An Adaptive pdating Prtcl fr Reducing Mving Object Database Wrlad Su Chen Beng Chin Oi Zhenjie Zhang Schl f Cmputing Natinal niversity f Singapre {chensu,ibc,zhenjie}@cmp.nus.edu.sg ABSTRACT In the last decade, spati-tempral database research fcuses n the design f effective and efficient indexing strucres in supprt f lcatin-based queries such as predictive range queries and nearest neighbr queries. While a variety f indexing techniques have been prpsed t accelerate the prcessing f updates and queries, nt much attentin has been paid t the updating prtcl, which is anther imprtant factr affecting system perfrmance. In this paper, we prpse a generic and adaptive updating prtcl fr mving bject databases with less number f updating messages between the bjects and database server, thereby reducing the verall wrlad f the system. In cntrast t the apprach adpted by mst cnventinal mving bject database systems where the exact lcatins and velcities last disclsed are used t predict their mtins, we prpse the cncept f Spati-Tempral Safe Regin t apprximate pssible fure lcatins. Spati-tempral safe regins prvide larger space f tlerance fr mving bjects, freeing them frm lcatin and velcity updates as lng as the errrs remain predictable in the database. T answer predictive queries accurately, the server is allwed t prbe the latest stas f sme mving bjects when their safe regins are inadequate in rerning the exact query results. Spati-tempral safe regins are calculated and ptimized by the database server with tw cntradictry bjectives: reducing update wrlad while guaranteeing query accuracy and efficiency. T achieve this, we prpse a cst mdel that estimates the cmpsitin f active and passive updates based n histrical mtin recrds and query distributin. We have cnducted extensive experiments t evaluate ur prpsal n a variety f ppular indexing strucres. The results cnfirm the viability, rbustness, accuracy and efficiency f ur prpsed prtcl.. INTRODCTION Spati-tempral databases, especially Mving Object Databases (MOD), are well sdied in the database cmmunity. Efficient disresident indexing strucres [, 9,,, 9, ] have been prpsed t supprt different types f queries fr lcatin-based services. Tw typical example queries are Which are the vehicles remaining in Central Par after minutes? r Wh will be the Permissin t cpy withut fee all r part f this material is granted prvided that the cpies are nt made r distributed fr direct cmmercial advantage, the VLDB cpyright ntice and the title f the publicatin and its date appear, and ntice is given that cpying is by permissin f the Very Large Data Base Endwment. T cpy therwise, r t republish, t pst n servers r t redistribute t lists, requires a fee and/r special permissin frm the publisher, ACM. VLDB, September -7,, Singapre Cpyright VLDB Endwment, ACM ----//. Figure : Architecre f a mving bject database system pliceman clsest t City Hall after minutes?. These queries can be frmalized with range query and nearest neighbr query n predicted lcatins f bjects mving in a tw-dimensinal (D) space. A mving bject database eeps trac f all bjects by receiving ccasinal lcatin disclsures frm bjects. In the meantime, the database server als answers all incming predictive queries. Fig. shws a typical architecre f a mving bject system. While mst existing sdies n mving bject databases fcus n index and query efficiency, less effrt has been made t address the issue n the updating mechanism/prtcl, which is anther crucial factr affecting the system wrlad and perfrmance even mre. Cnsidering the limited rm f technical advances n indexing strucres and query ptimizatin, it is nw imprtant t recnsider the design f the updating prtcl. This paper is the first attempt n a systematic investigatin n this pssibility. In particular, we prpse a new adaptive updating prtcl, which reduces the bject updating frequencies by maintaining nly apprximate mtins instead f exact nes in the database system. The design f the prtcl is n the basis f a careful analysis n the advantages and disadvantages f the existing prtcls. The first generatin f mving bject databases tracs bject lcatins nly. In particular, an bject reprts its lcatin t the database server every (lgical) timestamp. It is inadequate t mae predictin with lcatin infrmatin nly. Later, in rder t supprt predictive queries and reduce update frequency, mtin mdels are intrduced t apprximate bject mvements. Fr example, a linear mdel with bject s lcatin and velcity can be used t predict bject s lcatin at any fure time, assuming the bject mves with a fixed speed alng a straight line. Currently, bject updating prtcls fllw sme simple strategies, such as Tempral Bunded Strategy and Spatial Bunded Strategy. In the temp-
q q ral bunded strategy, mving bjects update their mtin mdels peridically after a fixed time interval. This scheme is neither effective nr efficient. An update is issued even if the mdel des nt change since the last update while the mvement predictin immediately becmes meaningless when the bject changes its velcity dramatically sn after the last update. On the ther hand, the spatial bunded strategy [] adaptively decides the update time, depending n the spatial errr incurred by the mvement mdel cnstructed n the previus update. Specifically, tw mtin mdels are stred n bth the bject and the database server. The mtin mdel at the bject side is always updated with current mvement, which is suppsed t be mre accurate. The bject regularly measures the errr f the mtin mdel which has been sent t the server previusly []. An immediate update is issued if the predictin errr is larger than sme specified threshld. In Fig., we present an example f the spatial bunded strategy. In the figure, slid pints (hllw pints resp.) represent the predicted lcatins f the ld mdel (new mdel resp.). When the distance between the slid pint and hllw pint in the near fure exceeds the tlerance, the bject updates its mvement mdel with the database server. T ensure the accuracy f predictive query, the database server is allwed t actively request the mdel update frm the bject, when the candidate bject des nt necessarily satisfy the query cnditin. Cmpared with the spatial bunded strategy [] intrduced abve, ur new generic updating prtcl is equipped with three majr feares t enhance the perfrmance. First, we adpt the linear mvement mdel instead f the cmplex high-rder mdels, which can be seamlessly integrated int database system with ptimized index strucres []. While achieving benefits n cmputatinal cst and index efficiency with the simple mtin mdel, ur empirical sdies shw that there is n significant difference n predictin quality even when mre cmplicated mtin mdel is emplyed instead. Secnd, ur prtcl allws apprximatin n bject mtin in bth spatial and velcity spaces, prviding better flexibility n the ning f the updating prtcl. In particular, ur prtcl relies n the new cncept f Spati-Tempral Safe Regin, r in shrt, which is a rectangle in spati-tempral space bunding the pssible lcatin and velcity. In Fig., we illustrate the basic idea f ur framewr fllwing the example in Fig.. Our system assigns sme rectangle t the bject as spatial safe regin, as well as sme maximal and minimal speeds as velcity safe regin. A series f rectangles, called Predicted Regins, can then be derived t bund the pssible lcatins f the bject n subsequent timestamps. An update is necessary if: ) the lcatin f the bject cannt be