Hui Xiao School of Environmental Science, Nanjing Xiaozhuang University, Nanjing , China

Transcription

1 doi:0.3/ Cotiuous knn Queries i Dyamic Road Networks Hui Xiao School of Evirometal Sciece, Najig Xiaozhuag Uiversity, Najig 7, Chia Abstract Cotiuous knn queries have bee widely studied i recet years. Most of the existig methods assume that road etwork is static; however real-time chages of traffic coditios i road etwork are always happeed. The existig solutios do t solve cotiuous KNN queries i dyamic road etworks. This paper studies the problem of cotiuous KNN queries i dyamic road etworks, traffic coditios of road segmets chage o differet time, weights of road segmets chage dyamically. We propose a query algorithm o cotiue k earest eighbors i dyamic road etworks, the CkNNDN algorithm. The algorithm is applied to query cotiuous k earest eighbors i dyamic coditio, miimize the cout of earest eighbors cadidates, ad reduce expesive costs of the shortest path computatio. I our tests, the algorithm shows good Effectiveess ad efficiecy. Keywords: Dyamic Road Networks, Cotiuous knn Queries, CKNNDN Algorithm, Network Vorooi Diagram, Travel Time.. INTRODUCTION With the progress of the positioig techology ad wireless commuicatio techology, the developmet of itelliget traffic ad locatio services as well as other systems is further improved, ad more ad more people eed to use services that are related to the locatio. With the deepeig of the research ad geeralizatio of the applicatio, cotiuous k earest eighbor queries (cotiuous k-nn, abbreviated CkNN) are studied, CkNN queries refer to the queries that for a give poit, providig k data objectives that are earest to the queries poit withi a period of time, while the queries poit itself is dyamic. For example, queries for 5 restaurats which are earest to ataxi i the movemet withi 5 miutes. CkNN queries eed to update the query results o a cotiuous basis. A large umber of scholars (Kim ad Chag, 03; Cheema, Zhag, Li, Zhag ad Li, 0; She, Zhu, Ye, Guo, Su ad Lee, 05)have studied the CkNN queries for the Euclidea space ad road etwork space. I the real world, the movemet of pedestrias or vehicles is usually restricted i the road etwork; hece obviously, CkNN queries of road etwork space are more compliat with the actual requiremets(cho, Ji ad Chug, 05; Huag, Che ad Lee, 009). At preset, there are two mai types of methods for the CkNN queries of the road etwork, oe method is the real-time cotiuous calculatio of k earest eighbor of the movig objects (Fa, Li ad Yua, 0; Kolahdouza ad Shahabi, 00; Cho ad Chug, 005), this method trasforms the cotiuous queries of the movig objects ito the sapshot queries at key poits, which iclude the odes of the road etwork, the result chagig poits of the earest eighbor, etc., its disadvatage is the ecessity to carry out a large umber of repeated computatio o the shortest path of the road for the queries poits ad the data poits; the aother method is to make use of the precomputatio algorithm (Kolahdouza ad Shahabi, 005; Huag, Jese ad Šalteis, 005), calculate ad store i advace the earest eighbors of the kow poits, so as to save the cost for the shortest path computatio. However, i most of the curret methods of CkNN queries i the road etwork, the static etwork distace is regarded as the stadard of measuremet, while i the real life, people who move i the road etwork are more cocered about the travel time i fact, which chages with the chage of the traffic coditios of the road etwork, hece it is chaged dyamically. I this paper, we adopted the etwork Vorooi diagram method, ad studied the cotiuous k earest eighbor queries i the dyamic road etwork. Compared with most of the curret methods that oly cosider the static road etwork, CkNNDN algorithm is of more practical value.. RELEVANT WORK I this sectio we summarize the related work of knn queries processig i the mobile object library. Accordig to the limitatio of the movemet space is limited, knn queries ca be divided ito Euclidea space ad spatial etwork space, the spatial etwork space limits the movig objects ad the data objects i the etwork, which is more i lie with the reality, ad also the curret research hot spot. knn queries ca maily be divided ito static knn queries ad cotiuous knn queries (CkNN)based o the movemet characteristics of the queries poits. CkNN queries usually require real-time respose, ad to be cotiued over a period of 35

