets the cst (,f ), travelig which have the legth weights f the edge, ad e.g.,, the time respectively. required t There travel are seve data bjects alg i the thisedge. example {,,..., 7 } O ad a sigle query bject q. Query q ad the data bjects are idicated Segmet: bysegmet triagles ( ad, ) is rectagles, a part f a respectively. edge betwee Give tw pits, tw pits, ad p, ad the pedge., the shrtest A edge csists f e r mre segmets. A edge is als csidered t be a segmet where the path distace dist(p, p ) is the miimum distace betwee p ad p. I Figure, the shrtest path des are the ed pits f the edge. The weight f a segmet (, frm q t is q 6 ; dist(q, ) =. Table presets ) is deted by W(s). the tatis used i this paper. 7 q Figure Figure. Example. f a rad etwrk. Figure presets a example f a udirected rad etwrk with seve des, t 7. Several edges ad segmets are Table illustrated. Summary with their f tatis respective used weights. i this Fr paper. example, the edge (, ) csists f segmets (, ) ad (, ), which have weights ad, respectively. There are Ntati seve data bjects i this example {,,, } Defiiti ad a sigle query bject q. Query q ad the G data = (N, E,W) bjects Graph are idicated mdel f rad by triagles etwrk ad rectagles, respectively. Give tw pits, p ad p, the dist(p shrtest s, p e ) Legth f the shrtest path frm p path distace (, ) is the s t p miimum e, where p distace s ad p e represet start ad ed pits, respectively betwee p p. I Figure, the i Nde i rad etwrk e(shrtest path frm t is ; (, ) s, e ) Edge i edge set E, where s ad e are start ad ed pits =. f the Table edge presets the tatis W(e) used i this Weight paper. f edge e( s, e ) q Query pit i rad etwrk k A umber that represets Table. qsummary ca be amg f tatis k umber used f clsest i this facilities paper. t a data bject O Set f bjects O = {,,..., } O + Ntati Set f aswer bjects O + = { +, +,..., + k } Defiiti O G = (N, E,W) Set f Graph -aswer mdel f bject rad Oetwrk = { k+, k+,..., O } IR + Ifluece Legth regi f the f shrtest aswerpath bjects frm (, ) t, where ad represet start ad ed pits, IR Ifluece respectively regi f -aswer bjects + Farthest Nde aswer i rad bject etwrk t pit p G, such that d(p, + f f ) = MIN(d(p, + ), d(p, + ),..., d(p, + + ) (, ) Nearest Edge -aswer i edge set bject E, where t pit s ad p G, e are such start that ad d(p, ed pits ) = MIN(d(p, f the edge ), d(p, ),..., d(p, ) p a W(e) Achr Weight pit f that edge crrespds (, ) t start pit f expasi UOq Set f Query useful pit bjects i rad required etwrk t cmpute safe regi f q Ω Safe exit pit where the safe ad -safe regi f q r itersects k A umber that represets q ca be amg k umber f clsest facilities t a data bject β Budary de crrespdig t start ( s ) r ed ( e ) pit f edge O Set f bjects ={,,, } A p Set f knns at pit p R Set f aswer bjects ={,,, } p Set f RkNNs at pit p Set f -aswer bject ={,,, } Ifluece regi f aswer bjects Ifluece regi f -aswer bjects Farthest aswer bject t pit, such that (, ) = ( (, ), (, ),, (, )

6 ISPRS It. J. Ge-If. 06,, 7 6 f 6.. Prblem Descripti I this paper, we primarily address the prblem f ctiuus mitrig f RkNN queries mvig queries ad data bjects i rad etwrks. T prvide a clear explaati, we use the rad etwrk example illustrated i Figure, where there are seve bjects, t 7, ad a query q i a rad etwrk. Fr explaati, we csider MRkNN queries where bth data bjects ad query bjects belg t the same data type. Hwever, ur methd ca als be exteded t mitr ctiuus BRkNNs queries. I the remaider f the paper, we use RkNN query t refer t a MRkNN query uless metied therwise. Let us assume that a mvig query requests e RNN (k = ) at a certai pit p. T btai e RNN, we traverse the rad etwrk frm the active edge that ctais pit q. Fr each data bject O. ecutered, we issue a verificati query verify(, k, q) that checks if it is a RNN. If there exists ather bject such that dist(, ) < dist(, q), the is t a RNN. Otherwise, is iserted it the RNN result set deted as R p. The expasi i each path termiates ce k bjects have bee fud i that directi. Fr btaiig a RNN at pit p, the simple methd is t repeat the prcedure executed at p. Hwever, the use f recmputati wheever query q r ay data bject chages its lcati sigificatly degrades the perfrmace f the algrithm. T address this issue, we itrduce the safe exit apprach. I the prpsed framewrk, the server cmputes the safe exit pits fr each mvig bject ad query bject. Because we are addressig a cmplex prblem where bth query bjects ad data bjects are mvig, it is ecessary t cmpute the safe exit pits fr each mvig data bject. The server maitais a set f mvig queries ad a set f mvig bjects. The query results will be the same util all the bjects lie iside their respective safe exit pits. Wheever a query r data bject leaves its safe exit pits, the server recmputes the RkNN ad safe exit pits fr the query ad data bjects. Hwever, the cmputati f safe exit pits fr all f the data bjects degrades the perfrmace f the algrithm. T address this issue, we devised pruig rules that allw us t cmpute the safe exit pits fr ly the uprued bjects. As metied previusly, we are addressig the prblem f ctiuusly mitrig RNN queries where bth query ad data bjects are mvig freely i a rad etwrk. Therefre, the fllwig evets ca chage the query result. Evet : The query bject q mves utside f its respective safe exit pits. Evet : A bject UO mves utside f its respective safe exit pits. Evet : The mvemet f prued bjects chages the RkNN set. Figure illustrates the request flw fr prcessig RkNN queries. Iitially, the query bject q (cliet) issues a RkNN query t server alg with the curret lcati (Step ). Up receivig a request, the server cmputes RkNN result set R k ad safe exit pits fr query ad uprued data bjects (Step ad ). The query results remai valid, util ay f the abvemetied three evets ccurs. T mitr Evet, the query bject checks its lcati agaist the received safe exit pits (Step ). If query bject travels beyd ay safe exit, cliet re-issues a RkNN query t the server fr a updated result ad its safe exit pits. A server mitrs the mvemet f data bjects t verify Evets ad (Step 6). If either Evet r ccurs, the server recmputes the query results ad tifies the cliet.

7 (k = ) at a certai pit p. T btai e RNN, we traverse the rad etwrk frm the active edge that ctais pit q. Fr each data bject ecutered, we issue a verificati query verify(, k, q) that checks if it is a RNN. If there exists ather bject such that (, ) < (, ), the is t a RNN. Otherwise, is iserted it the RNN result set deted as Rp. The expasi i each path termiates ce k bjects have bee fud i that directi. Fr btaiig a RNN at pit p, the simple methd is t repeat the prcedure executed at p. Hwever, the use f recmputati ISPRS It. wheever J. Ge-If. query 06, q, r 7 ay data bject chages its lcati sigificatly degrades the perfrmace f 7 f 6 the algrithm. T address this issue, we itrduce the safe exit apprach. Cliet. Checks if its lcati is iside safe exit pits. Issues RkNN query (q, k). Returs Rk ad Ω f q Server. Cmpute the result set Rk. Cmpute the safe exit pits (Ω) f q ad 6. Mitrs data dd bjects. bif Evet r Evet Figure. Request flw flw fr prcessig fr prcessig RkNN queries. RkNN queries. I the prpsed framewrk, the server cmputes the safe exit pits fr each mvig bject ad. Safe Exit Algrithm fr Mvig RkNN Queries ad Mvig Objects query bject. Because we are addressig a cmplex prblem where bth query bjects ad data Ibjects this secti, are mvig, weit develp is ecessary techiques t cmpute t the mitr safe exit pits mvig fr each RkNN mvig queries data bject. ad mvig The bjects i server maitais a set f mvig queries ad a set f mvig bjects. The query results will be the a rad etwrk. I Secti., we preset a algrithm fr cmputig the safe regi fr mvig same util all the bjects lie iside their respective safe exit pits. Wheever a query r data bject bject q. leaves Theits mitrig safe exit pits, f data the server bjects recmputes is described the RkNN i Secti ad safe exit.. pits We preset fr the query the extesis ad f the prpsed data apprach bjects. Hwever, t ther variats cmputati f RkNN f safe exit queries pits ifr Secti all f the.. data Fially, bjects degrades Secti the. aalyzes the time ad perfrmace space cmplexities f the algrithm. f CORE-X. T address this issue, we devised pruig rules that allw us t cmpute the safe exit pits fr ly the uprued bjects... Safe Regi fr Mvig q I this secti, we preset a ew safe exit algrithm that addresses the issue fr mvig RkNN queries ad mvig bjects i a rad etwrk. Algrithm depicts the skelet f the prpsed safe exit algrithm fr cmputig safe regis. It csists f three phases: () determiig the useful bjects that ca ctribute t the safe regi; () cmputig the ifluece regi f the useful bjects; () cmputig the safe exit pits fr the query bjects. Algrithm : Cmputati f Safe Regis (skelet) Iput: O: data bjects, q: query bject, k: iteger umber Output: SR: Safe Regi : Object set O radetwrk(o, q, k) : Object set O + { + O dist(, q) < dist(, k+ )} : Object set O { O dist(, q) > dist(, k )} : While O + r O is -empty d : Object = pickbject(o +, O ) 6: If O + the 7: IR + cmputeir(o +, O, k) 8: SR SR IR + 9: Else 0: IR cmputeir(o +, O, k) : SR SR IR : Ed while : Retur SR Algrithm presets the skelet f the prpsed idea. The algrithm begis by determiig the aswer ad -aswer bjects. The details f the methdlgy are explaied i Secti... The, i Phase, the algrithm cmputes the ifluece regi f the aswer bjects ad -aswer bjects (Secti..). Fially, it cmputes the safe regi by perfrmig itersecti ad set differece peratis the rad segmets (Secti..).