bunded by the predicted regin at the current timestamp, r ) the mvement inferences n the current lcatin and velcity vilate sme predicted regins in the fure. In Fig., the bject is safe at time t + since it stays in the predicted regin and its updated mvement predictin remains bunded by the fllwt q q q Figure : Example f spatial bunded strategy Figure : Example f spatial-tempral safe regin ing predicted regins. At time t +, althugh its new lcatin is still within the predicted regin, updating is inved t ensure that the predictin errr n the server side remains within a tlerable bund. Third, ur updating prtcl taes bth histrical mtin recrds and query distributin int cnsideratin, rendering a cst ptimizatin mdel and an autmatic ning mechanism highly adaptive t the changing wrld. Recall the example in Fig.. Given the lcatins f the recent queries shwn in the figure, {q, q, q, q, q }, careful design n the avids the ptential verlaps between the predicted regins and ppular querying areas in the spatial space. This enables ur framewr t utperfrm existing slutins n the maximizatin f update savings. We summarize the cntributins f this paper as fllws:. We present a generic and adaptive updating prtcl fr the purpse f reducing the number f update messages while guaranteeing the efficiency and accuracy f predictive queries.. We prpse a cst mdel t estimate the update wrlad incurred by specific mving bject(s).. We prpse cst-based ptimizatin strategies t reduce the updating frequencies.. We evaluate the perfrmance f ur prpsal with a variety f index strucres.. PRELIMINARIES Assume there are n mving bjects, i.e., O = {,,..., n}, being mnitred in the system. Fllwing the cmmn assumptin n spati-tempral indexing, the time axis is sliced int snapshts at discrete times, i.e., T = {,,..., t,...}. The exact physical lcatin f bject i at timestamp t is dented by a vectr li t = (li.x, t li.y). t Similarly, the velcity f i at t is dented by vi t = (vi.x, t vi.y). t With the linear mvement mdel, the predicted lcatin f i at time s t is estimated as pli s, i.e., pli s.x = li.x t + vi.x t (s t) and pli s.y = li.y t + vi.y t (s t). Befre delving int the details f ur updating prtcl, we first intrduce the cncept f Spati-Tempral Safe Regin, as defined belw. DEFINITION.. Spati-Tempral Safe Regin () Given a mving bject i, a Spati-Tempral Safe Regin () fr i is represented by a ple R( i) = (, V R, t r, t e), where is a rectangle in the physical space (i.e., the space where the bject mves), V R is a rectangle in the velcity space, t r is the reference time, and t e is the expiry time. Given an R( i), the spatial rectangle is bunded by.x,.x,.y,.y n the tw dimensins respectively. Similarly, the velcity rectangle V R f R( i) is bunded by
6 y Q 6 y t l t l R ( ) x 6 t l R ( ) x 6 Figure : Example f Figure : Examples f cnsistency verificatin V R t r t e R( ) [.,.] [.,.] [.,.] [, ] R( ) [, ] [, 6] [., ] [., ] R( ) [, 6] [, ] [, ] [.,.7] Table : Details n the in Fig. V R.x, V R.x, V R.y, V R.y. Initively, relaxes the lcatin f i at reference time t r, and V R apprximates the pssible velcities f i between t r and t e. A predicted regin, as defined belw, infers the pssible lcatins f i at time t befre the expiry time t e. DEFINITION.. Predicted Regin Given an R( i) = (, V R, t r, t e) and inferring time t (t r t t e), the predicted regin f i at time t, P t i = P.x, P.x P.y, P.y, is the maximal spatial rectangle expanded frm with respect t V R, where P.x =.x + V R.x (t t r) P.x =.x + V R.x (t t r) P.y =.y + V R.y (t t r) P.y =.y + V R.y (t t r) The definitin abve assumes that the inferring time t is n earlier than the reference time t r, which can be easily relaxed. When t < t r, the predicted regin is calculated with the reverse velcity bunding rectangle, which is V R.x, V R.x V R.y, V R.y. The predicted regin is P.x =.x + V R.x (t t r) P.x =.x + V R.x (t t r) P.y =.y + V R.y (t t r) P.y =.y + V R.y (t t r) An R( i) = (, V R, t r, t e) is cnsistent with the mving bject i at time t t e, if bth f the fllwing tw cnditins hld: ) Current lcatin l t i remains in the predicted regin P t i inferred frm R( i); and ) The predicted lcatin pl s i remains in the predicted regin P s i fr any t < s t e. Nte that the cnsistency nly depends n the lcatins, which remains valid even when the velcity f i at time t is ut f the velcity rectangle V R. As an example, Tbl. lists the s f three bjects {,, } which are illustrated in Fig.. Based n Definitin. and., we can derive the predicted regins f the bjects at timestamp t =, as shwn in Fig.. Accrding t the abve definitins, it is straightfrward t verify the cnsistency f the lcatin with the given n the client side, i.e., the mving bject, by simply checing the predicted lcatins f the bject at every timestamp befre the expiry time t e. Since we adpt the linear mdel n deriving the predicted regins, the verificatin prcess can be simplified by checing the predicted lcatin (r exact current lcatin) at nly three timestamps. In particular, the bject first tests if the current timestamp is beynd the expiry time t e. If the has already expired, the algrithm rerns a negative answer immediately. Otherwise, the bject then checs if the current lcatin li t and the predicted lcatin pl te i are bth cvered by the predicted regins Pi t and P te i respectively. Finally, if the current timestamp t is ahead f the reference time t r, we als need t see if the predicted lcatin at t r is adequately cvered by the lcatin rectangle. The verificatin algrithm is summarized in Algrithm in Appendix B. In Fig., we give three examples t shw why the cnditins abve are sufficient t prve the validity f a. Tw s, R( ) and R( ), are shwn in the figure. The slid dts dente the current lcatins and hllw dts represent the predicted lcatins. The reference time f R( ) is the current timestamp. Since the current lcatin l t is inside the lcatin rectangle and the predicted lcatin is cvered by the predicted regin, R( ) remains valid. The reference time f R( ), n the ther hand, is after the current timestamp. The predicted regins between t and t e are thus extended bacwards accrding t the lcatin predictin frmula. If we test nly the lcatins f the current timestamp and expiry time, sme false psitive may wrngly pass the verificatin, e.g., the predictin based n l. t Hwever, when the lcatin at the reference time is als verified, all false psitives are pruned, as the mvement predictin at l t implies. In this paper, we fcus n the prcessing f predictive range queries. Given a querying rectangle QR in lcatin space and the querying time t q, the predictive range query rerns all the bjects with predicted lcatins in QR at time t q. Althugh we cnstrain ur discussin in range query thrughut the paper, it is easy t see that ther queries, e.g., -nearest neighbr query, can be answered with a series f range queries [9].. PDATING PROTOCOL Generally speaing, ur updating prtcl cnsists f tw types f updates: the active update and the passive update. In the database, each bject i is always assciated with ne (and nly ne)