2 time, thus it is more complicated tha the static knn, with larger amout of calculatio ad also the difficult problem i knn queries. knn queries of the spatial etwork takes the etwork distace as the uit of measuremet of the queries distace, compared with the knn queries uder the Euclidea space, the etwork distace knn queries are of more practical sigificace. Papadias et al.(papadias, Zhag, Mamoulis ad Tao, 003)proposed IER algorithm (Icremetal Euclidea Restrictio) ad INE algorithm (Icremetal Network Expasio). IER applies Euclidea distace as the miimum boudary trimmig the cadidate queries results of the queries, while INE algorithm makes use of the Dijkstra algorithm to acquire the knn result o the extesio of the Network distace. Shahabi etc.(shahabi, Kolahdouza ad Sharifzadeh, 003) put forward a method of graph embeddig techology, which coverts road etwork to the high dimesioal Euclidea space, so as to obtai the approximate knn results. Hui Xiao(Xiao ad Yag, 008) studied the knn pre computatio method i the road etwork. Cotiuous knn query is a importat variat of knn queries, which is widely applied i the locatio services etc. The mai difficulties of this kid of queries is to fid out the right poit of divisio i movig path of the queries poits which knn results eed to be updated, ad to avoid the iappropriate repeated computatio o knn. UBA algorithm (Upper Boud Algorithm) (Kolahdouza ad Shahabi, 00)performs static knn algorithm at the locatio that eeds update, which has higher efficiecy tha the method to simply reduce the umber of cadidate knn. Cho et al. proposed UNICONS algorithm (uique cotiuous search algorithm)(cho ad Chug, 005), it will divide the path of the queries poit ito several sub paths through the itersectio poit, the at the edpoit of each subpath, perform the static knn algorithm, fially, each subpath knn is combied ito the knn result of the path of the queries poits. UBA algorithm ad UNICONS algorithm is maily applied to the situatio whe the queries target is static object, ad whe the queries target is dyamic object, the algorithm is sigificatly degraded. Mouratidis et al.(mouratidis, Yiu, Papadias ad Mamoulis, 006)proposed IMA algorithm (icremetal moitorig algorithm), which takes ito accout the weight chage i the road etwork, ad reassesses the queries wheever ay update occurs,the queries make use of the result of the last query, so as to avoid the repeated treatmet o the uchaged queries results. Compared with the direct computatio knn method, the adoptio of pre computatio knn queries for the processig ca save a huge amout of expesive shortest path computatio. Curretly, this method has also attracted the attetio of may scholars, icludig the applicatio of Vorooi diagram for NN pretreatmet, which is a relatively effective way. Kolahdouza et al. proposed VN3 queries algorithm (Vorooi-based etwork earest eighbor)(kolahdouza ad Shahabi, 005) to perform the static knn queries processig. Whe NN queries are carried out, just the executio of the simple search operatio is eough; whe knn queries are carried out ( k ),results are obtaied through the ispectio of the adjacet areas. Huag et al.(huag, Jese ad Šalteis, 005) proposed a blaket method that is differet from the Vorooi diagram - islad method. This method is similar to the method of usig Vorooi diagram, but i differet blaketig mode, the odes are associated with the earby poits of iterest, which are pre computed ad the earest eighbor poits of iterest are stored. Defiitio of Problem. I this sectio, we formally defie the cotiuous K earest eighbor queries problem i the dyamic road etwork. Assumig that the objects i motio are always movig o the road etwork, the poits of iterest i query durig the movig process of the objects i motio (such as hotels, restaurats, hospitals, etc.) are located o the road. The dyamic road etwork is demostrated i the weighted graph related to time, the edge correspods to the road segmet, the ode correspods to the road itersectio poit, ad the weights of the edge correspods to the travel time o the road segmet, accordig to the differet traffic coditios i differet time, the edge has differet weights. Defiitio Dyamic Road Network is defied as G D (V, E), which represets the set of odes, E represets the set of edges. Each edge e coects two odes ( v, v ), ad the weight of e is cofirmed by subsectio weightig fuctio w, () t which is related to time, ad it shows the travel time at momet t from vi vj vi, to v j at the edge ofe. Defiitio Road Network Distace is defied as the shortest path distace coectig ay two poits i dist p, q. the dyamic road etwork, as represeted by N Defiitio 3 Cotiuous K Nearest Neighbor Queries i the Dyamic Road Network is defied as K earest eighbor targets withi the queries time iterval of T for the movig object m o i the dyamic etwork D d, d, d3,..., d, the movig object m o has G. Assumig that the target set for the queries is D movemet i the dyamic etwork G, the k earest eighbor targets withi the time iterval of T ca be represeted as T ', T,, T,...,, T i, T D D D D D D D, D ' D. Targets D, D,..., D correspod to the i T i j, T, T, T 355