8 ISPRS It. J. Ge-If. 06,, 7 8 f 6... Determiig Useful Objects This phase aims at determiig ptetial bjects that culd ctribute t the cmputati f the safe regi. The gal is t retrieve a small set f data bjects t reduce the cmputati verhead. I ur study, the data bjects are divided it tw categries: aswer bjects (deted by O + ) ad -aswer bjects (deted by O ). Defiiti. A bject is called a aswer bject if dist(, q) dist(, ) where is ay ther bject i the rad etwrk. Similarly, we ca geeralize this defiiti fr RkNN: a bject is called a aswer bject if dist(, q) dist(, k+ ), where k+ is the (k+)th NN bject f. That is, we ca say that all aswer bjects are RkNNs f query q, therefre, + R k. Defiiti. A bject is called a -aswer bject if dist(, q) > dist(, ), where is ay ther bject i the rad etwrk. Similarly, we ca geeralize this defiiti fr RkNN: a bject is called a -aswer bject if dist(, q) > dist(, k ), where k is the kth NN bject f. That is, we ca say that all -aswer bjects are t RkNN f query q, therefre, / R k. A simple methd fr retrievig the R k set is t traverse the etwrk frm q, ad fr each data bject O ecutered, issue a earest-eighbr query. If q NN(), q is the clsest bject t. Csequetly, + R k. Hwever, this apprach must evaluate all the data bjects because the size f R k is t fixed ad the rad etwrk may ctai pits that are far frm q. T avid uecessary rad etwrk explrati, we preset the pruig lemma. Befre presetig the lemma, it is ecessary t defie clsed des. A de is called a clsed de if there exists a bject such that dist(, ) < dist(, q). The bject is called a blckig bject because it causes de t be a clsed de. I Figure, de is a clsed de because dist(, ) < dist(, q), which makes a blckig bject. Lemma. A bject cat be the RNN f q if the shrtest path betwee q ad ctais a clsed de with a blckig bject, where =. Prf. Let us assume that there exists a clsed de the shrtest path betwee ad q. The shrtest distace betwee ad q is dist(, q) = dist(, ) + dist(, q). Let be the blckig bject ad dist(, ) = dist(, ) + dist(, ). As we kw dist(, ) < dist(, q), therefre, dist(, ) < dist(, q). Therefre, cat be a RNN f q. I Figure, the data bject cat be a RNN f q because the shrtest distace betwee ad q passes thrugh. Because dist(, ) = ad dist(, q) =, data bject is clser t tha q. The abve lemma ca be easily exteded t RkNN. A bject cat be the RkNN f q if the shrtest path betwee q ad ctais a de ad there exist k data bjects such that fr every bject, dist(, ) < dist(, q), where =. Algrithm presets the pseud cde fr determiig the aswer bjects. CORE-X traverses the etwrk arud q i a similar fashi t Dijkstra s algrithm; usig Lemma, it elimiates the des that may t lead t RNNs. The algrithm begis by explrig the active edge where query bject q is lcated ad expads the etwrk i a icreasig rder f distace frm the query bject q. Each etry i the queue has the frm p a, edge, where p a idicates the achr pit the edge. Fr a active edge, q becmes the achr pit. Otherwise, either f the budary des f the edge, i.e., s r e, becmes the achr pit. If the desired umber f aswer bjects is t fud a active edge, the edges adjacet t the budary des are equeued. The edges are ppped i a icreasig rder f distace frm q. The traversal f the edges is termiated whe the queue is exhausted. Lie iitializes a queue by isertig the active edge. If a edge ctais a data bject

9 ISPRS It. J. Ge-If. 06,, 7 9 f 6, we must verify whether RkNN(q). Thus, the algrithm issues a verify(, k, q) query (Lie 0). The verificati query checks if q is amg the knns f data bject by applyig a rage-nn query arud bject with the rage set t dist(, q). If q = knn(), is the RkNN f q. Therefre, is added t the result set R k (Lie ). If the edge des t ctai ay data bject, the algrithm ctiues its expasi ad equeues the adjacet edges f the budary de. Algrithm : Aswer bject(q, k) Iput: q: query lcati, k: iteger umber Output: R k : query result (aswer bjects) : queue /* queue is a pririty queue with edges rdered by distace t q*/ : A k /* set f aswer bjects*/ : visited /*stres ifrmati f visited edges */ : queue.push(q,edge active ) /* edge active idicates active edge */ : While queue is t empty d 6: p a, edge queue.pp() 7: If p a, edge / visited the 8: visited visited {edge} 9: If edge ctais a data bject 0: knn(): verify(, k, q) : If q discvered by verificati : R k R k : Else : queue.push β, edge : Ed while 6: Retur R k Let us csider the example preseted i Figure, where query q requested e RNN (k = ). We will csider this example thrughut this secti. The algrithm starts expasi frm the active edge, which is e(, ). Because data bject is discvered that edge, edges adjacet t the budary des will be added t the queue. The edges adjacet t the clsest budary de are equeued first. Therefre, edges e{(, ), (, 6 )} are equeued. Data bject 6 is first discvered (, ), which triggers the verificati query verify( 6,, q). Because q = NN( 6 ), 6 is added t the result set R k. Recall that by Lemma, de is a clsed de with bject 6 as the blckig bject. Therefre, all the ther bjects fr which the shrtest distace passes thrugh except 6 are autmatically prued. The, data bject is discvered e(, 6 ), by verificati f, q = NN( ); therefre, the search als termiates i this directi. Next, the edges adjacet t are added t the queue, e{(, ), (, ), (, 7 )} ad data bject is retrieved e(, ). The data bject is verified ad added t the result set because q = NN( ). The search termiates as further expasi is required i this directi because de is a clsed de. Next, we determie the -aswer bjects that ca ctribute t the safe regi. Useful -aswer bjects UO O are bjects fr which ay + = NN( ). I ther wrds, UO are RNNs f aswer bjects. RNNs f Useful Objects (UO) ca be determied by the same algrithm makig these mdificatis: () the achr pit is + istead f q ad () verificati query verify(, k, q) is mdified t verify(, k, + ). The algrithm reuses the results f the rage-nn queries issued at the data bjects t avid multiple verificati f the same data bjects. Whe the cmputati is cmpleted, the cached query results are remved t reduce the memry csumptis. Fially, we ca cclude that useful bjects are: () all aswer bjects ad () RNNs f aswer bjects. Nw, let us determie the UO bjects fr k = i the give example i Figure. Fr explaati, Table displays the knns f each data bject. Frm Table it is clear that ad 6 are bth aswer bjects because q is e f the NNs. Nw, frm the -aswer bjects, we ca see that is the RNN f because = NN( ). Similarly, 7 is

10 ISPRS It. J. Ge-If. 06,, 7 0 f 6 the RNN f 6. Hece, the set f useful bjects is UO = {,, 6, 7 }. Data bjects,, ad have bee prued. Table. Summary f NNs fr O.. Data Object NNs Distace (q, ) (, 6) (, ) (6, 6) (, ) (, ) (, ) (, ) (, ) (, ) 6 (q, 7 ) (, ) 7 ( 6, ) (, 7)... Cmputig Ifluece Regi fr Useful Objects After we retrieve the set f useful bjects, the ext step is t cmpute the ifluece regi f the aswer ad -aswer bjects. Ifluece Regi f Aswer Objects The ifluece regi f the aswer bjects is defied as: IR + = { p dist ( +, p ) dist ( +, k+ )} () Here, k+ detes the (k+)th earest eighbr f. By defiiti, the ifluece regi f the aswer bjects ctais all the pits fr which q = NN( + ). That is, it ctais all f the pits where bject remais +. The ifluece regi f aswer bjects ca be cmputed by explrig the etwrk arud the aswer bject i a maer similar t that explaied i Secti... The explrati termiates with the discvery f the (k+)th earest eighbr f the aswer bject. The ifluece regi will be idicated by rage(, d), where d is the dist(, k+ ). Figure idicates the ifluece regi f bject fr the example sceari discussed abve. The expasi f the rad etwrk starts frm util it fids k+, which is NN i this example. Object 7 is NN f ad dist(, 7 ) = 8. The algrithm will issue a rage(, 8) query that marks all f the pits as the ifluece regi f as shw i Figure. Similarly, the ifluece regi f 6 is idicated i Figure. The data bject is the NN f 6 ad dist( 6, ) = 7. Therefre, the rage( 6, 7) query marks all the pits withi the distace f 7 frm 6. The bld lies i Figures ad idicate the ifluece regi f ad 6, respectively. Ifluece Regi f N-Aswer Objects The ifluece regi f -aswer bjects is defied as: IR = { p dist (, p ) dist (, k )} () The ifluece regi f -aswer bjects ca be cmputed frm the distace betwee the -aswer bject ad kth bject. Here k detes the kth earest eighbr f. I ther wrds, the ifluece regi f -aswer bjects ctais all the pits where bject remais. The ifluece regi f -aswer bjects is cmputed i the same maer as the ifluece regi f aswer bjects, the ly differece is that the algrithm explres ad cmputes the distace t the knn bject istead f (k + )NN. Csider bject 7 i Figure : NNs f 7 = ( 6, ) with weights (, 7). I this example, is the secd NN f 7 ad dist( 7, ) = 7. Therefre, the ifluece regi f bject 7 is 7 uits frm 7 i every cected directi. Similarly, we ca cmpute the ifluece regi f. The bld lies i