R( i), which is ept in the database as well as the memry f the client device. In the fllwing, we discuss the tw updates in detail. Active pdate On each timestamp, the bject i checs if the new mtin predictin mdel with its current lcatin and velcity remains cnsistent with its R( i), using Algrithm. If there is any incnsistency, it issues an active update t the database server cnsisting f its current lcatin and velcity. At the server side, the database system cntinuusly listens t any incming active updates frm the bjects. If ne f the mving bjects i updates its lcatin and velcity at time t, the system renews the recrd f the bject in the database. The system then calculates a new fr i based n the the updated recrd. The new is sent t the bject i, while the crrespnding recrd in the database is als refreshed accrdingly. An utline f the update prcedure can be fund in Algrithm in Appendix B. Query Prcessing and Passive pdate While active updates are initiated by the bjects themselves, passive updates are issued when the database prcesses predictive range queries. Typically, predictive range queries in mst mving bject databases are prcessed using a filter-and-refine apprach, which determines candidate bjects based n their predicted regins and verifies them by prbing passive updates if necessary. A candidate set is cnstructed first by retrieving all bjects verlapping with the query range QR at query time t q, based n the predicted regins f the s. Fr each candidate bject i, if the predicted regin f i is cvered by the query range, the bject can be safely included int the query result. Otherwise, a request is sent t the bject fr an update n its current lcatin and velcity, which will be used n the server side t mae a mre accurate predictin. The bject is subsequently listed in the query result if the new predicted lcatin is still in the query range. A general framewr fr answering range queries based n the cncept f s is presented in Algrithm in Appendix B. Let us recall the example shwn in Fig., and see hw predictive range queries can be answered with s. If a range query is issued in the rectangle regin QR = [.,.] [.,.] at querying time t q =, the predicted regins can be calculated accrding t the inference equatins abve. Fr bject, fr example, the predicted regin at timestamp t =, P, is the rectangle [, ] [,.]. Since P is cvered by the query regin cmpletely, is a psitive result if it remains cnsistent with its current. On the ther hand, there is n verlap between P, implying that is a negative result. nlie the ther r, the case f is mre cmplicated, since the predicted regin P partially verlaps with QR. T clarify if is in the query result r nt, the system sends an update request t fr its current mtin parameters, i.e., the lcatin and velcity. T put the updating prtcl in use, there are tw issues t reslve. First, t minimize the wrlad f the system, the calculatin algrithm plays an imprtant rle in finding an ptimal (Step in Algrithm ). Recall the example in shwn Fig., which demnstrates that verall update cst, including bth active and passive updates, can be reduced if we extend the predicted regins as much as pssible withut t many verlapping query ranges. We will expand n this idea with further details in Sec.. and the crrespnding ptimizatin techniques fr better design in Sec... Secnd, cnsidering different index strucres in mving bject databases, bjects are stred and searched in different ways, even with the same linear mtin mdel. Therefre, it remains unclear s far hw existing database systems supprt the search fr all bjects whse crrespnding predicted regins verlap with the query range. This is imprtant in query prcessing fr efficient retrieval f candidate bjects in the filter step (Step in Algrithm ). In Sec.C in the appendix, we answer this questin by shwing that it des nt tae much effrt t mdify existing mving bject indexing strucres t supprt these queries.. OPTIMIZATION THNIQES. Cst Mdel In this sectin, we present a cst mdel estimating the prbable validity f a given. As intrduced in Sec., there are tw types f updates, namely Active pdate and Passive pdate. Either active update r passive update leads t a new. We use P a(r( i)) and P p(r( i)) t dente the prbabilities f active update and passive update happening n R( i) befre the expiry time t e. An remains valid until the expiry time t e with prbability: P valid (R( i)) = ( P a(r( i))) ( P p(r( i))) () Obviusly, a gd shuld maximize P valid. Initively, a larger and a larger V R lead t lwer prbability f P a(r( i)) but higher prbability f P p(r( i)), and vice versa. T ptimize Equatin, it is imprtant t estimate bth prbabilities first. Active pdate Prbability An active update is issued by bject i if the previus is n lnger cnsistent with the current lcatin and velcity. Given an R( i), P a(r( i)) is the prbability f incnsistency happening befre the expiry time t e f R( i). Withut nwing the exact fure trajectry f i, it is hard t estimate P a(r( i)). If we assume the previus mtin mdel desn t change, R( i) will always be valid until t e; n the ther hand, it is hard t indicate the pssible changes in the mtin withut any additinal nwledge n the fure trajectry. Hwever, if all the similar histrical trajectries are recrded in the database, the accumulated statistical infrmatin prvides prbability estimatin n the active update. nfrnately, this slutin is impractical due t the high cst in bth strage and prcessing n the trajectries. T facilitate effective and efficient statistical estimatin, the database system maintains a set f update recrds, as defined belw, t simulate the histrical trajectries f bjects. DEFINITION.. pdate Recrd An update recrd is a ple (R( ),, l, v, ) where is the identity f the assciated mving bject, t u is the time when the update recrd is generated, l and v are the lcatin and velcity f at t u, and R( ) = (, V R, t r, t e) is the latest f befre the update time t u. The update recrds are maintained in a separate table in the database, called the pdate Recrd Table. A recrd is inserted int the table when: ) an update frm is received due t the vilatin n R( ), r ) the previus R( ) expires. In the first case, the lcatin and velcity at update time are written int the recrd. In the secnd case, NLL values are inserted instead. This implies that each issued in the past has a recrd ept by the database system in the update histry table. Next, we intrduce the cncept f Recrd Cverage t evaluate the rbustness f a new with respect t similar update recrds in the update recrd table. Specifically, an R( i) cvers an update recrd = (R( ),, l, ), if, v
y.... x... vy R ( ). R( ). i i.... v.. v... Figure 6: Cverage f n update recrds.. v. v vx. ) R( i). cvers R( )., and ) R( i).v R cvers bth R( ).V R and v. Withut ambiguity, we use R(i) t dente the cverage relatinship, in which is a specific update recrd. In Fig.6, fr example, we present an example n the cverage relatinship with an R( i) and fur update recrds {,,, }. The lcatin rectangle f the update recrds are shwn with blac thin lines in the spatial space n the left. Similarly, the velcity rectangle V R and the velcity at update time v are pltted in the velcity space n the right. Given the R( i) mared with red thic lines in bth spaces, R( i) cvers by the definitin abve. is nt cvered by R( i) since the updated velcity v f is ut f the velcity rectangle f R(i). If the R( i) cvers an update recrd, it is able t remain cnsistent until the expiry time, if i fllws the same trajectry f when was recrded fr. This mtivates the fllwing definitin f Cverage Rate t apprximate the prbability f an active update n the new R( i). DEFINITION.. Cverage Rate Given an R( i) and a grup f similar update recrds NN, the cverage rate f R( i) is measured by { i NN R( i ) i } NN We nw discuss the similar