3 time iterval that each earest eighbor target keeps the same. Ad for ay d' DiT, ad d D D T ', the, ', dist mo d dist mo d. N 3. CkNNDN Algorithm N I this sectio we specifically illustrate the CkNNDN algorithm. Accordig to the NVD geeratig NV p, ad the method, the earest eighbor queries object of the queries poit ca be directly foud i queries ca be completed just by fidig the NV p cotaiig the queries poit m i NVD. Whe k>, the earby uit iformatio of NV p ca be utilized to acquire the other earest eighbor queries object set. Whe the road travel time weight chages, the pre computed etwork distace ca be utilized to obtai the cadidate earest eighbor queries object set ad acquire the accurate result set. The queries object NVD eeds to be pre NV p i NVD also eeds to be pre computed ad stored. computed, ad the O the basis of the classificatio of the queries objects, the NVD of all kids of queries objects eeds to be calculated respectively. I the dyamic road etwork, the travel time of the road segmet is chaged dyamically, while NVD is established based o a sigle road segmet weight, ad the road segmet travel time has both the maximum ad miimum. If NVD is established accordig to the maximum of the road segmet travel time i the road etwork, every chage of the travel time will ifluece the etwork distace i the etire etwork, thus the etwork distace betwee each poit has to be re calculated. While applyig the miimum of the travel time to establish NVD, whe the travel time chages, it oly ifluece the relevat part of the poits, ad thus it is oly required to recalculate part of the etwork distace of part of the poits, therefore, we establish NVD with the miimum of the road segmet travel time, which is called LNVD (Light Network Vorooi Diagram). 3.. LNVD Properties The followig are the NVD geometric properties, which are the basis of CkNNDN algorithm. NV p, the the earest Property -Let the query poit q be located i the etwork Vooroi uit i eighbor poit of q geerated is pi; Prove:As kow by the defiitio of NV p, the distace from all the poits withi the etwork of the Vooroi uit to the geerated poit pi of the uit is closer tha the distace to the geerated poit pj of other uits.,,... NV m, Property - Let M m m m P be the k earest eighbor of query poit q withi k the mk is the adjacet geeratio poit of the first NV uits. M m, m,... m P be the k earest eighbor of query poit Property -3 I the Vooroi diagram, let k q, the the shortest path from q to mk oly passes the Vorooi uit withi,... k passes the commo edge withi NV m,... NV m k. For specific proof please refer to[,5] Properties - I the LNVD diagram, let i, j P p,... p k ad ' ',... ' k '' '',... '' i NV m NV m, ad oly R be the edge betwee the adjacet odes i ad j, P p p are the k earest eighbor poits of iterest of i ad j respectively, ad P p p m is the poit located i the edge of R i, j, the the k earest eighbor poit of iterest o R i, j is the subset of P P' P ''. Prove: The property ca be proved by reductio to absurdity. Assume that the earest eighbor of the, ' '' R,, from the query q o the edge i j R is m kow coditios ' '' m p P P P. Whe poit pm is located betwee i j p P P P ca be obtaied, which is cotrary to the assumptio, hece the property is prove; whe poit pm is located outside R i, j, from the query poit q to pm, it must pass oe of the otes i ad j. Assume R q, pm passes i, the N, m N, i N i, m N i, m N i,,, dist q p dist q dist p. However, dist p dist p is kow, as p is the earest eighbor of i, it ca be kow that dist q p dist q p, amely, the distace from p to q is closer tha the distace from pm to q, which is N m N cotrary to the assumptio, hece the property is prove. 356

4 Property -5 I the LNVD diagram, let edge AB be located at the etwork Vooroi uit NV p i, if the weight of edge AB chages, it will oly affect AB as the shortest travel path poits, ad these poits eed to be recalculated. Prove: I the LNVD diagram, the poits o the edge are divided ito two categories, oe category is to take AB as the poits with the shortest travel path, ad the other category is ot to take AB as the poits with the NV p is based o the shortest travel path from shortest travel path. The geeratio of etwork Vooroi uit the geerated poits at the edge to the geerated poit p, for the poits that take AB as the shortest travel path, whe the travel time from the poits at the edge to the geerated poits chages, it will