11 aswer ad -aswer bjects. aswer ad -aswer bjects. Ifluece Regi f Aswer Objects Ifluece Regi f Aswer Objects The ifluece regi f the aswer bjects is defied as: The ifluece regi f the aswer bjects is defied as: = { (, ) (, )} (),, () Here, k+ detes the (k+)th earest eighbr f. By defiiti, the ifluece regi f the aswer ISPRS It. J. Ge-If. Here, 06, k+ detes, 7 the (k+)th earest eighbr f. By f 6 bjects ctais all the pits fr which = ( defiiti, the ifluece regi f the aswer ). That is, it ctais all f the pits where bjects ctais all bject remais the pits fr which.. That is, it ctais all f the pits where bject remais The ifluece. regi f aswer bjects ca be cmputed by explrig the etwrk arud the Figures ad 6 The idicate ifluece theregi ifluece f aswer regi bjects f ca be ad cmputed 7, respectively. by explrig the The etwrk NNs arud fr aswer the aswer bject i a maer similar t that explaied i Secti... The explrati termiates with bjects aswer bject i a maer similar t that explaied Secti... The explrati termiates with chage whethe the discvery query f bject the (k+)th mves earest utside eighbr the ifluece f the aswer regi, bject. whereas The ifluece the regi casewill f be -aswer the discvery f the (k+)th earest eighbr f the aswer bject. The ifluece regi will be bjects, the result idicated chages by rage(, whe d), where thed query is the (, bject mves ). idicated by rage(, d), where d is the, iside a ifluece regi. That is, the NNs Figure idicates the ifluece regi f. bject fr the example sceari discussed abve. f the aswer bject Figure remai idicates the the same ifluece util regi thef query bject bject fr the lies example iside sceari the ifluece discussed abve. regi ad fr The expasi f the rad etwrk starts frm util it fids k+, which is NN i this example. -aswerthe Object bjects, expasi 7 is NN thef NNs the rad f ad remai etwrk (, thestarts ) =8. same frm The util util algrithm query it fids will issue bject k+, which a ( lies utside is NN i,8) query the this example. that ifluece marks regi. Object 7 is NN f ad Nte that ce all f athe ifluece pits as the regi ifluece is cmputed,, regi 8. The algrithm will issue a f as it shw remais Figure valid. Similarly, as lg the as,8 query that marks qifluece ad UO regi lie withi f their all f the pits as the ifluece regi f as shw i Figure. Similarly, the ifluece regi f respective safe is idicated i Figure. The data bject is the NN f 6 ad ( regis. Hece, recmputati f the ifluece regi is ly, ) =7. Therefre, the required whe either q r ( is idicated i Figure. The data bject is the NN f 6 ad,7) query marks all the pits withi the distace f 7 frm. The, 7. Therefre, the bld lies i Figures UO leave their ad respective idicate,7 query the safe marks ifluece regis. all the pits withi the distace f 7 frm regi f ad, respectively.. The bld lies i Figures ad idicate the ifluece regi f ad, respectively. ISPRS It. J. Ge-If. 06,, 7 7 f 6 7 ISPRS It. J. Ge-If. 06,, 7 f 6 Ifluece Regi f N-Aswer Objects q s s The Ifluece ifluece Regi regi 6 f f N-Aswer -aswer bjects Objects q is defied s s as: 6 The ifluece regi f -aswer = { ( bjects, ) ( is defied as:, )} () s The ifluece regi f -aswer = { ( bjects ca, ) ( be cmputed, )} frm the distace betwee the () s -aswer The ifluece bject ad regi kth bject. f -aswer Here k detes bjects the ca kth 6 be earest cmputed eighbr frm f the. distace I ther wrds, betwee the the ifluece -aswer regi bject f ad -aswer kth bject. bjects Here k ctais detes all the 6 the kth pits earest where eighbr bject f. I remais ther wrds,. The the ifluece regi regi f -aswer f -aswer bjects Figure Figure bjects is. Ifluece. cmputed ctais regi i all the the f same f pits maer.. where as bject the ifluece remais regi. f The aswer Figure. Ifluece regi f ifluece bjects, regi the f ly -aswer differece bjects is that is the cmputed algrithm i explres the same. ad maer cmputes as the the ifluece distace regi t the f knn aswer bject bjects, istead the f ly (k + )NN. differece is that the algrithm 7 explres ad cmputes the distace t the 7 knn Csider bject istead bject 7 f i (k Figure + )NN. : NNs f =(, ) with weights (, 7). I this example, is the secd Csider NN f bject ad 7 i ( Figure, : ) NNs =7. Therefre, f 7 =( the, ) ifluece with weights regi (, f 7). bject I this 7 example, is 7 uits is 7 frm the 7 secd i every NN cected f ad directi. (, Similarly, ) =7. Therefre, we ca cmpute the ifluece the ifluece regi f regi bject f 7. is The 7 bld uits lies frm i 7 Figures i every ad cected 6 idicate directi. the ifluece Similarly, regi we ca f cmpute ad 7, respectively. the ifluece The regi NNs f fr. aswer The bld q bjects lies i chage Figures whe ad the 6 query idicate bject the ifluece mves utside 6 q regi the f ifluece ad 7, respectively. regi, whereas The NNs i the fr case aswer f 6 -aswer bjects chage bjects, whe the the result query chages bject whe mves the query utside bject the ifluece mves iside regi, a ifluece whereas regi. i the case That f is, -aswer the NNs f bjects, the aswer the result bject chages remai whe the same the query util bject the query mves bject iside lies a iside ifluece the regi. ifluece That regi is, the ad NNs fr f -aswer the bjects, remai the NNs the remai same util the same the s query util query bject bject lies iside lies utside the ifluece the ifluece regi ad regi. fr Nte -aswer that ce bjects, a ifluece the NNs regi remai is cmputed, the same util it remais query valid bject as lies lg utside as q ad the UO 6 ifluece lie withi regi. their Nte respective that ce safe a regis. ifluece Hece, regi recmputati 6 is cmputed, f it remais the ifluece valid as regi lg is as ly q ad required UO lie whe withi either their q respective r UO leave safe Figure their regis. respective. Ifluece Hece, safe regi recmputati regis. f 6. f the ifluece regi is ly required whe either q r UO Figure. Ifluece f 6. Figure leave their. Ifluece respective regi safe regis. f s 7 6 s q 6 s q 6 6 s Figure. Ifluece regi f. Figure. Ifluece f. Figure. Ifluece regi f. Figure 6. Ifluece regi f 7. Algrithm cmputes the ifluece regi f aswer bjects ad -aswer bjects. Algrithm cmputes the ifluece regi f aswer bjects ad -aswer bjects s 7 s 7 s8 q 6 s8 q Figure Figure 6. Ifluece 6. regi f f

12 Table summarizes the cmputati f the safe exit pits fr the query bject q i Figure. I the previus secti, we cmputed the ifluece regi IR f all useful bjects, which was described by set f segmets. T btai the safe regi, we must perfrm itersecti ad set differece peratis the rad segmets. We btai segmets (, ), {(, ), (, )} frm the itersecti f IR() ad IR(6). As ad 7 are -aswer bjects, the safe regi must exclude the IR() ad IR(7). Thus, the safe regi wuld be the segmets {(, ), (, )}. The set f budary pits s ad s becmes the set f safe exit pits. Figure 7 presets the safe exit pits f q. ISPRS It. J. Ge-If. 06,, 7 f 6 Algrithm cmputes the ifluece regi f aswer bjects ad -aswer bjects. Algrithm : Cmputati f Ifluece Regi (O +, O, k) Iput: O + : aswer bjects set, O : -aswer bjects set, k: iteger umber Output: IR + ifluece regi f aswer bjects, IR ifluece regi f -aswer bjects : IR +, IR /* Ifluece Regi f aswer bjects */ : Fr each O + d : Expad the rad etwrk util k+ ; : d dist(, k+ ) : Mark rage(, d); 6: IR + IR + IR + ; 7: Ed fr /* Ifluece Regi f -aswer bjects */ 8: Fr each O d 9: Expad the rad etwrk util k ; 0: d dist(, k ) ; : Mark rage(, d); : IR ISPRS IR It. IR J. Ge-If. ; 06,, 7 f 6 : Ed fr : Retur IR + Algrithm ad IR : Cmputati f Ifluece Regi (,, k) Iput: : aswer bjects set, : -aswer bjects set, k: iteger umber Output: ifluece regi f aswer bjects, ifluece regi f -aswer bjects... Cmputati : f Safe, Exit Pits /* Ifluece Regi f aswer bjects */ The safe regi : SR Fr each f a query d q is defied as fllws: : Expad the rad etwrk util k+; : (, ) : Mark rage(, d); SR = { IR + IR } () 6: ; where IR + 7: Ed fr detes the ifluece regi f aswer bjects ad IR /* Ifluece Regi f -aswer bjects */ detes the ifluece regi f -aswer bjects. 8: Frm Fr each the d defiiti, we ca see that ay pit that lies i the itersecti f the 9: Expad the rad etwrk ifluece regi f aswer bject O + util k; 0: (, ); is regarded as a safe regi ad that ay pit p that lies i the ifluece regi: f -aswer Mark rage(, bject d); O shuld be excluded. : I a rad etwrk, the safe regi ; : Ed fr is expressed by a set f segmets. I Figure, recall that ad 6 are aswer bjects : Retur ad ad 7 are useful -aswer bjects. By applyig the abve frmula, the safe regi... cacmputati be expressed f Safe as: Exit Pits The safe regi SR f a query q is defied as fllws: SR = {(IR( ) IR( 6 )) (IR( ) IR( 7 ))} () = { } () Table summarizes where detes the cmputati the ifluece regi f the f aswer safe exit bjects pits ad fr detes the query the ifluece bject regi q i Figure f. I the -aswer bjects. Frm the defiiti, we ca see that ay pit that lies i the itersecti f the previus secti, we cmputed the ifluece regi IR f all useful bjects, which was described by set ifluece regi f aswer bject is regarded as a safe regi ad that ay pit p that lies i the f segmets. T ifluece btai regi thef safe -aswer regi, bject we must shuld perfrm be excluded. itersecti ad set differece peratis the rad segmets. I a rad Weetwrk, btaithe segmets safe regi e( is expressed, ), s{( by a set, s f ), segmets. (, s )} I frm Figure the, recall itersecti that f IR( ) ad 6 are aswer bjects ad ad 7 are useful -aswer bjects. By applyig the abve frmula, ad IR( 6 ). Asthe safe ad regi 7 are ca be -aswer expressed as: bjects, the safe regi must exclude the IR( ) ad IR( 7 ). Thus, the safe regi wuld be the segmets s{(s, ), (, s )}. The set f budary pits s ad s = ( ) ( ) ( ) ( ) () becmes the set f safe exit pits. Figure 7 presets the safe exit pits f q. 7 s q 6 s 7 6 Figure 7. Safe exit pits f q. Figure 7. Safe exit pits f q.