update recrd set NN as shwn in the abve definitin. T get all update recrds related t the current mving bject i, the system retrieves all update recrds in NN, if the lcatin rectangle i. and velcity rectangle i.v R f the recrd i cver the lcatin l tr i time t r respectively. As a summary, we have P a(r( i)) = Passive pdate Prbability and velcity v tr i {i NN R(i) i} NN at reference A passive update is issued when it is nt sufficient t decide if an bject meets the query with its current stred in the database, i.e., the predicted regin partially verlaps with the query regin. T estimate the number f passive updates fr a given R( i), it is necessary t predict the prbability f the event f partial verlap. T simplify the mdel and save cmputatinal cst, we relax the prbability by including any verlap event even if the predicted regin is cmpletely cvered by the query range. This relaxatin des nt greatly affect estimatin errr since the query range is usually nt large enugh t cver many predicted regins. Fllwing the existing assumptins n the perfrmance analysis f range queries [,, 9], we assume the querying lcatin and querying time fllw the unifrm distributin in spatial space and tempral space. The prbability f a predicted regin verlapping any range query at time t is thus prprtinal t the vlume f the predicted regin. Fr an R( i) issued at update time t u, the ttal vlume f all predicted regins at timestamps between t u and t e is dented by V l(r( i)). By using the fllwing ntatins t replace the side lengths f lcatin rectangle and velcity rectangle, i.e., LD x =.x V R.x, LD y =.y V R.y, V D x = V R.x V R.x, and V D y = V R.y V R.y, the ttal vlume can be further simplified: V l(r( i)) = t e t=t u (LD x + V D x(t t r)) (LD y + V D y(t t r)) If the expected number f queries happening at each timestamp is N and the vlume f the whle spatial space is S, the prbability f nt meeting any range query is apprximated by the rati f ttal vlume with respect t the expected query vlume. P p(r( i)) = max V l(r(i)) N, (t t r)s. Calculatin f Optimal In this sectin, we present an ptimizatin methd t find an R( i) t minimize the expected update cst. Given the cst mdel presented abve, the estimatin n the active update prbability P a(r( i)) depends n the number f update recrds cvered by R( i). This implies that there are nly NN different values fr the pssible active update prbability fr R( i). Each pssible value is assciated with a grup f cvered recrds. This mtivates ur ptimizatin technique f mdeling the recrd cvering with a series f discrete events. T find the ptimized, an initial R( i) is first created with minimal and minimal V R cvering nly l t i and v t i f i respectively. The ptimizatin prcedure executes iteratively. In each iteratin, it tries t expand the t cver ne mre update recrd frm the remaining uncvered recrds in NN. If the estimated update cst des nt further decrease after sme iteratins, the ptimizatin prcedure stps and rerns the final. The detailed algrithm fr the ptimizatin prcedure is presented in Algrithm in Appendix B. In Fig.7, we illustrate an example f the ptimizatin algrithm, using the data shwn in Fig.6. The red square pints are the lcatin and velcity f the bject i at time t r. At the beginning f the algrithm, the R( i) is initialized with the minimum square cvering the red squares in bth spaces. Since the inclusin f any update recrd has the same reductin effect n P a(r( i)), the ptimal update recrd t cver next acally has the minimum increase n the passive update prbability P p(r( i)). By testing all update recrds, is selected accrding t the selectin criterin. The R( i) grws in bth spatial and velcity spaces t cver the update recrd, as shwn in Fig.7(a). In the secnd iteratin, as shwn in Fig.7(b), the update recrd is piced since the decrease n P a(r( i)) is still larger than the increase n P p(r( i)). The algrithm terminates after the secnd iteratin, when there is n ther expansin that can further reduce the estimated cst. Nte that this algrithm wrs in a greedy manner. Hence, it des nt guarantee cnvergence t the glbal ptimum. The retrieval f the similar update recrd set NN discvers all update recrds cvering bth the lcatin and velcity f the current bject. T efficiently supprt such a retrieval prcess, an index is built n the update recrds with respect t their lcatin rectangle and velcity rectangle. Given the D index strucre, NN is simply retrieved with the issuance f a pint query at lcatin l t i and velcity v t i. The cmputatin f P p(r( i)) taes cnstant time since the ttal vlume V l(r( i)) can be summed up quicly by the frmula. In each iteratin, all the remaining update recrds are
y y............ vy.. x. v. (a) In first iteratin.... vy. x. v. (b) In secnd iteratin. v.. v........ Figure 7: Example f ptimizatin algrithm. v. v. v vx.. v vx. tested in sequence. This leads t O(m ) cmplexity in the wrst case, if m update recrds are retrieved frm the D index strucre.. Reducing Cmputatin Cst The calculatin algrithm runs in quadratic cmplexity in terms f the number f update recrds. Thus, the cmputatin cst n the s can be very high if every single bject update runs the cnstructin methd. T reduce cmputatin cst, there are tw simple methds, namely Static, and Glbal Dynamic. T distinguish frm the basic strategy with independent cmputatin f fr each bject, we call the basic slutin Persnal Dynamic. With static strategy, there is a grup f fixed parameters { l, v, t}. l and v are rectangles in the spatial and velcity space, cvering the rigins, respectively. t is a psitive cnstant value that specifies the length between reference time and expiry time. Fr any updating mving bject i with lcatin l t i, velcity v t i and time t, the lcatin rectangle fr the R( i) fr i is cmputed by mving l aligning l t i with the rigin, i.e.,.x = l t i.x + l.x.x = l t i.x + l.x.y = l t i.y + l.y.y = l t i.y + l.y Similarly, the velcity rectangle V R in R( i) is als cnstructed by expanding the velcity with margins in v n bth dimensins. The expiry time f R( i) is t + t. This strategy is suppsed t incur minimal cmputatin cst, since the parameters are never updated after the specificatin at the beginning. As an example, if the parameter set is { l = (, ) (, ), v = (.,.) (.,.), t = }, bject i updates at time t r = with lcatin li t = (, ) and velcity vi t = (, ), the new R( i) is cnstructed with = (9, ) (, 6), V R = (.8,.) (.9,.) and t e =. With the strategy f glbal dynamic, each glbal parameter set { l, v, t} is valid nly in an interval n the time axis. There is ne and nly ne valid glbal parameter set at any timestamp t. The bject updates are handled with the glbal parameters valid at the update time, as is dne with the static strategy. There is sme cmputatin cst incurred by this strategy t calculate a new parameter set when the previus glbal parameter set is expiring. The cmputatin n parameter re-cmputatin can be run ffline when the system has free CP cycles fr use, nt affecting the perfrmance f the database system. In Glbal Dynamic Strategy a glbal expansin plan is designed and updated frm time t time, with evlutin n the index strucre. T mae Algrithm applicable n the search f ptimal expansin plan, we present sme mdificatins t the riginal algrithm. In the riginal algrithm, the update recrds cvering the updating lcatin f the current bject, NN, are first retrieved. On the glbal expansin ptimizatin algrithm, hwever, there des nt exist such