ievitably cause the NV p. For the poits that do ot take AB as the shortest travel path, due to recalculatio of the correspodig the fact that the geeratio of NV p withi LNVD are all based o the coditio of the miimum of the weight of the travel time for the edge, the the greater the weight of the edge AB, the value of the travel time passig the edge AB will ievitably icrease, therefore, the edge AB with the ew weight will ot be take as the shortest travel path. To sum up, the propositio is prove. p 6 f p 5 f p 3 f f 6.5 p 3 3 /q 5 f 9 f 0 f f 6 7 p p 3 p 7 f 5 p 8 f 7 f 8 Figure. Network NVD Diagram Figure is a etwork NVD diagram example, p,... p 8 is the set of the geerated poits, which ca be correspodig to the queries objects i the real world, such as hotels, museums, etc.; the polygo eclosed by NV p thi solid lies i the diagram is the etwork Vorooi uit i NV p ; the dotted lies withi the represets the road segmets, ad the heavy solid lie represets the query path mff of the movig objects. 3.. CkNNDN Algorithm The NV p withi LNVD is composed by areas with shared earest eighbor poits, ad the earest poit of iterest is a sigle value, therefore, i the processig of knn queries, for the case that k, the processig algorithm is relatively simple, accordig to the queries path ad the itersectio poit of the queries NV p divides the queries path ito a umber of queries segmets, ad from the characteristics of path ad NVD, the earest eighbor of each queries segmet ca be kow. Take figure for example, if k, the queries path ca be divided ito m, f f, f f,. The earest eighbor of segmet m, f is p, the earest eighbor of f f is p, ad the earest eighbor of f, is p. If the weight of the travel time o, the road segmet chages, NV p eeds to be recalculated, ad to obtai the ew queries segmet ad the earest eighbor value, with the specific processig method similar as the case whe k, please refer to the cotet below. CkNNDN algorithm cosists of two parts, the first part is for the cotiuous k earest eighbor queries i the Vorooi diagram, ad the secod part is the update of the weight of the travel time o the road segmet. NV p where the start poit of the query is located Step : Search for the i Accordig to the locatio of the query start poit of the movig objects, acquire the earest eighbor query object for the query poit. As ca be kow from property -, the movig objects i the figure move from poit to f, ad its earest eighbor query object is always p. Step : Acquire the queries path ad NVP itersectio poit I cotiuous k earest eighbor queries, k earest eighbor chages with the chage of the locatio of the movig objects i the queries path, ad this chagig locatio poit is called a dividig poit. Both edges of the itersectio poit of the queries path ad NVP are areas with differet poits of iterest respectively, if the 357

5 queries path pass the boudary of NVP, the earest eighbor will defiitely chage, therefore, the itersectio poit of the queries path ad the NVP must be the dividig poit. I figure, f is the dividig poit of NV p. NV p ad NV p, ad f is the dividig poit of NV p ad Step 3: Acquire the Key Poit We will iquire the path i accordace with the dividig poit of the itersectio of the query path ad NVP, the odes of the road segmets, as well as the start poit of the query as the key poits. As ca be kow from the property -, the ode of the road segmets is the key iformatio to compute the earest eighbor. Figure will segmet the query path based o the key poits ito (q, f), (f, ), (, f) ad (f, ), ad, are the odes of the road segmets. Step : Search for the k Nearest Neighbor of the Key Poit The k earest eighbor queries of the key poit i essece is the static queries processig i the Vorooi diagram, which is similar to the VN3 method[,5], however, i the pre computatio table what is stored i the VN3 method is the distace betwee the boudary poits i each NVP ad the distace betwee the boudary poit ad the geerated poit, while our method stores the distace betwee all the boudary poits ad all the geerated poits, which further saves the computatio time for the distace. The earest eighbor ca be acquired directly from the i NV p where the key poit is located, ad the rest k- earest eighbor query is a iterative treatmet process. First of all, accordig to the ature of the Vorooi diagram, acquire the k earest eighbor cadidate set, so as to avoid the meaigless computatio o a large umber of impossible poits, ad the calculate the shortest etwork path for the cadidate set, ad the obtai the required k earest eighbor. Step 5: Calculate the Dividig Poit ad k Nearest Neighbor of the Queries Path After acquirig the queries path segmets, it is