13 ISPRS It. J. Ge-If. 06,, 7 f 6 Table. Cmputati f safe regi f q. O Ifluece Regi IR( ) IR( ) {IR( ) IR( )} {IR( ) IR( )} e{(, ), (, ), (, 7 )}s{(, s ), (, s ), (, s ), ( 7, 7 )} 6 e(, ), (, s ), (, 7 )s{(, s ), (, ), (, s 6 )} e(, )s{(, s ), (, s )} e(, )s{(, ), (, s ), (, s 6 ), (, )} e(, )s{(, s ), (, s )} s{(s, ), (, s ), (, s )} e(, 7 ), (, 7 ), (, )s{(, ), (, s 7 ), (, s 8 )} e(, )s{(, s ), (, s )} s{(s, ), (, s )}

14 ISPRS It. J. Ge-If. 06,, 7 f 6.. Mitrig f Data Objects As metied previusly, we are studyig a special case where bth data bjects ad query bjects ca mve radmly i a rad etwrk. The RkNN fr a query q ca be chaged by the mti f data bjects. I this secti, we preset techiques t ctiuusly mitr the data bjects. As discussed i Secti.., the data bjects are divided it types: () UO bjects ad () prued bjects. Sice the mvemet f UO is critical ad a slight mvemet ca chage the results, we cmpute the safe exit pits f all the UO, whereas fr prued bjects, we use the mitred etwrk based the ifluece regi cmputed i Secti... The cmputati f safe exit pits f all the prued bjects will als icrease the cmputati cst.... Cmputati f Safe Exit Pits f Useful Objects (UO) Recall that aswer bjects are thse bjects whse query bject q is e f its knn bjects. This meas that a aswer bject lies withi the safe exit pits as lg as its knn bjects remai the same. Similarly, -aswer useful bjects are thse bjects fr which ay f the aswer bjects are amg its knn. This meas that all useful bjects lie iside the safe exit pits as lg as their respective knns are the same. Therefre, we must mitr the earest eighbrs f mvig bjects i a rad etwrk. We use the apprach f Ch et al. [], Safe Exit Algrithm (SEA), which ca efficietly cmpute the safe exit pits f a mvig earest eighbr query a rad etwrk. First, we frmally defie a set f safe exit pits fr a mvig NN query i the rad } etwrk. Let Ω be the set f safe exit pits fr a knn query pit q ad O = {,,..., O be the set f bjects f iterest t q. Assume that the aswer set (i.e., O + ) f q ad its -aswer set (i.e., O ) are O + = { +, +,..., } { } + k ad O = k+, k+,...,, respectively. The, it hlds that O d(q, + ) d(q, ) fr a aswer bject + O + ad a -aswer bject O. Fially, Ω is defied as fllws: { Ω = ω G MAX ( d ( ω, + ) ( ) ( )) ( ) ( ) ( ) }, d ω, +,..., d ω, + k = MIN(d ω, k+, d ω, k+,... d ω, ) where MI N() ad MAX() retur the miimum ad maximum values f the iput array, respectively. That is, a safe exit pit ω is the ceter pit, i.e., AX(d ( ω, + ) ( ),..., d ω, + ( ) ( ) k ) = MIN(d ω, k+,..., d ω, ), betwee the farthest aswer bject ad the earest -aswer bject. The fllwig tw mai lemmas are preseted t determie whether a safe exit pit exists i the segmet. Lemma. If A β O (β,p a ) = A pa, there is a safe exit pit ω i the segmet. Prf. Please refer t []. This lemma idicates that a safe exit pit i a segmet exists if the set f aswer bjects at β is t equal t the set f aswer bjects at p a. Lemma. If A β O (β,p a ) = A pa, there is safe exit pit i the segmet. Prf: Please refer t []. This lemma idicates that a safe exit pit i a segmet des t exist if the set f aswer bjects at β is equal t the set f aswer bjects at p a. Fr this purpse, we itrduce ad + f. Fr simplicity, let us assume that A β O (β,p a ) A pa crrespds t O = {,,..., O } ad A p a crrespds t O + = { +, +,..., + O + }. The, at a pit p [ ] β, p a, is referred t as the earest -aswer bject t p such that d(p, ) = MIN(d(p, ), d(p, ),..., d(p, )). Similarly, at a pit p [ ] β, p a, + f is referred t as the farthest aswer bject frm p such that d(p, + f ) = MAX(d(p, + ), d(p, + ),..., d(p, + + )). The midpit betwee ad + f becmes a safe exit pit ω. That is, d(ω, ) = d(ω, + f ). ()

15 ISPRS It. J. Ge-If. 06,, 7 f 6 We w discuss the cmputati f the safe exit pits fr the aswer bject i the example rad etwrk give i Figure. Table summarizes the cmputati f the safe exit pits fr data bject fr the example sceari i Figure. Recall that we are csiderig k =. The aswer bjects f data bject are A = {q, }. ISPRS It. J. Ge-If. 06,, 7 Table. Cmputati f safe exit pits fr. f 6 withi ( Segmet/Edge, ) is explred p a idividually. A pa As shw i A fi Table, fr O (fi (,p a ), Safe ), exit = pit, ={, }, ={, s(, }, ) ad i e( (,, ) ) ={ }. By A Lemma = {q, }, A = ( {q, )} = ;{ } therefre, there e is safe exit pit withi s(, ) ( i e(,,). Similarly, ) fr A segmet = {q, } ( A, ) = there {q, } is safe { } exit pit e based Lemma. Therefre, e(, the ) edges adjacet A t = {q, are } explred A = {q, with 6 } = {q}. The edge ω (, ) will be e(, ) A = {q, } A = {, } { } ω explred ext. As idicated i Table, fr ( e(, 7 ) A = {q, } A, 7 = ), {q, = 7 }, {q} ={, }, ω ={, }, ad (, ) ={ }. By Lemma, i.e., (, ), a safe exit pit exists at the edge. Fr each pit (, ), will be selected frm the aswer bjects i ={, }, whereas will be SEA starts explrati frm the active edge ctaiig bject. Because e(, ) is the active selected frm the -aswer bjects i (, ) ={, }. As illustrated i Figure 8, sequece, the lcati f is the achr pit. Each f the tw segmets s(, ) ad s(, ) = because fr every pit (, ), (, ) > (, ), whereas = because fr withi e(, ) is explred idividually. As shw i Table, fr s(, ), p a =, A = {q, }, every pit ( A =, ), (, {q, }, ad O (, ) = ) < (, ). The safe exit pit ω { }. By Lemma, A O (, ) = is the midpit betwee A ; therefre, there is safe exit ad 6. That is, (, ) = (, ), where (, ) = + ad (, ) = +7 pit withi s(, ). Similarly, fr segmet s(, ) there is safe exit pit based Lemma. fr 0< <. Csequetly, =., which meas that the distace frm t ω is.. Therefre, the edges adjacet t are explred with = p a. The edge e(, ) will be Next, we determie a safe exit pit i the edge (, ). As shw i Table, fr (, ), explred ext. As idicated i Table, fr e(, ), p a =, A = {q, }, A = {q, 6 }, ad =, ={, }, ={, }, ad (, O (, ) = {q}. By Lemma, i.e., A O (, ) = ) ={ }. By Lemma, i.e., (, A, a safe exit pit exists at the edge. ), a Fr each pit p e(, ), + safe exit pit exists i the edge. Fr each pit ( f will be selected frm the aswer bjects, ), i A will be selected frm the = {q, }, whereas aswer will be bjects i selected frm ={, the -aswer }, whereas bjects will be selected frm the -aswer bjects i i A O (, ) A = { 6, }. As illustrated i Figure (, ) 8, + f = ={ because }. As idicated i Figure 9, fr every pit e(, ), dist(p, = because fr every pit ) > dist(p, q), whereas ( = 6 because, ), (, ) > fr every pit (, p e( ), because, ), dist(p, there is 6 ) < ly dist(p, e -aswer ). The safe exit bject pit fr ω ( is, the ), midpit therefre, betwee =. The ad safe 6. That exit is, pit dist(ω ω, ) is = the dist(ω midpit, 6 ), where betwee dist(ω q, ad ) =. x + That ad is, dist(ω (, ), ) = = ( x + 7 fr, 0 < ), x where <. Csequetly, (, ) = + x =., ad which ( meas, ) that = +0 the distace fr frm 0 < <7. t ω Csequetly, is.. =., which meas that the distace frm t ω is q 6 q.. q 6 Figure 8. Determiati f safe exit pit ω Figure 8. Determiati f safe exit pit.. Next, we determie a safe exit pit i the edge e(, ). As shw i Table, fr e(, ), p a =, A = {q, }, A = {, }, ad O (, ) = { }. By Lemma, i.e., A 6 O (, 6 ) = A, a safe exit pit exists i the 0 edge. Fr each pit p e(, 6 ), 0 + f will be selected 0 frm the 0 aswer bjects i A = {q, }, whereas will be selected frm the -aswer bjects i A 6 O (, 6 ) A = { }. As idicated i Figure 9, + f = q because fr every pit e(, 6 ), dist(p, q) > dist(p, ), because there is ly e -aswer bject fr e(, 6 ), therefre, =. The safe exit pit ω is the midpit betwee q ad q. That is, dist(ω q, q) = dist(ω, ), where dist(ω, q) = x + ad dist(ω, ) = x + 0 fr 0 < x < 7. Csequetly, x =., which meas that the distace frm. t ω is... Figure 9. Determiati f safe exit pit. Similarly, we cmpute a safe exit pit i the edge (, ). Figure 0 displays safe exit pit

16 q q.. q Figure 8. Determiati f safe exit pit. ISPRS It. J. Ge-If. 06,, 7 6 f q q.. Figure Figure Determiati f safe exit pit ω. Similarly, we we cmpute a safe exit piti ithe the edge edge ( e(, )., Figure 7 ). Figure 0 displays 0 displays safe exit safe pit exit pit ω ωi i the the edge edge ( e(,,). 7 ). Safe Safe exit exit pit pit ωis isthe the midpit betwee ad ad7. 7. That That is, is, dist(ω (, ), = ) dist(ω = (, 7, ), ), where where dist(ω (, ), = ) = + x + ad dist(ω (,, 7 ) ) = = +6 x + 6fr fr 0< 0 <. x. ISPRS It. J. Ge-If. 06,, 7 6 f 6 Csequetly, =, which meas that the distace frm Csequetly, x = which meas that the distace frm t is t ω is Figure 0. Determiati f safe exit pit ω..... Mitrig f Prued Objects Table. Cmputati f safe exit pits fr. Recall that Segmet/Edge there are three differet evets that ca affect the RkNN ( results., ) Safe Sectis exit pit. ad.. (, ) (, ) ={, } ={, } { } preset the ctiuus mitrig f query ad UO, respectively. I this secti, e we discuss the (, ) (, ) ={, } ={, } { } e mitrig f prued bjects. T avid ay additial cmputati, we are usig the precmputed (, ) ={, } ={, } { } ifluece regi ( f, UO ) as the mitrig regi ={, } fr prued ={ bjects., } By{ defiiti, } the ifluece regi f a aswer bject (, is ) valid util q is a knn ={, f the } aswer ={, bject, } whereas { } i the case f a -aswer bject, the ifluece regi remais valid util q is t a knn f the -aswer bject. Therefre, it is... ecessary Mitrig mitr f Prued theobjects bjects that are eterig r leavig the ifluece regi because the mvemet Recall f that bjects there i are ad three ut the differet ifluece evets regi that ca ca chage affect the the RkNN results. results. Therefre, Sectis we. ly ad start mitrig.. preset the the mvemet ctiuus f amitrig prued bject f query ce it ad eters UO, r respectively. leaves the ifluece I this secti, regi. we discuss the Let mitrig be a prued f prued bject lcated bjects. utside T avid the ifluece ay additial regi f cmputati, ay UO. Thewe mitrig are usig f bject the will precmputed t triggerifluece uless itregi etersf the UO ifluece as the mitrig regi fregi ay UO. fr Oce prued it eters bjects. the By ifluece defiiti, regi, the theifluece mitrig regi willf start a aswer ad thebject results is set valid will util lyq chage is a knn whe f the theaswer NN fbject, ay UOwhereas chages. i the If the mvemet case f a f-aswer a prued bject bject, des the t ifluece chage the regi NN set remais f ayvalid UO, the util the q is result t set a knn remais f the valid ad-aswer it is t ecessary bject. Therefre, t call theit determiig is ecessary UOmitr phase agai. the bjects The that determiig are eterig UOr phase leavig willthe ly beifluece called whe regi the because NN set chages. the mvemet f bjects i ad ut the ifluece regi ca chage the results. Therefre, we ly start mitrig the mvemet f a prued bject ce it eters r.. leaves Extesis the ifluece regi. Let be a prued bject lcated utside the ifluece regi f ay UO. The mitrig f... bject RkNN will Queries t trigger i Directed uless it Rad eters Netwrks the ifluece regi f ay UO. Oce it eters the ifluece regi, Previus the mitrig sectis discussed will start the ad mitrig the results set f will RkNN ly queries chage iwhe a rad the etwrk NN f ay that UO are represeted chages. If bythe bidirectial mvemet graphs. f a prued I this bject secti, des we t idetify chage the mdificatis NN set f ay required UO, the tthe exted result the prpsed set remais techique valid ad t ait directed is t ecessary rad etwrk t call where the determiig each rad segmet UO phase has agai. a particular The determiig rietati. UO First, phase we will discuss ly be the called mdificatis whe the NN required set chages. fr the cmputati f the safe exit pits f q. The majrity f the chages are required t be implemeted i Phase (i.e., determiig useful.. Extesis... RkNN Queries i Directed Rad Netwrks Previus sectis discussed the mitrig f RkNN queries i a rad etwrk that are represeted by bidirectial graphs. I this secti, we idetify the mdificatis required t