similar update recrd set since the ptimizatin is nt lcatin dependent. Thus, instead f finding a grup f update recrds in the table, all recrds are utilized in the glbal parameter search by aligning all f the centers f s in the update recrds t the rigin. After the alignment peratin, a viral mving bject is created with lcatin at the rigin and the velcity at the average speeds f the bjects n bth dimensins. The riginal algrithm is then applied n the viral bject and the aligned recrds. The final expansins n bth spatial and velcity spaces are recrded and stred in the parameter set { l, v, t}.. EXPERIMENTAL EVALATION We nw reprt experimental results that evaluate effectiveness and efficiency f the based updating prtcl. We first sdied the perfrmance f the basic prtcl under different parameters and then cmpared it with anther state-f-art update mechanism. Finally, we evaluated the perfrmance f different strategies n different indexes. Experimental Settings: Three surces f real and semi-real datasets were used in ur experiments: [] is a real dataset prvided by R-tree Prtal[], which cntains trajectries f 76 trucs mving in Athens metrplitan area (see Fig.6(a) in Appx.D.). The trucs update at a rate f sec. [] is anther real dataset as a part f the e-curier datasets[]. e-curier eeps trac f the mvement f all its curiers all ver K (see Fig.6(b) in Appx.D.). The curiers reprt their lcatins (GPS recrds) every sec. We crawled e-curier fr ne wee and extracted 87 bjects (i.e., trajectries) that mved nnstp fr ver ts. [] Due t the lac f large real mving bject datasets, we used Brinhff generatr[] t generate a set f synthetic mvements based n the real rad map f Singapre (see Fig.6(c) in Appx.D.). We generated datasets f different sizes and used the ne cntaining K bjects by default. In Appx.D., Tbl. lists the specificatins f the three data surces abve, including the data space, maximum bject speed and the mapping frm physical time t lgical time (a timestamp). The query lad f the experiments cnsists f a given number f predictive range queries. Each f these range queries is square-sized, with a preset side length. Since the datasets differ in data space size, we use qlen t represent the percentage f the side length f the query ver the length f the entire data space. In the experiments, qlen is varied frm.% t % (e.g., m t 8m fr datasets). Queries fllw the same distributin as the bjects. Specifically, the lcatin f a randmly piced bject is used as the center f the query. The average predictive time f queries varies
Precisin.9.8.7.6.. 6 6 6 δ l (a) Query precisin Recall.9.8.7.6.. 6 6 6 δ l (b) Query recall # f regular updates 7 6 6 6 6 δ l (c) # f active updates # f updates 7 6 6 6 6 δ l (d) # f all updates Figure 8: Effect f δ l n perfrmance # f updates 7 6 # f updates 8 7 6 # f updates 8 7 6 # f updates 8 7 6 6 6 6 6 6 6 δ v δ v (a) # f active updates (b) # f all updates Figure 9: Effect f δ v n perfrmance 6 6 6 6 6 6 δ t δ t (a) # f active updates (b) # f all updates Figure : Effect f t n perfrmance frm ts (current query) t 6ts. The query frequency qfqy varies frm t 6, which means the crrespnding wrlad cntains. t queries per timestamp. Tbl. in Appx.D. summarizes the parameters and their values used in the experiments, where default values fr variable parameters are shwn in bld. All the prgrams are implemented in C++ and run n a PC with.ghz Intel Cre Du CP,.GB RAM and GB SATA dis. Sdy f the basic strategy We first sdied the perfrmance f the basic strategy withut regard f the underlying index. We investigated the effect f the three elements f an : the spatial and velcity rectangles l and v; the tempral length t, i.e., the length f time between expiry time and reference time. A static glbal parameter set is used in each experiment in this sectin, where l.x = l.y = δ l l.y = l.y = δ l v.x = v.y = δ l v.y = v.y = δ l The values f δ l, δ v, and t are listed in Tbl.. Fig.8-Fig. shws the effect f δ l, δ v and t n update wrlad and the quality f query result. It is wrth nting that query precisin and recall are nt affected by the size f s. During query prcessing, an bject is added t r excluded frm the results if its predicted regin is fully cntained r disjint with the query regin. Otherwise, a passive update is inved. The query precisin and recall are primarily decided by the predictability f data itself, i.e., predicting based n the mtin f the bject at query issuing time. As δ l increases, meaning a larger initial spatial regin, the number f active updates decreases at the expense f mre passive updates, resulting in an increase in the ttal number f updates. As shwn in Fig.9, δ v has a similar effect n update times. With a larger δ v, the predicted regin expands faster. is mre liely t enclse the lcatin f the bject and fewer active updates are incurred. On the ther hand, a larger predicted regin has higher prbability t intersect with the query regin and mre passive updates are required as a result. The tempral length f s t affects the update perfrmance differently. The number f active updates and the ttal number f updates bth decrease with a lnger tempral length t. When t is lnger than 6ts, the number f updates d nt change much with t. Fr a small t, the expires quicly, resulting in a large number f active updates. Mtin Functins We next investigated the effect f different update plicies n update and query perfrmance. We cmpared the update plicy with the recursive functin mdel prpsed tgether with the -tree[]. A client (bject) eeps h histrical recrds and derives a recursive mtin functin frm the h recrds. A D dimensinal plynmial functin is used t apprximate the recursive mtin functin. Fr an update, the plynmial functin is sent t the server and the system can predict bject lcatin using the plynmial functin. An active update is issued if the distance between the current lcatin f the bject and the lcatin cmputed frm the last reprted plynmial functin is larger than an errr bund de in the next H timestamps. In ur experiments, h, D, de and H are set t 8,, 6m and ts respectively. The query prcessing f the is similar t that f the, i.e., the system ass fr an update (passive) if it cannt determine if the bject is inside the query regin r nt. Fig.- Fig. shws the results n and datasets while varying the query predictive time qpdt. Fr bth and, the methd results in a higher query precisin, while the plicy has a higher query recall. The differences in precisin and recall grw with query predictive time. On, when qpdt = 6, the precisin f is abut % less than that f, hwever, the recall is abut.6 times f that f. On, when qpdt = 6, the difference in query precisin is less than., while difference in query recall is as large as.. Cnsidering the update lad, the number f active updates are less affected by the query predictive time fr bth methds and the effectively reduces the number f active updates. The number f ttal updates increases with the query predictive time. Since the predicted regin is larger with lnger predictive time, the bjects are mre liely t issue passive updates. When qpdt = 6, the update times f plicy is % less than that f. When qpdt = 6, n, the update times f is slightly higher than that f. As shwn in Tbl., ne timestamp f () crrespnds t sec(sec). We als tested n datasets and investigated the effect f query side length, query frequency and etc. The results are mitted due t space limitatin, which can be fund n [6].