ecessary to calculate the dividig poit withi each segmet. Through step the k earest eighbor of the two edpoits for each segmet has already bee obtaied, as ca be kow from the property -, if the k earest eighbors of the two edpoits of the segmet are ot the same, there will defiitely be a dividig poit withi the segmet, ad the k earest eighbor of the dividig poit must be geerated from the k earest eighbor of the edpoit. Figure has demostrated the chages of 3 earest eighbor whe the object i motio moves o the road segmet f i figure Figure.Queries Poit ad the Nearest Neighbor Distace Value Step 6: Termiatio k earest eighbor is acquired ad the algorithm is termiated Experimet I order to verify the algorithm proposed i this paper, we made a series of experimets to verify the performace of the algorithm. The realizatio of the algorithm process adopts the C++ laguage programmig, with the operatig system of Widows XP professioal editio, CUP of Itel CPU Core.6 Hz, ad memory of G RAM. The experimetal data is Najig city road etwork ad the poits of iterest data, all levels of accumulative toads are totally,59, 0 the dyamic etwork iformatio of the roads are acquired based o realtime traffic iformatio o GOOGLE ad make use of poits of iterest with differet desity ad distributio (such as hotels, hospitals, etc.) to carry out the 0 experimet, ad compare the 3differece of the performace 5 of CkNNDN algorithm ad IMA algorithm for differet k values ad differet legth of the queries path uder the coditio that the travel time o the road segmet ca be updated dyamically. Dyamic etwork has more 358

6 Iterestig Poits Access Number Resose Time(Secods) practical sigificace tha the static etwork; therefore, we maily compare the CkNN performace differece of the two algorithms uder the dyamic etwork. Based o the fact that the performace of the earest eighbor queries of the Vorooi diagram is better tha the earest eighbor queries algorithm geerally based o the etwork distace, but so far, there is o earest eighbor queries algorithm based o Vorooi diagram i the dyamic etwork, therefore, we chose the algorithm i this paper to compare with the IMA algorithm based o the etwork distace. 3.. The Ifluece of K Value I the experimets, we compared the CkNNDN algorithm, VCkNN algorithm ad IMA algorithm i the dyamic road etwork, differet k values will affect the performace of the algorithm. figure 3 shows the respose time of the queries of differet k values whe other coditios are the same. We tested six values of k whe k = (,3,5,0,5,0). The results show that CkNNDN algorithm uder the coditios of all values of k is better tha the IMA algorithm, ad with the icrease of the k value, the performace is more superior to the tred of the other two algorithms. For example, whe the k =, CkNNDN algorithm is basically the real-time acquired result, which is because through the defiitio of the Vorooi diagram, the earest eighbor of the queries poit ca directly be obtaied, amely, to determie whether the road segmet has itersectio with the polygo. The dyamic chages of the road etwork, through the optimizatio treatmet to the chages of the weight value of the road segmets, ca also quickly acquire the result of the earest eighbor of the queries poit at the ew weight. With the icrease of the k value, we ca see from the figure that the growth speed of the respose time of IMA algorithm is far greater tha that of CkNNDN algorithm, as CkNNDN algorithm is ot blidly the expasio of searches, but makig full use of the earest eighbor iformatio of the pre-populated Vorooi diagram to carry out the search, so as to decrease the umber of access to the road etwork odes, thus improve the respose time to the queries k value DNCKNN IMA Figure 3. The Compariso of the Queries Respose Time of Differet k Values 3.5. The Ifluece of the Queries Distace We compared the ifluece of differet queries distace to the queries performace of the algorithm, Figure compares the differece of the performace of CkNNDN algorithm ad IMA algorithm at k=3, uder differet queries distace from 3 km to 5 km. As ca be see from the figure, with the icrease of the queries distace, the access umbers of the poits of iterest for both algorithms have icreased, which is because as the queries distace icreases, the algorithm obviously eeds to ru more queries o the poits of iterest. Whe the queries distace icreases from 3 km to 5 km, which is icreased by 5 times, the access umbers of CkNNDN algorithm to the poits of iterest have also icreased from 5 times to 90 times, icreased by 6 times. It ca be see that the loger the queries distace of IMA algorithm is, the tred of the icrease of the access umbers to the poits of iterest is greater. We experimet the queries distace rage of four kids of situatios 3 km, 5km, 0 km ad 5 km respectively, ad the IMA access umbers of the poits of iterest are all sigificatly higher tha the access umbers of the CkNNDN algorithm, ad also shows that with the icrease of the queries distace, the access umbers of the two kids of algorithms demostrate the tred of eve wider differece km 5km 0km 5km DNCKNN IMA Figure. Compariso of Two Algorithms with Differet Queries Legth (k=3) 359