17 ISPRS It. J. Ge-If. 06,, 7 7 f 6 bjects) ad are autmatically addressed by redefiig dist(p, p ). Give tw pits p ad p, the shrtest path distace dist(p, p ) is the miimum distace frm p t p. By applyig this mdified defiiti, the categrizig f aswer ad -aswer bjects remais uchaged. The defiiti f the clsed de will be same. Csequetly, Lemma remais valid. Algrithm, ca be used ISPRS It. J. Ge-If. 06,, 7 7 f 6 i a similar maer. The ly differece is that t determie the UO, we explre the etwrk i the reverse reverse directi, directi, i.e., i.e., a a edge edge e( (,, ) is) csidered is csidered the the adjacet adjacet edge edge f f if if ad ad ly ly if if the the edge is bidirectial is bidirectial r ther directi the directi f the f edge the edge is frm is frm t t.. Phase Phase is the is same, the same, except except t determie t determie k+ r k we r expad we expad the radthe etwrk rad etwrk i the directi i the f the directi edges. f Further, edges. thefurther, cmputati the cmputati f the ifluece f the ifluece regi fregi + is the f same, is the except same, the except distace the dist( distace +, k+ ( ) is the, shrtest path frm + t k+ istead f the shrtest path frm + ) is the shrtest path frm t istead f the shrtest path t k+ frm. Similarly, t the cmputati f ifluece regi f is the same, except the distace dist(. Similarly, the cmputati f the ifluece regi f the same, except, k ) is the the distace shrtest path ( frm, t k istead f the shrtest path betwee ) is the shrtest path frm t istead ad f the k. Phase shrtest ispath the same betwee as fr aad directed. radphase etwrk. is the Fially, same as wefr discuss a directed the mdificatis rad etwrk. ifially, the cmputati we discuss f the the mdificatis safe exit pits the f the cmputati UO. SEA fuctis f the safe i aexit similar pits fashi f the except UO. SEA it ly fuctis explres i a the similar etwrk fashi i the except directi it ly f the explres edges. The the etwrk remaider the fdirecti the methdlgy, f the edges. i.e., The Lemma remaider, Lemma f the methdlgy,, ad the defiiti i.e., Lemma f, Ω, remais Lemma uchaged., ad the defiiti f Ω, remais uchaged RkNN RkNN Queries Queries i i Dyamic Dyamic Rad Rad Netwrks I this I this secti, secti, we exted we exted CORE-X CORE-X t dyamic t dyamic rad etwrks rad etwrks where where the etwrk the etwrk distace distace chages chages depedig the traffic cditis. The query result r safe regi f query r data bject depedig the traffic cditis. The query result r safe regi f query r data bject may may frequetly be ullified by updatig the weights f sme edges, eve thugh the query bject q frequetly be ullified by updatig the weights f sme edges, eve thugh the query bject q r data r data bject remais withi their respective safe regi. Thus, we itrduce a cmpressed safe bject remais withi their respective safe regi. Thus, we itrduce a cmpressed safe regi t regi t reduce the frequecy f evaluatis f the safe regi i a dyamic rad etwrk. I this reduce the frequecy f evaluatis f the safe regi i a dyamic rad etwrk. I this secti, secti, the safe regi i a static rad etwrk is deted as the rigial safe regi,, whereas the safe regi i a static rad etwrk is deted as the rigial safe regi, SR r, whereas the safe the safe regi i a dyamic rad etwrk is deted as the cmpressed safe regi,, which is regi clearly i a a dyamic subset f rad etwrk is deted as the cmpressed safe regi, SR c, which is clearly. T simplify the explaati, we study the impact f a dyamic rad a subset etwrk f SR with r. respect T simplify t q ly. the explaati, we study the impact f a dyamic rad etwrk with respect t Ulike q ly. the rigial safe regi, the cmpressed safe regi is determied withi ly the active sequece. Ulike the Figure rigial illustrates safe regi, the thedifferece cmpressed betwee safe regi ad is determied, where withi ly is the theactive sequece. edge. The Figure bld lie illustrates i Figure the a differece idicates that betwee fr SR r, ad, SR, ad c, where are the safe is the exit active pits. edge. I Thedyamic bld lierad Figure etwrks, a idicates the safe regi that fris SR cmpressed r, s, s, ad t active s areedge the safe exit as pits. idicated I dyamic i bld lies rad etwrks, i Figure theb. safe regi is cmpressed t active edge as idicated i bld lies i Figure b. s s q s (a) Origial safe regi q (b) Cmpressed safe regi Figure. Differece betwee rigial safe regi ad cmpressed safe regi SR r ad cmpressed Figure. Differece betwee rigial safe regi ad cmpressed safe regi. (a) safe regi SR c. (a) Origial safe regi SR r ; (b) Cmpressed safe regi SR c. Origial safe regi ; (b) Cmpressed safe regi. Next, Next, we we itrduce itrduce a critical a critical regi regi t mitr mitr the the validity validity f the f the safe safe regi regi effectively whe whe the weight the weight f a edge f a isedge chaged. is chaged. The critical The critical regi regi CR is defied CR is defied as CR as = IR + = IR, where, where IR + ad IR ad are the are ifluece the ifluece regi regi f aswer f aswer ad -aswer ad -aswer bjects, bjects, respectively. respectively. By defiiti, By defiiti, the critical the regi critical ctais regi allctais the pits all the p pits such that {, cr such that p cr {IR +, IR }. As illustrated }. As illustrated i Figure i a, Figure there a, are twthere dataare bjects tw data adbjects ad aad query bject ad a q. query Fr k bject =, q. it is Fr bvius =, it that is bvius is a aswer that is bject a aswer bject because = ( ) ad is a -aswer bject because = ( ). Thus, the critical regi ctais all the pits { ( ), ( )}, deted as the bld lie i Figure b with ad as budary pits. The safe regi f q may be affected whe the weight f a edge assciated with the critical regi chages. Therefre, the critical regis f q are stred ad idexed t prmptly verify whether a chage affects the safe regi. If the ifluece regi f ay