.8.8.8.8 Precisin.6. Precisin.6. Recall.6. Recall.6.. 8 6 6 (a). 8 6 6 (b) Figure : Effect f predictive time n precisin. 8 6 6 (a). 8 6 6 (b) Figure : Effect f predictive time n # f active updates # f updates (K) 8 6 6 (a) # f updates (K) 6 8 6 6 (b) Figure : Effect f predictive time n # f active updates # f updates (K) 8 6 8 8 6 6 (a) # f updates (K) 7 6 6 8 6 6 (b) Figure : Effect f predictive time n # f ttal updates Therefre, a predictive time f qpdt = 6 means t find the bjects in specific regin after mins(mins). Based n the reasn abve, althugh ur experiments shw that incurs higher updating csts than n when query predictive time is larger than 6, we argue that the predictin is meaningful nly n a clse fure. In summary, ur achieves gd query precisin and utperfrms update methd with regard t query recall. In additin, can effectively reduce update wrlad when query predictive time is in a reasnable range. 6. CONCLSION In this paper, we prpse a generic updating prtcl fr mving bject databases that is independent f the underlying index strucre. By utilizing the cncept f spati-tempral safe regin (), bjects actively send mtin updates including their lcatins and velcities t the database server nly when the predictin errr f their current mvement is n lnger bunded. T guarantee the accuracy f query predictin, the database server ass bjects fr their latest mtin, if they are ptential results f the query. T minimize the number f update messages between mving bjects and database system, we present a cst mdel that analysis the apprximate cst depending n the recent update recrds stred in the system. Based n the cst mdel, an effective ptimizatin technique is designed t reduce the expected update cst. We carefully evaluate three different implementatin strategies, including static, dynamic glbal and dynamic persnal. Experiments n the TPR-tree and the B x -tree shw that ur prpsed prtcl significantly imprves the accuracy f query results and reduces the number f update messages. 7. ACKNOWLEDGEMENT The wr was in part supprted by Singapre MDA grant R-- -76-79. 8. REFERENCES [] ecurier. http://api.ecurier.c.u/. [] R-tree Prtal, http://www.rtreeprtal.rg/. [] T. Brinhff. A Framewr fr Generating Netwr-Based Mving Objects. GeInfrmatica, 6(): 8,. [] S. Chen, C. S. Jensen, and D. Lin. A benchmar fr evaluating mving bject indexes. PVLDB, ():7 8, 8. [] S. Chen, B. C. Oi, K.-L. Tan, and M. A. Nasciment. St b-tree: a self-nable spati-tempral b + -tree index fr mving bjects. In SIGMOD, pages 9, 8. [6] S. Chen, B. C. Oi, and Z. Zhang. Capring the mtins with adaptive updating mdel in mving bject database. http://www.cmp.nus.edu.sg/ chensu/stsr-tr.pdf. [7] B. Gedi and L. Liu. Mbieyes: Distributed prcessing f cntinuusly mving queries n mving bjects in a mbile system. In EDBT, pages 67 87,. [8] H. Hu, J. Xu, and D. L. Lee. A generic framewr fr mnitring cntinuus spatial queries ver mving bjects. In SIGMOD, pages 79 9,. [9] C. S. Jensen, D. Lin, and B. C. Oi. Query and pdate Efficient B + -Tree Based Indexing f Mving Objects. In VLDB, pages 768 779,. [] D. Lin, C. S. Jensen, B. C. Oi, and S. Saltenis. Efficient indexing f the histrical, present, and fure psitins f mving bjects. In MDM, pages 9 66,. [] M. F. Mbel, X. Xing, and W. G. Aref. Sina: Scalable incremental prcessing f cntinuus queries in spati-tempral databases. In SIGMOD, pages 6 6,. [] S. Saltenis, C. S. Jensen, S. T. Leutenegger, and M. A. Lpez. Indexing the Psitins f Cntinuusly Mving Objects. In SIGMOD, pages,. [] Y. Ta, C. Falutss, D. Papadias, and B. Liu. Predictin and indexing f mving bjects with unnwn mtin patterns. In SIGMOD, pages 6 6,. [] Y. Ta, D. Papadias, and J. Sun. The TPR*-Tree: An Optimized Spati-Tempral Access Methd fr Predictive Queries. In VLDB, pages 79 8,. [] Y. Ta, D. Papadias, J. Zhai, and Q. Li. Venn sampling: A nvel predictin technique fr mving bjects. In ICDE, pages 68 69,. [6] O. Wlfsn, A. P. Sistla, S. Chamberlain, and Y. Yesha. pdating and querying databases that trac mbile units. Distributed and Parallel Databases, 7():7 87, 999. [7] W. Wu, W. Gu, and K.-L. Tan. Distributed prcessing f mving -nearest-neighbr query n mving bjects. In ICDE, pages 6, 7. [8] X. Xing, M. F. Mbel, and W. G. Aref. Sea-cnn: Scalable prcessing f cntinuus -nearest neighbr queries in spati-tempral databases. In ICDE, pages 6 6,. [9] M. L. Yiu, Y. Ta, and N. Mamulis. The b dual -tree: indexing mving bjects by space filling curves in the dual space. VLDB J., 7():79, 8. [] X. Yu, K. Q. Pu, and N. Kudas. Mnitring -nearest neighbr queries ver mving bjects. In ICDE, pages 6 6,. [] M. Zhang, S. Chen, C. S. Jensen, B. C. Oi, and Z. Zhang. Effectively indexing uncertain mving bjects fr predictive queries. In VLDB, 9. [] Z. Zhang, R. Cheng, D. Papadias, and A. K. H. Tung. Minimizing cmmunicatin cst fr cntinus syline maintenance. In SIGMOD, 9. [] Z. Zhang, Y. Yang, A. K. H. Tung, and D. Papadias. Cntinuus -means mnitring ver mving bjects. IEEE Trans. Knwl. Data Eng., (9): 6, 8.