7 Resose Times(Secods) 3.6. The Ifluece of the Desity of the Poits of Iterest I this sectio, we studied the ifluece of the desity of poits of iterest o the queries, the desity of poits of iterest refers to the umber of the poits of iterest distributed uder the same area. Figure 5 shows the differece of the performace betwee CkNNDN algorithm ad IMA algorithm i the queries of differet desity poits of iterest, i which the desity of the restaurats (68), hotels (9), hospitals (03), ad parks (55) declies i tur. For IMA algorithm, with the decrease of the desity of the poits of iterest, the respose time icreases gradually. For CkNNDN algorithm, with the decrease of the desity of poits of iterest, the respose time reduces gradually, ad i all cases i the experimet, CkNNDN algorithm is better tha the IMA algorithm, ad the smaller the desity of poits of iterest is, the more differece the performace of the algorithm shows. The reaso that causes the tred i the figure is because the mechaisms of the two kids of algorithms are differet, IMA algorithm performs queries o the adjacet roadsi tur, ad the more the poits of iterest are, the greater the possibility there is to obtai the result. However, CkNNDN algorithm is maily based o the pre costructed Vorooi diagram, the less the poits of iterest are, the less layers of the Vorooi diagram of the poits of iterest idex tree hierarchical structure has, the faster the queries speed will be, ad with the icrease of the idex tree hierarchical structure layers, the queries speed has also bee affected Restaurat Hotel Hospital Park DNCKNN IMA Figure 5. The Compariso of Queries Performace of Differet Desity of Poits of Iterest. CONCLUSION I this paper, we proposed a cotiuous k earest eighbor queries algorithm (CkNNDN algorithm) i the dyamic road etwork, which ca deal with the real-time updated road iformatio data, ad avoid road coditios such as traffic jam, so as to achieve the result of the optimum k earest eighbor i real time. CkNNDN algorithm ca fid out the dividig poit i the cotiuous queries path with high efficiecy, acquire the prelimiary result of the cotiuous k earest eighbor, through the real-time updated road segmet weight, ad make adjustmet to the k earest eighbor result set, so as to esure that the k earest eighbor is always effective i real time. Based o the detailed aalysis of four kids of situatios of the road segmet weight chages, accordig to whether the chaged segmet is k earest eighbor segmet, ad whether the weight icreases or decreases, differet methods of optimizatio are preseted respectively. Through the experimetal compariso with IMA algorithm that supports the dyamic road etwork, it ca be see that CkNNDN algorithm i a variety of experimetal coditios (differet k values, differet queries legth, differet desity of poits of iterest, differet road segmet update frequecy) is superior to IMA algorithm. Ackowledgmets The work was supported by the Natioal Natural Sciece Foudatio of Chia (Grat 305). REFERENCES CheemaM. A, ZhagW, LiX, ZhagY,LiX. (0) Cotiuous reverse k earest eighbors queries i euclidea space ad i spatial etworks, The VLDB Joural, (), pp Cho H. J, Chug C. W. (005) A efficiet ad scalable approach to CNN queries i a road etwork, Proceedigs of the 3st iteratioal coferece o Very large data bases, pp Cho H. J, Ji R, Chug T. S. (05) A collaborative approach to movig k-earest eighbor queries i directed ad dyamic road etworks, Pervasive ad Mobile Computig,7, pp Fa P, Li G, Yua L. (0) Cotiuous K-Nearest Neighbor processig based o speed ad directio of movig objects i a road etwork, Telecommuicatio systems,55(3), pp Huag X, Jese C. S, Šalteis S. (005) The islads approach to earest eighbor queryig i spatial etworks, Lecture Notes i Computer Sciece, 3633, pp Huag Y. K, Che Z. W, Lee C. (009) Cotiuous k-earest eighbor query over movig objects i road etworks, Advaces i Data ad Web Maagemet, 56, pp. 7-38,. Kim H. I, Chag J. W. (03) k-nearest Neighbor Query Processig Algorithms for a Query Regio i Road Networks, Joural of Computer Sciece ad Techology,8(), pp

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