18 ISPRS It. J. Ge-If. 06,, 7 8 f 6 ISPRS It. J. Ge-If. 06,, 7 8 f 6 because UO is q updated, = NN( the ) ad critical is regi a -aswer f q is als bject updated because accrdigly. = NN( Clearly, ). Thus, chages thet critical edges regi that ctais are t all assciated the pits with p cr the critical {IR( regi ), IR(ca )}, be deted safely igred. as the bld lie i Figure b with c ad c as budary Figure pits. a The idicates safe regi the critical f q may regi be affected by bld whe lies the at time weight. f Suppse, a edgebecause assciated f heavy with the critical traffic regi cditis, chages. the weights Therefre, f tw the edges critical regis ad f q are are stred chaged adat idexed time tfrm prmptly t verify 6 whether as ISPRS displayed It. a J. chage Ge-If. i Figure 06, affects, b. 7 thethe safeupdate regi. t edge If the ifluece, which regi is assciated f aywith UO is a critical updated, regi, the8 critical may f 6 regi affect f the q is safe als regi. updated Hwever, accrdigly. the update Clearly, t edge chages, which t edges is t that part aref t a critical assciated regi, with des the critical t UO affect is regi updated, the ca safe be the regi safely critical ad igred. regi thus ca f q be is safely als updated igred. accrdigly. Clearly, chages t edges that are t assciated with the critical regi ca be safely igred. Figure a idicates the critical regi by bld lies at time. Suppse, because f heavy traffic cditis, the weights f tw edges ad are chaged at time frm t 6 as displayed i Figure b. The update t edge, which is assciated with a critical regi, may affect the safe regi. Hwever, the update t edge, which is t c part f a critical regi, des t affect the safe regi ad thus ca be safely igred. q c q (a) Rad Netwrk Example (b) Critical c Regi f q Figure. Differece betwee rigial safe regi ad cmpressed safe regi. (a) Rad Figure. Differece betwee rigial safe regi SR r ad cmpressed safe regi SR c. (a) Rad Netwrk Example; (b) Critical Regi f q. Netwrk Example; (b) Critical Regi f q. q Figure a idicates the critical regi by bld lies at time t i. Suppse, c q because f heavy traffic cditis, the (a) Rad weights Netwrk f tw Example edges ad are chaged(b) at Critical time t j Regi frm f q t 6 as displayed i Figure b. The update Figure. Differece betwee t edgerigial, which safe regi is assciated ad with cmpressed a critical safe regi, may. (a) affect Rad the safe c c regi. Netwrk Hwever, Example; the update (b) Critical t edge Regi f, which q. is t part f a critical regi, des t affect the safe regi ad thus ca be safely igred. c q c (a) Critical Regi f q at (b) Updates t ad at Figure. Updatig the weights f tw edges ad where <. (a) Critical Regi f q at ; (b) Updates t ad at Bichrmatic RkNN Queries q c q c Ulike (a) the Critical mchrmatic Regi f case q at where all bjects are f the same type, i bichrmatic RNN, we (b) Updates t ad at distiguish betwee tw types f bjects O ad S. Fr a query bject f type S, the bjective is t Figure. Updatig the weights f tw edges determie data bjects f type O fr which the query ad pit where is their clsest <. (a) Critical Regi f Figure. Updatig the weights f tw edges ad where t i < t j. (a) bject Critical i Regi the set f S. qwith at at this fudametal ; (b) Updates t differece ad betwee at. t i ; (b) Updates t ad at t j. MRkNN ad BRkNN, CORE-X maitais the same methdlgy t maage bth types f query. This secti presets the chages required i the... Bichrmatic RkNN Queries... prpsed Bichrmatic techique RkNN t prcess Queries BRkNN queries.. Ulike I determiig the mchrmatic the UO case phase, where ly all bjects f are type f S the are same csidered type, i as bichrmatic blckig bjects RNN, we t distiguish Ulike prue the betwee the mchrmatic etwrk. types Therefre, case f bjects where a clsed O all ad bjects de S. Fr ca are a query be f redefied the bject same f as type type, a de S, ithe fr bichrmatic bjective which there is RNN, t wedetermie distiguish exists data betwee a bjects tw f type types such O fr fthat bjects which (, the O ad ) query < (, ). S. Fr pit a query is The their bject remaider clsest fbject type Lemma S, i the the bjective set remais S. With is t determie this fudametal uchaged. data bjectsdifferece The f type O fr betwee f which type O MRkNN the that query lie i ad pit the uprued BRkNN, is their clsest CORE-X etwrk bject are maitais csidered the setthe S. With as same UO this fudametal methdlgy bjects, differece t whereas maage betwee the bth bjects types MRkNN f f type query. ad S i BRkNN, the This uprued secti CORE-X presets etwrk maitais are the called chages the same active required methdlgy bjects. i the t maage prpsed. bth Similar techique types t f MRkNN, t query. prcess This UO BRkNN bjects sectiare queries. presets categrized the chages it aswer required ad -aswer i the prpsed bjects. techique Aswer t prcess. BRkNN bjects I determiig queries. fr BRkNN the queries UO phase, ca be ly defied bjects as f a type bject S are fr csidered which (, as blckig ) < (, bjects t ) where prue sk+ the is etwrk. the (k +)th Therefre, NN bject a clsed f i de dataset ca S. be Similarly, redefied a -aswer as a de bject fr which is defied there. I determiig as exists a bject a bject the such UO that phase, such (, ly that ) bjects (, > (, ) f < (, ). type ), where S are sk csidered The is the remaider kth NN as blckig bject f Lemma f bjects i dataset remais t prue S. the etwrk. uchaged. Therefre, bjects a clsed f type deo ca that belie redefied the uprued as a deetwrk fr which are there csidered exists a as UO bject bjects, whereas the bjects f type S i the uprued etwrk are called active bjects.. Similar t MRkNN, UO bjects are categrized it aswer ad -aswer bjects. Aswer bjects fr BRkNN queries ca be defied as a bject fr which (, ) < (, ) where sk+ is the (k +)th NN bject f i dataset S. Similarly, a -aswer bject is defied as a bject such that (, ) > (, ), where sk is the kth NN bject f i dataset S. 6 c q c

19 ISPRS It. J. Ge-If. 06,, 7 9 f 6 s S such that dist(, s) < dist(, q). The remaider f Lemma remais uchaged. The bjects f type O that lie i the uprued etwrk are csidered as UO bjects, whereas the bjects f type S i the uprued etwrk are called active bjects.. Similar t MRkNN, UO bjects are categrized it aswer ad -aswer bjects. Aswer bjects fr BRkNN queries ca be defied as a bject fr which dist(, q) < dist(, s k+ ) where s k+ is the (k +)th NN bject f i dataset S. Similarly, a -aswer bject is defied as a bject such that dist(, q) > dist(, s k ), where s k is the kth NN bject f i dataset S.. The ifluece regi f a aswer bject is defied as IR + = {p dist( +, p) dist( +, s k+ )}, where dist( +, s k+ ) is the distace betwee the aswer bject ad (k+)th NN i set S. Similarly, the ifluece regi f a -aswer bject is defied as IR = {p dist(, p) dist(, s k )}, where dist(, s k ) is the distace betwee the -aswer bject ad kth earest bject i set S. Phase remais uchaged fr BRkNN.. The cmputati f safe exit pits fr the UO remais the same. The ly differece is fr BRkNN, the farthest aswer bject ad earest -aswer bject belg t set S. Therefre, the safe exit pit is the ceter pit betwee s + f ad s.. The safe regi f a active bject ca be cmputed i a similar fashi as q because bth belg t same data type... Aalysis f Time ad Space Cmplexities I this secti, we aalyze bth time ad space cmplexities f the CORE-X algrithm. Recall that we are studyig RkNN queries i a udirected rad etwrk that is represeted as G = (N, E, W). Let N p k ad Ep k be the sets f des ad edges f pit p withi the knn bjects, respectively. First, we aalyze the time cmplexity f CORE-X. CORE-X retrieves a set f UO bjects by ( traversig the rad ) etwrk frm query lcati q, which has a time cmplexity f T E q k + N q k lgn q k because a edge is visited at mst twice, first t determie the aswer bjects ( ) N q ad the t determie the useful -aswer bjects. Here, time cmplexity f T k lgn q k refers t the time cmplexity f the shrtest path algrithm frm a query pit t krnn bjects [9]. Next, CORE-X cmputes the safe exit pits usig the retrieved UO bjects. Fr each data bject UO, the ifluece regi ( f data bject ) is searched t discver safe exit pits, E q which has a time cmplexity f T k + N q k lgn q k. Thus, the time cmplexity f CORE-X is ( T ( E q k + N q k lgn q k ) + E UO ( q k + N q k lgn q k ). ) The verall time cmplexity f CORE-X ca be ( ) imprved t T ( E q k + N q k ) + UO ( E q k + N q k ) usig the preprcessed rderig f des i static rad etwrks. Hwever, i the case f dyamic rad etwrks, the preprcessig techique is aturally isigificat because updatig the weight f sme edges fte ivalidates the preprcessed ifrmati. The search space f CORE-X becmes a regi withi a distace f k RNNs f q ad (k+)nns f the data bjects because it is ecessary t explre the (k+)th NN t cmpute the ifluece regi f a aswer bject. Fially, the space cmplexity f CORE-X is + O UO.. Perfrmace Evaluati I this secti, we evaluate the perfrmace f the prpsed algrithm (CORE-X) thrugh simulati experimets. We describe ur experimetal settigs i Secti. ad preset ur experimetal results fr static ad dyamic rad etwrks i Sectis. ad., respectively... Experimetal Settigs Our experimets were perfrmed usig a real rad etwrk [0] fr Sa Jaqui Cuty, Califria, USA, which ctais 8,6 des ad,87 edges. The rad etwrk ctais a set f queries ad a set f data bjects that ca mve radmly i the etwrk. We simulate mvig O q k k+

20 ISPRS It. J. Ge-If. 06,, 7 0 f 6 bjects (bth query ad data bjects) usig the etwrk-based mvig-bject geeratr []. I each experimet, we evaluated the perfrmace by varyig a sigle parameter withi the rage idicated; all ther parameters were set t the default values idicated i bld. All queries were mitred ctiuusly fr 600 timestamps. Table lists the default parameters used i ur experimets. Table. Experimetal parameter settigs. Parameter Rage Number f data bjects (N Data ) 0, 0, 00, 0, 00, 0 ( 000) Number f queries (N qry ) 00, 800, 00, 600, 000 Number f requested RNNs (k), 0,, 0, Speed f bjects (V bj ) 0, 0, 60, 80, 00 (km/h) Percetage f mvig bjects (R bj ) 0, 0, 60, 80, 00 Percetage f updated edges (R upd ) 0,, 0,, 0 We evaluated the perfrmace f CORE-X usig the fllwig measures: () ttal amut f server CPU time per timestamp, which idicates the query prcessig time ad () ttal cmmuicati cst as the ttal umber f pits (i.e., the lcati updates set by data ad query bjects, ad the query results ad safe exit pits retured by the server) trasferred betwee cliets ad the server. The ttal query prcessig time is dmiated by the cmputatis (i.e., determiig UO, cmputati ifluece regi ad safe exit pits) perfrmed at the server ed, therefre we ly measure the CPU time the server. The battery pwer ad wireless badwidth csumpti typically icrease with the amut f data trasferred betwee bjects(cliets) ad servers, thus we use the amut f trasferred data as a metric t evaluate the cmmuicati cst. Fr static rad etwrks, we cmpared CORE-X with the state-f-the-art algrithm SAC [8] ad a baselie algrithm. SAC ctiuusly mitrs RkNN queries by assigig a safe regi t query ad data bjects, whereas baselie algrithm recmputes the results at each timestamp. Fr dyamic rad etwrks, we cmpared the prpsed algrithm with ly the baselie because SAC des t address dyamic rad etwrks. All algrithms were implemeted i Java ad executed a desktp PC,.80 GHz, Itel Cre i with GB memry... Experimetal Results fr Static Rad Netwrks Figure a presets the perfrmace f baselie, SAC, ad CORE-X with respect t the umber f bjects. Observe that all the three algrithms are sesitive t the icrease i the umber f bjects because the algrithm must address updates frm a large umber f bjects. The perfrmace f SAC degrades with a large umber f data bjects because filterig ad verificati f mre data bjects are required. The query prcessig time f CORE-X icreases with the icrease i data bjects maily because the ifluece regi ad ω fr a large umber f bjects must be calculated. Hwever, CORE-X scales better tha baselie ad SAC. Figure b displays the cmparis f cmputati cst f CORE-X, SAC, ad baselie with differet values f N qry. Experimetal results reveal that the cmputati time f all algrithms icreases as the umber f queries icreases. Hwever, CORE-X ad SAC csumed csiderably less CPU time tha baselie because the query results are valid as lg as the query ad data bjects remai i their respective safe regi. Figure c, studies the effect f the prprti f query ad data bjects that are mvig. Perfrmace f all algrithms degrades as the percetage f mvig bjects icreases. The cmputati time f CORE-X icreases maily because with data mbility, the safe regi expires mre frequetly ad the algrithm must recmpute the results frequetly. Figure d demstrates the ifluece f the speed f data bjects ad query the perfrmace f CORE-X, SAC, ad baselie algrithms. The experimetal results reveal that the baselie ad SAC algrithms icur cstat cmputati csts. The perfrmace f SAC is t sigificatly affected by the speed because cadidate bjects must be verified at each timestamp, regardless f their speed. Cversely, the perfrmace f CORE-X gradually decreases as the speed f the bjects