APPENDIX A. RELATED WORK A. Mving Object Indexing Generally speaing, the index strucres fr mving bjects can be divided int tw categries, namely data-partitin based strucres and space-partitin based strucres. The TPR-tree [] and TPR -tree [] are typical examples f data-partitin based index strucres. Given the lcatins and velcities f mving bjects at their respective update times, the bjects are inserted int a multidimensinal index after transfrmatin int sme standard reference time. On each intermediate nde in the tree strucre, the maximal bunding rectangle (MBR) is used t bund the lcatins f the bjects in the subtree at the reference time. The maximal and minimal speeds f the bjects alng bth dimensins are als recrded. Given a range query at the querying time, the query prcessing algrithm fllws the traditinal pruning strategies in the R-tree. In particular, when visiting an intermediate nde in the index, the MBR expands with respect t its maximal and minimal speeds. If the expanded MBR des nt intersect with the query range, this nde can be safely pruned. Otherwise, its child ndes are pushed int a queue fr further examinatin. When the querying time is faraway frm the reference time f the index, the MBRs have t be expanded accrdingly and the resultant MBRs will have a much higher lielihd f intersecting with the query, causing many paths t be traversed. This is the majr drawbac f datapartitin based index strucres. Amng space-partitin based indexes, the B x -tree and the B dual - tree are the tw representative strucres. In the B x -tree, spatial space is split int small cells, and the cells are mapped t a nedimensinal space with sme space filling curve, such as Z-curve r Hilbert curve. T prcess a range query, the B x -tree first transfrms the query range int a sequence f cells in the space. These cells are used as queries, and during tree traversal, the query cells are expanded since the B + -tree des nt mae use f any MBRs. There are tw lgical sub-trees rlling with time, each f which is respnsible fr the updates happening in an interval time T, with T as the maximal update interval. nlie the B x -tree, B dual partitins bth spatial and velcity spaces int cells, with a similar index strucre built n the spati-tempral cells. Sme extensins have been further develped t enhance the effectiveness and efficiency f these strucres. In [], a frest f B x -trees is cnstructed t allw queries n bth predicted mvements and histrical mvements. In [], clusters are dynamically identified in each phase, and different granularity f cells is used imprve query efficiency. The index is aut-ned dynamically based n the bject mvement. In [], we shw that uncertain mdels n the mving bjects can be easily incrprated int the B x -tree, rerning mre meaningful results n predictive queries. Besides predictive queries fr range and nearest neighbr search, sme research sdies are devted t ther queries, e.g. range aggregatin []. A. Cntinuus Query Optimizatin Cntinuus query prcessing is als ne f the ht tpics in the database cmmunity. Different frm predictive queries, cntinuus query tries t eep accurate results n range query and nearest neighbr query n the current timestamp, with minimal cmmunicatin and update csts. In [6], fr example, Wlfsn et al. prpsed a general framewr, prviding a mechanism t render apprximate trajectries fr cntinuus query prcessing. In [], a scalable hash-based framewr is prpsed fr -Nearest Neighbr (-NN) and range queries mnitring n mving bjects, with shared executin mechanism n incremental evaluatins f the queries. In [8], a generic framewr is frmulated fr an energy-efficient mnitring scheme n mving bjects fr the range query and the -NN query, with safe regins cnstructed fr each bject f all cncurrent queries mnitred n the server side. In [] and [8], grid-based techniques are explited t reduce the number f messages and enhance quic prcessing f -NN queries n mving bjects. While the prblems cnsider nly queries n static lcatin, mre recent wrs sdy the query prcessing prblem when queries are mving arund, e.g. [7, 7] Besides wr that address traditinal range queries and -NN queries, sme extensins have been prpsed t supprt mre cmplicated queries n mving bjects. In [], a safe-regin based methd is prpsed t mnitr -means clustering, utilizing sme efficient lwer bund cmputatin technique n the lcal ptimum f -means clustering. In [], cst mdels and ptimizatin techniques are deplyed t evaluate and maintain syline queries with minimal cmmunicatin csts between bjects and central server. B. ALGORITHMS In this appendix, we shw the detailed pseudcde fr algrithms referred in the paper. Algrithm shws the cnsistency verificatin prcedure as intrduced in Sec.. Algrithm Cnsistency Verificatin (timestamp t, current lcatin li, t current velcity vi, t current R( i) = (, V R, t r, t e)) : if t > t e then : Reprt t the server and update with new : Cmpute the predicted regin Pi t with respect t R( i) : if li t is ut f Pi t then : Rern FALSE 6: Cmpute the predicted regin P te i w.r.t. R( i) 7: Cmpute predicted lcatin pl te i 8: if pl te i is ut f P te i then 9: Rern FALSE : if t < t r then : Cmpute the predicted lcatin pl tr i : if pl tr i is ut f then : Rern FALSE : Rern TRE w.r.t. l t i and v t i w.r.t. l t i and v t i Algrithm utlines the prcessing f an bject update in Sec.. Algrithm Object pdate (bject i, current lcatin l tr i ), current velcity v tr i, reference time tr) : Calculate a new R( i) = (, V R, t r, t e) depending n l tr i and v tr i : Send R( i) t i : Renew the recrd f i in the database with R( i), l tr i and v tr i Algrithm shws the prcessing f predictive range query as presented in Sec.. Algrithm summarizes the ptimizatin methd fr cnstructing a new fr an bject (Sec.).