21 SAC degrades with a large umber f data bjects because filterig ad verificati f mre data bjects are required. The query prcessig time f CORE-X icreases with the icrease i data bjects maily because the ifluece regi ad ω fr a large umber f bjects must be calculated. Hwever, CORE-X scales better tha baselie ad SAC. Figure b displays the cmparis f cmputati cst f CORE-X, SAC, ad baselie with ISPRS It. J. Ge-If. 06,, 7 f 6 differet values f Nqry. Experimetal results reveal that the cmputati time f all algrithms icreases as the umber f queries icreases. Hwever, CORE-X ad SAC csumed csiderably icreases less CPU because time tha bjects baselie leave because their respective the query results safe regis are valid mre as lg frequetly. as the query Figuread edata shws bjects the query remai prcessig their respective time f CORE-X, safe regi. SAC, Figure ad baselie c, studies as a fucti the effect f f k umber the prprti f requested f query RNNs. ad Experimetal data bjects that results are idicate mvig. that Perfrmace the cmputati f all algrithms time f alldegrades algrithms as icreases the percetage with af icrease mvig f bjects the k icreases. value. This The is cmputati expected because time f with CORE-X a icreases k, maily mre data because bjects with are data required mbility, t the be explred safe regi adexpires verified. mre Hwever, frequetly CORE-X ad the utperfrms algrithm SAC must ad recmpute baselie. the results frequetly. N data N qry ISPRS It. J. Ge-If. 06,, 7 f 6 (a) Effect f Ndata (b) Effect f Nqry R bj V bj (c) Effect f Rbj (d) Effect f Vbj k (e) Effect f k Figure Figure.. Cmparis Cmparis f f cmputatial cmputatial cst. cst. (a) (a) Effect Effect f f Ndata; N (b) Effect f Nqry; (c) Effect f Rbj; (d) data ; (b) Effect f N qry ; (c) Effect f R bj ; (d) Effect Effect f Vbj; f V (e) Effect f k. bj ; (e) Effect f k. Figure d demstrates the ifluece f the speed f data bjects ad query the I Figure a, we study the effect f the umber f bjects the cmmuicati cst. This figure perfrmace f CORE-X, SAC, ad baselie algrithms. The experimetal results reveal that the illustrates that the umber f messages set by all algrithms teded t icrease as the umber baselie ad SAC algrithms icur cstat cmputati csts. The perfrmace f SAC is t f bjects icreased. It is clear, hwever, that the safe regi-based algrithms SAC ad CORE-X sigificatly affected by the speed because cadidate bjects must be verified at each timestamp, sigificatly utperfrmed baselie. The prpsed algrithm demstrates superir perfrmace regardless f their speed. Cversely, the perfrmace f CORE-X gradually decreases as the speed cmpared t SAC because the cliet-server cmmuicati is t required whe the query ad data f the bjects icreases because bjects leave their respective safe regis mre frequetly. Figure bjects remai withi the safe exit pits, whereas i SAC, cadidate bjects are required t reprt e shws the query prcessig time f CORE-X, SAC, ad baselie as a fucti f k umber f their lcati t the server fr verificati wheever they chage their lcati. requested RNNs. Experimetal results idicate that the cmputati time f all algrithms icreases Figure b illustrates the effect f the umber f queries the cmmuicati cst. with a icrease f the k value. This is expected because with a icrease i k, mre data bjects are Baselie icurs cstat cmmuicati cst, regardless f differet values f N required t be explred ad verified. Hwever, CORE-X utperfrms SAC ad baselie. qry. Cversely, the cmmuicati cst f CORE-X ad SAC icreases with a icrease i the umber f queries. I Figure a, we study the effect f the umber f bjects the cmmuicati cst. This The cmmuicati cst f SAC icreases maily because mre data bjects are required t be verified figure illustrates that the umber f messages set by all algrithms teded t icrease as the at each timestamp, which icreases the umber f server-iitiated updates. CORE-X ly verifies the umber f bjects icreased. It is clear, hwever, that the safe regi-based algrithms SAC ad useful bject whe it leaves its safe regi, which sigificatly reduces the cmmuicati cst. CORE-X sigificatly utperfrmed baselie. The prpsed algrithm demstrates superir Figure c presets the perfrmace treds f CORE-X, SAC, ad baselie as a fucti f query perfrmace cmpared t SAC because the cliet-server cmmuicati is t required whe the ad data bject mbility. As expected, baselie perfrms mst prly because it must update each query ad data bjects remai withi the safe exit pits, whereas i SAC, cadidate bjects are data bject at every timestamp. The cmmuicati cst f SAC is higher tha CORE-X because f required t reprt their lcati t the server fr verificati wheever they chage their lcati. its expesive verificati methd. Figure d shws the ttal cmmuicati cst f CORE-X, SAC, Figure b illustrates the effect f the umber f queries the cmmuicati cst. Baselie icurs cstat cmmuicati cst, regardless f differet values f Nqry. Cversely, the cmmuicati cst f CORE-X ad SAC icreases with a icrease i the umber f queries. The cmmuicati cst f SAC icreases maily because mre data bjects are required t be verified at each timestamp, which icreases the umber f server-iitiated updates. CORE-X ly verifies the

22 ISPRS It. J. Ge-If. 06,, 7 f 6 ad baselie with respect t speed f data ad query bjects. This figure idicates similar treds t Figure d. SAC icurs cstat cmmuicati cst because the server-iitiated request t verify the cadidate bjects des t deped the speed. Fr CORE-X, the bjects reach a safe exit pit earlier whe ISPRS It. thej. Ge-If. speed 06, is icreased,, 7 which icreases the cmmuicati cst. As idicated i Figure f e, 6 the cmmuicati csts f CORE-X, SAC, ad the baselie algrithm icrease with k. Bth CORE-X Bth ad SAC CORE-X perfrm ad better SAC tha perfrm the baselie better tha methd. the baselie Hwever, methd. CORE-X Hwever, sigificatly CORE-X utperfrms sigificatly SAC utperfrms all cases because SAC fr f all lw cases verificati because f cst. lw verificati cst. N data N qry (a) Effect f Ndata (b) Effect f Nqry R bj V bj (c) Effect f Rbj (d) Effect f Vbj k (e) Effect f k Figure Figure.. Cmparis Cmparis f f the the cmmuicati cmmuicati cst. cst. (a) (a) Effect Effect f f Ndata; N (b) Effect f Nqry; (c) Effect f Rbj; data ; (b) Effect f N qry ; (c) Effect f R (d) Effect f Vbj; (e) Effect f k. bj ; (d) Effect f V bj ; (e) Effect f k... Experimetal Results fr Dyamic Rad Netwrks.. Experimetal Results fr Dyamic Rad Netwrks Figure 6 demstrates cmparis f the query prcessig time fr dyamic rad etwrk Figure 6 demstrates a cmparis f the query prcessig time fr a dyamic rad etwrk where the Rupd (%) values f all edges chage their weight at each timestamp. The updated edges are where the R upd (%) values f all edges chage their weight at each timestamp. The updated edges are selected radmly, irrespective f the lcatis f data ad query bjects. The legth f a updated selected radmly, irrespective f the lcatis f data ad query bjects. The legth f a updated edge is selected radmly frm 0. t 0 times the rigial legth. edge is selected radmly frm 0. t 0 times the rigial legth. Figure 6a presets the query prcessig time as fucti f the percetage f updated edges Figure 6a presets the query prcessig time as a fucti f the percetage f updated edges (Rupd) per timestamp. Here, =0 idicates a static rad etwrk. Naturally, the query (R upd ) per timestamp. Here, R upd = 0 idicates a static rad etwrk. Naturally, the query prcessig prcessig time f baselie is almst cstat regardless f the value f Rupd because query bjects i time f baselie is almst cstat regardless f the value f R upd because query bjects i baselie baselie issue RkNN queries at each timestamp. The query prcessig time f CORE-X icreases issue RkNN queries at each timestamp. The query prcessig time f CORE-X icreases with R upd ; this with Rupd; this is maily because the safe regi f a query ad data bject must be updated if a is maily because the safe regi f a query ad data bject must be updated if a edge is assciated edge is assciated with the critical regi f the data r query bject. Observe that fr, with the critical regi f the data r query bject. Observe that fr R upd, CORE-X perfrms CORE-X perfrms csiderably better tha baselie, whereas fr =0, the query prcessig csiderably better tha baselie, whereas fr R upd = 0, the query prcessig time f CORE-X is time f CORE-X is greater tha baselie. greater tha baselie.

23 ISPRS It. J. Ge-If. 06,, 7 f 6 Figure 6b illustrates the effect f the umber f data bjects the query prcessig time. The cmputati time fr bth algrithms typically icreases with N data. I this case, CORE-X sigificatly utperfrms baselie. I Figure 6c, we study the effect f N qry the cmputati cst. Experimetal results reveal that bth algrithms are sesitive t a icrease i the umber f query bjects. The query prcessig time f CORE-X icreases maily because the cmputati f the critical regi ad safe regi fr mre bjects is required. Figure 6d idicates the tred that cmputati cst f bth algrithms icreases with a icrease i query ad data bject mbility. Hwever, CORE-X scales sigificatly better tha baselie. Figure 6e is the query prcessig time f CORE-X ad baselie as a fucti f the speed f the data ad query bjects. Baselie has a early stable query prcessig time. Hwever, the cmputati cst f CORE-X icreases with V bj because, as the bjects mve faster, they reach their respective safe exit pits earlier, resultig i mre frequet updates f the query results. Figure 6f demstrates the ifluece f the k umber f requested RNNs the perfrmace f bth algrithms. As expected, the cmputati f bth algrithms icreases with a icrease f the k value. With CORE-X, the critical regi f mre data bjects eedig t be cmputed icreases the size f the critical regi. Csequetly, the umber f edges assciated with the critical regi als icreases, which elevates ISPRS It. thej. query Ge-If. prcessig 06,, 7 time. f 6 R upd N data (a) Effect f Rupd (b) Effect f Ndata N qry R bj (c) Effect f Nqry (d) Effect f Rbj V bj k (e) Effect f V bj (f) Effect f k Figure 6. Cmparis f the cmputatial cst. (a) Effect f Rupd; (b) Effect f Ndata; (c) Effect f Nqry; Figure 6. Cmparis f the cmputatial cst. (a) Effect f R upd ; (b) Effect f N data ; (c) Effect f N qry ; (d) Effect f Rbj; (e) Effect f Vbj; (f) Effect f k. (d) Effect f R bj ; (e) Effect f V bj ; (f) Effect f k. Figure 6b illustrates the effect f the umber f data bjects the query prcessig time. The cmputati Figure 7 presets time fr a cmparis bth algrithms f cmmuicati typically icreases cst f CORE-X with Ndata. adi baselie this case, usigcore-x the same cditis sigificatly as thse utperfrms i Figure baselie. 6. Figure I Figure 7a illustrates 6c, we study the effect the effect f Rf upd Nqry the the cmmuicati cmputati cst. cst f bth Experimetal algrithms. results The reveal perfrmace that bth algrithms f baselie are issesitive t sigificatly t icrease affected i the byumber R upd. f Hwever, query thebjects. cmmuicati The query cst prcessig f CORE-X time icreases f CORE-X withicreases R upd because maily thebecause critical regis the cmputati f data ad f query the bjects critical are regi updated ad mre safe frequetly regi fr asmre R upd icreases. bjects is Csequetly, required. Figure the6d safeidicates regis f the the tred query that ad data cmputati bjects are reevaluated. cst f bth algrithms icreases with a icrease i query ad data bject mbility. Hwever, CORE-X scales sigificatly better tha baselie. Figure 6e is the query prcessig time f CORE-X ad baselie as a fucti f the speed f the data ad query bjects. Baselie has a early stable query prcessig time. Hwever, the cmputati cst f CORE-X icreases with Vbj because, as the bjects mve faster, they reach their respective safe exit pits earlier, resultig i mre frequet updates f the query results. Figure 6f demstrates the ifluece f the k umber f requested RNNs the perfrmace f bth

24 ISPRS It. J. Ge-If. 06,, 7 f 6 I Figure 7b, we study the ifluece f the cardiality f the data bjects the cmmuicati csts. Experimetal results idicate that CORE-X perfrms better tha baselie because i CORE-X, server-cliet cmmuicati is ly required whe the query ad data bjects leave their safe regis r whe the critical regis f the data ad query bjects are updated. Figure 7c shws the cmmuicati cst f CORE-X ad baselie with respect t the umber f query bjects. Observe that the cmmuicati cst f baselie des t deped the umber f query bjects because each data bject reprts its lcati wheever it chages its lcati. O the ther had, the cmmuicati cst f CORE-X icreases because fr a large umber f query bjects, mre data bjects are required t be verified. Figure 7d exhibits the effect f query ad data bject mbility the cmmuicati csts. Clearly, the umber f trasferred pits icreases with a icrease i R bj. Hwever, CORE-X csistetly prvides imprved perfrmace cmpared t baselie. Figure 7e illustrates the effect f the speed f the data ad query bjects cmmuicatis csts. I CORE-X, the umber f trasferred pits icreases with V bj fr the same reas as discussed earlier i the case f Figure d. As idicated i Figure 7f, the cmmuicati csts f CORE-X ad the baselie algrithm icreases with k. This is maily because the umber f data bjects that require verificati ad mitrig icreases ISPRS with It. J. k. Ge-If. 06,, 7 f 6 R upd N data (a) Effect f Rupd (b) Effect f N data N qry R bj (c) Effect f Nqry (d) Effect f Rbj V bj k (e) Effect f Vbj (f) Effect f k Figure 7. Cmparis f the cmmuicati cst. (a) Effect f Rupd; (b) Effect f Ndata; (c) Effect f Figure 7. Cmparis f the cmmuicati cst. (a) Effect f R Nqry; (d) Effect f Rbj; (e) Effect f Vbj j; (f) Effect f k. upd ; (b) Effect f N data ; (c) Effect f N qry ; (d) Effect f R bj ; (e) Effect f V bj j ; (f) Effect f k. I Figure 7b, we study the ifluece f the cardiality f the data bjects the 6. Cclusis cmmuicati csts. Experimetal results idicate that CORE-X perfrms better tha baselie because i CORE-X, server-cliet cmmuicati is ly required whe the query ad data bjects Weleave prpsed their safe a ew regis algrithm whe called the critical CORE-X regis fr the f the efficiet data ad prcessig query bjects f ctiuus are updated. reverse k earest Figure eigbr 7c shws (RkNN) the queries cmmuicati i radcst etwrks f CORE-X where ad bth baselie query with ad respect data t bjects the umber are mvig. f The prpsed query bjects. apprach Observe is based that the cmmuicati safe exit pits, cst f which baselie cades sigificatly t deped imprve the umber t ly f the cmputati query bjects cst but because alseach thedata cmmuicati bject reprts its cst lcati betwee wheever server it chages ad query its lcati. bject. O Mrever, the we preseted ther had, pruig the cmmuicati techiques ad cst ifluece f CORE-X regi-based icreases because mitrig fr a large tumber avid prcessig f query f bjects, mre data bjects are required t be verified. irrelevat data bjects. CORE-X ca effectively cstruct a safe regi i a dyamic rad etwrk by Figure 7d exhibits the effect f query ad data bject mbility the cmmuicati csts. itrducig a cmpressed safe regi ad critical regi. The results f experimets cducted usig Clearly, the umber f trasferred pits icreases with a icrease i Rbj. Hwever, CORE-X csistetly prvides imprved perfrmace cmpared t baselie. Figure 7e illustrates the effect f the speed f the data ad query bjects cmmuicatis csts. I CORE-X, the umber f trasferred pits icreases with Vbj fr the same reas as discussed earlier i the case f Figure d. As idicated i Figure 7f, the cmmuicati csts f CORE-X ad the baselie algrithm icreases with k. This is maily because the umber f data bjects that require verificati ad

25 ISPRS It. J. Ge-If. 06,, 7 f 6 real datasets cfirm that the prpsed algrithm sigificatly reduces the cmputati cst ad the cmmuicati cst cmpared t a baselie ad state-f-the art algrithm (SAC). There are several prmisig directis f future research. First, RNN queries ca be studied fr privacy-aware systems t esure the lcati privacy f a query bject frm a attacker. Further, this study ca be exteded fr ucertai data bjects, which may t have exact lcatis. It is imprtat t determie their apprximate lcatis ad develp accuracy buds the query results. Ackwledgmets: We thak aymus reviewers fr their valuable cmmets ad suggestis. Muhammad Attique, Rize Ji ad Tae-Su Chug were supprted by Basic Sciece Research Prgram thrugh the Natial Research Fudati f Krea (NRF) fuded by the Miistry f Educati (0RAAA0096 ad NRF-0RAA0). Hyug-Ju Ch was supprted by the Natial Research Fudati f Krea (NRF) grat fuded by the Krea gvermet (MSIP) (N. NRF-06RAB00979). Fially, this wrk was partially supprted by Leaders Idustry-Uiversity Cperati Prject. Authr Ctributis: All authrs sigificatly ctributed t the mauscript. Muhammad Attique iitiated the idea, implemeted the experimets, ad wrte the mauscript. Muhammad Attique ad Hyug-Ju Ch desiged the sluti ad experimets. Rize Ji critically reviewed the paper ad revised the mauscript. Tae-Su Chug reviewed the mauscript ad supervised the research. Cflicts f Iterest: The authrs declare cflict f iterest. Refereces. Ciuz, D.; Bua, A.; D Urs, M.; Palmieri, F. Distributed classificati f multiple mvig targets with biary wireless sesr etwrk. I Prceedigs f the th Iteratial Cferece Ifrmati Fusi (FUSION), Chicag, IL, USA, 8 July 0; pp. 8.. Bua, A.; D Urs, M.; Prisc, G.; Felac, M.; Meliadò, E.F.; Mattei, M.; Palmieri, F.; Ciuz, D. Mbile sesr etwrks based autmus platfrms fr hmelad security. I Prceedigs f the Tyrrheia Wrkshp Advaces i Radar ad Remte Sesig (TyWRRS), Naples, Italy, September 0; pp Ch, H.; Kw, S.; Chug, T. A safe exit algrithm fr ctiuus earest eighbr mitrig i rad etwrks. Mb. If. Syst. 0, 9, 7. [CrssRef]. Ch, H.; Ryu, K.; Chug, T. A efficiet algrithm fr cmputig safe exit pits f mvig rage queries i directed rad etwrks. If. Syst. 0,, 9. [CrssRef]. Wag, H.; Zimmerma, R. Prcessig f ctiuus lcati-based rage queries mvig bjects i rad etwrks. IEEE Tras. Kwl. Data Eg. 0,, [CrssRef] 6. Yug, D.; Yiu, M.; L, E. A safe-exit apprach fr efficiet etwrk-based mvig rage queries. Data Kwl. Eg. 0, 7, 6 7. [CrssRef] 7. Zhag, J.; Zhu, M.; Papadias, D.; Ta, Y.; Lee, D. Lcati-based spatial queries. I Prceedigs f the 00 ACM SIGMOD Iteratial Cferece Maagemet f data, Sa Dieg, CA, USA, 0 Jue 00; pp.. 8. Cheema, M.; Li, M.X.; Zhag, Y.; Zhag, W.; Li, X. Ctiuus reverse k earest eighbrs queries i Euclidea space ad i spatial etwrks. VLDB J. 0,, [CrssRef] 9. Kr, F.; Muthukrisha, S. Ifluece sets based reverse earest eighbr queries. I Prceedigs f the 000 ACM SIGMOD iteratial cferece Maagemet f data, Dallas, TX, USA, 6 8 May 000; pp Stai, I.; Agrawal, S.; Abbadi, A. Reverse earest eighbr queries fr dyamic databases. I Prceedigs f the ACM SIGMOD Wrkshp Research Issues i Data Miig ad Kwledge Discvery, Dallas, TX, USA, May 000; pp... Ta, Y.; Papadias, D.; Lia, X. Reverse knn search i arbitrary dimesiality. I Prceedigs f the Thirtieth Iteratial Cferece Very Large Data Bases, Trt, Japa, August September 00; pp Klahduza, M.; Shahabi, C. Vri-based k earest eighbr search fr spatial etwrk databases. I ACM SIGMOD Wrkshp Research Issues i Data Miig ad Kwledge Discvery, Prceedigs f the Thirtieth Iteratial Cferece Very Large Data Bases, Trt, Japa, August September 00; ACM: New Yrk, NY, USA; pp

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