Algrithm Range Query Prcessing (Query range QR, query time t q) : Find the bject set O O that the predicted regin P tq i verlaps with QR fr any i O : fr each i O d : if P tq i is ttally cvered by QR then : Include i in the query result : else 6: Send a prbe request t i fr current lcatin and velcity 7: Cmpute the new pl tq i with new lcatin and velcity 8: if pl tq i RQ then 9: Include i in the query result : Rern the cmplete query result Algrithm Optimizatin (current lcatin l t i, current velcity v t i, expiry time t e) : Search all update recrds cvering li t and vi t and stre them in NN. : Initialize R( i) with = {li}, t V R = {vi} t and t e : Initialize cvered update recrd set CR = : Initialize the cst Cst(R( i)) = : while Cst(R( i)) des nt cnverge d 6: Set ptimal expansin recrd as NLL 7: Set ptimal expanded R as NLL 8: fr each update recrd j in NN d 9: Cnstruct R by expanding R( i) t cver j : Estimate P a(r ) and P p(r ) : if the cst f R is smaller than R then : Replace R with R : Replace with j : if j is nt NLL then : Replace R( i) with R 6: Mve frm NN t CS 7: Rern R( i) C. PROTOCOL IMPLEMENTATION WITH EXISTING INDEX STRCTRES Our prpsed prtcl can be seamlessly implemented with almst all existing indexing strucres n predictive queries fr mving bjects. In this sectin, we fcus n incrprating ur prpsed prtcl int tw ppular data strucres: the TPR-tree (and its variants) and the B x -tree. While these index strucres feare in different aspects, e.g., query prcessing, etc., we discuss primarily the strage issue f s in these strucres. C. TPR-tree and its variants The ptimizatin algrithm (Algrithm ) can be embedded directly int the TPR-tree. In the TPR-tree and its variants such as the TPR -tree, spati-tempral bunding rectangles are used t summarize the pssible lcatins and velcities f a grup f mving bjects. T replace exact lcatin and velcity with as the underlying bject representatin in the TPR-tree, we nly need t mae sme minr mdificatins n the leaf ndes f the TPR-tree. Such changes d nt affect the intermediate ndes in the TPR-tree, since s nly enlarge the spatial tempral bunding bxes f these intermediate ndes. This enables us t equip general s n TPR-tree, withut any cnstraint n the lcatin rectangles and the velcity rectangles. y.... x.... vy... v.. v..... v. v Figure : Initial lcatin and velcity rectangle fr B bdual -tree vx. Since the verall strucre f the TPR-tree remains the same, the existing update algrithms n the TPR-tree can be reused withut any mdificatin. Similarly, the querying algrithm can be left unchanged since we can regard every mving bject as the traditinal spatial tempral bunding bx. The cncurrency cntrl mechanism, i.e. the RLin-tree, cmmnly used by the R-tree indexes remains applicable. C. B x -tree T extend the ptimizatin technique frm the TPR-tree t ther index strucres, such as the B dual -tree and the B x -tree, recall the difference between the TPR-tree and the ther tw index strucres n strage. In particular, the B x -tree discretizes the spatial space and the B dual -tree discretizes bth spatial and velcity space. T facilitate the extensin, the nly mdificatin f the algrithm is n the initializatin f the. In particular, the at the beginning is initialized by expanding the lcatin and velcity t the minimal cells cntaining them. In Fig., we present the initial fr the same mving bject update in Fig.7. If the widths f the cells in the spatial space and the velcity space are and. respectively, the new befre the first iteratin in Algrithm is cnstructed with = (, ) (, ) and V R = (.,.) (.,.). The subsequent expansin iteratins fllw exactly the same implementatin f the ptimizatin algrithm fr the TPRtree. In the B x -tree, spatial space is divided int small cells f equal width n bth dimensins. In the riginal B x -tree, the crdinates f the bject are transfrmed int the ID f the cell cntaining it. This implies that the lcatin rectangle in s stred in the B x - tree must als be discretized befre insertin. There are tw pssible slutins t supprt ur prtcl with the B x -tree. The first ptin is t allw s t ccupy a few spatial cells. With this ptin, multiple cpies f each may be stred in different leaf ndes in the tree, prviding mre flexible spatial cnstraints but incurring extra prcessing csts n queries. The secnd ptin requires the lcatin rectangle fr every t cver exactly ne cell in the partitined spatial space. This leads t sme implementatin n the B x -tree, n which every mving bject resides nly in ne leaf nde. Hwever, it sacrifices sme f the ning ability n the lcatin rectangle if this strategy is adpted. In the empirical sdies, we emply the secnd implementatin. Given a mving bject waiting fr a new, the lcatin rectangle is fixed depending n the space partitin f the B x -tree. With such strage implementatin, bth the updating and querying algrithms n the B x -tree are simply adpted withut any mdificatin. Other ptimizatins, such as bject gruping and cell size ning [], can als be applied directly; the Blin-tree cncurrency cntrl that is used by the B x -tree remains applicable fr handling
cncurrency peratins. D. EXPERIMENTS D. Experimental Settings Fig.6 shws the maps f rad netwrs fr each datasets. (a) : Athens metrplitan Table : Specifics f data surces Data surce Space (lng side) 7.6m 879.m.m Maximum speed.m/s.m/s m/s nit f timestamp sec sec sec Table : Experimental parameters and values Parameter Setting Datasets,, Time duratin ts Data size K, K, K, K, K Query side length qlen.%,.%, %, %, % qpdt ts, ts, 6s, 6ts, 6ts Query frequency qfqy,, 6, 6, 6 t ts, ts, ts, 8ts, 6ts, 6ts, 6ts δ l m, m, 6, 6m, 6m m/ts, m/ts, 6m/ts, 6m/ts, 6m/ts δ v and the B x -tree as explained in Appx.C. In the remaining part f this sectin, / TPR dentes using the B x -tree/tpr-tree with static ; / dentes using the B x -tree/tpr-tree with glbal dynamic ; -P / TPR-P dentes using the B x - tree/tpr-tree with persnal dynamic. CP time (sec) 8 6 TPR-P TPR -P Number f messages (e+6) TPR-P TPR -P (b) : K Object cardinality (K) (a) Executin time Object cardinality (K) (b) # f all updates Number f active updates 8 6 TPR-P TPR -P Number f passive updates TPR-P TPR -P Object cardinality (K) (c) # f active updates. Object cardinality (K) (d) # f passive updates Figure 7: Effect f data size (c) : Singapre Figure 6: Maps f varius data surces Tbl. shws sme specificatins n different data surces. Tbl. lists parameters and there values used in the experiments in Sec., where the default values are shwn in bld. D. Mre experimental results We nw prceed t sdy the perfrmance f different strategies as intrduced in Sec., including the static, the glbal dynamic and the persnal dynamic. We implement all the three strategies n tp f bth the TPR-tree We first examined the scalability f all update strategies by varying the bject cardinality frm K t K. Fig.7 illustrates the ttal prcessing time and the number f updates. The ttal prcessing includes all cmputatins n queries, cmputatin and updates f the mving bjects, in cnsecutive timestamps. Cmparing t the static and persnal dynamic strategies, glbal dynamic strategy largely decreases the amunt f active updates, while adds a number f passive updates. The persnal dynamic strategy, n the ther hand, reduces the number f passive updates at the expense f mre active updates. This is because that the with glbal dynamic strategy, the size f s is much larger than thse generated by persnal dynamic strategy. Secnd, we examined the effect f varius query parameters n the perfrmance f different update strategies. Frm Fig.7, we have already seen that the requires the highest exe-