Predictive Join Processing between Regions and Moving Objects

Transcription

1 Prditiv Join Proing btwn Rgion and Moving Objt Antonio Corral, Manul Torr, Mihal Vailakopoulo 2, and Yanni Manolopoulo 3 Dpt. of Languag and Computing, Univrity of Almria, 42 Almria, Spain. aorral,mtorr@ual. 2 Dpt. of Informati with Appliation in Biomdiin, Univrity of Cntral Gr, Papaiopoulou 2-4, 35, Lamia, Gr, and Dpt. of Informati, Alxandr TEI of Thaloniki, 574 Gr. mvailako@ug.gr 3 Dpt. of Informati, Aritotl Univrity, 5424 Thaloniki, Gr. manolopo@d.auth.gr Abtrat. Th family of R-tr i uitabl for indxing variou kind of multidimnional objt. TPR*-tr ar R-tr bad trutur that hav bn propod for indxing a moving objt databa,.g. a databa of moving boat. Rgion Quadtr ar uitabl for indxing 2- dimnional rgional data and thir linar variant (Linar Rgion Quadtr) i ud in many Gographial Information Sytm (GIS) for thi purpo,.g. for th rprntation of tormy, or unny rgion. Although, both ar tr trutur, th organization of data pa, th typ of patial data tord and th arh algorithm applid on thm ar diffrnt in R-tr and Rgion Quadtr. In thi papr, w xamin a patio-tmporal problm that appar in many pratial appliation: proing of prditiv join btwn moving objt and rgion (.g. diovring th boat that will ntr a torm), uing th two famili of data trutur a torag and indxing mhanim, and taking into aount thir imilariti and diffrn. With a thorough xprimntal tudy, w how that th u of a ynhronou Dpth-Firt travral ordr ha th bt prforman balan (on avrag), taking into aount th I/O ativity and rpon tim a prforman maurmnt. Kyword: Moving objt, TPR-tr, R-tr, linar quad-tr, qury proing, join. Introdution Th rnt advan of thnologi in mobil ommuniation and global poitioning ytm hav inrad ur attntion to an fftiv managmnt of Supportd by th Alman d Dato Epaio-Tmporal baado n Ontologia projt (TIN C5-3), fundd by th Spanih Minitry of Sin and Thnology.

2 information on th objt that mov in 2-dimnional pa. Tho moving objt nd thir urrnt poition, whih an b forwardd priodially to th ur for diffrnt purpo (.g. making diion, t.). Thi poition information i patio-tmporal, in patial loation of objt hang with tim. A databa that tor information for a larg numbr of objt loation hanging with tim i alld a moving objt databa. Ur quri iud on moving objt databa an b atgorizd into two typ: pat-tim quri and futur-tim quri [7]. Th pat-tim qury rtriv th hitory of dynami objt movmnt in th pat, whil th futurtim qury prdit movmnt of dynami objt in th futur [7]. In thi papr, w diu join proing of futur-tim quri btwn rgion and moving objt. Spatial data ar olltd and tord in two main gnri format, alld vtor and ratr. Th bai unit of patial data in th vtor format orrpond to dirt ral world fatur rprntd by point, lin or polygon. In th ratr altrnativ, th bai unit of patial data tak th form of a quar grid ll, mbddd within a grid of qually izd pixl (pitur lmnt). On th othr hand, th patial a mthod an b laifid aording to th two following approah. Firt, data-drivn patial a mthod ar organizd by partitioning th t of patial objt, and th partitioning adapt to th ditribution of th objt in th mbdding pa. An xampl of thi approah i th R-tr. Sond, pa-drivn patial a mthod ar bad on partitioning of th mbdding two-dimnional pa into ll of pifi hap and iz, indpndntly of th ditribution of th patial objt (objt ar mappd to th ll aording to om gomtri ritrion). An xampl of thi approah i th Quadtr. Th book [6] and [2] provid xllnt information our for th intrtd radr about Quadtr and R-tr, rptivly. Thr ar a numbr of variation of th R-tr all of whih organiz multidimnional data objt by making u of th Minimum Bounding Rtangl (MBR) of th objt. W will onntrat on a mthod that hav th apability of daling with antiipatd futur-tim quri of moving objt or point (dynami point of viw). Gnrally, to upport th futur-tim quri, databa tor th urrnt poition and vloiti of moving objt a linar funtion. Up to now, for proing urrnt and futur-tim quri, vral indxing mthod hav bn propod blonging to th R-tr family and th TPR*-tr [9] i th mot widly-ud indx trutur for prditing th futur poition of moving point, whih an b ud for futur-tim quri. Th Rgion Quadtr i a pa-drivn patial a mthod, whih i uitabl of toring and manipulating 2-dimnional rgional data (or binary imag). Morovr, many algorithm hav bn dvlopd bad on Quadtr. Th mot widly known ondary mmory altrnativ of thi trutur i th Linar Rgion Quadtr [6]. Linar Quadtr hav bn ud for organizing rgional data in GIS [6]. Th ontribution of thi papr onit in th following:

3 . W prnt prditiv join proing thniqu btwn two diffrnt a mthod, TPR*-tr for moving objt (vtor data) and Linar Rgion Quadtr for rgion (ratr data), in ordr to anwr futur-tim (prditiv) quri apparing in pratial appliation, lik Rtriv all th boat ovring by a torm within hour. To th bt of our knowldg, thi i th firt tudy of patio-tmporal join btwn diffrnt data format (vtor and ratr). 2. W ditinguih btwn two typ of uh futur-tim quri, dpnding on th rquird rult: futur-tim-intrval and futur-tim-paramtrizd join quri, btwn vtor and ratr data. 3. W prnt a dtaild xprimntal omparion of th altrnativ mthod and highlight th prforman winnr, for ah xprimntal tting. Th papr i organizd a follow. In Stion 2, w rviw th rlatd litratur and motivat th rarh rportd hr. In Stion 3, a brif dription of th TPR*-tr and th Linar Rgion Quadtr ar prntd. In Stion 4, w prnt th algorithm that prform th prditiv join proing btwn rgion and moving objt. In Stion 5, a omparativ prforman tudy of propod algorithm i rportd. Finally, in Stion 6, onluion on th ontribution of thi papr and futur work ar ummarizd. 2 Rlatd Work and Motivation In gnral, th patial join ombin two t of patial objt bad on a patial prdiat (uually ovrlap). Rntly, an xhautiv analyi of vral thniqu ud to prform a patial join taking into aount a filtr-and-rfinmnt approah ha bn publihd in [7]. Rgarding patial join ovr tati patial objt, w an laify th patial join mthod in thr atgori, dpnding on whthr th t of patial objt involvd in th qury ar indxd or not. Whn both t ar indxd, th mot influntial and known algorithm for joining two datat indxd by R*-tr wa prntd in [2], whr additionally vral thniqu to improv both CPU and I/O tim hav bn tudid. Thi algorithm follow a Dpth-Firt ynhronizd tr travral ordr. A bradthfirt ynhronizd tr travral vrion to rdu I/O ot wa prntd in [4]. In th a of jut on t bing indxd, vral patial join algorithm hav bn propod in th litratur, and th mot rprntativ on ar [9, 3, ]. In th lat atgory, whn both t ar not indxd, th mot rlvant publiation ar [4,, 6]. W mut highlight that, whn both t ar indxd, but with inompatibl typ of indx, uh a by R-tr (hirarhial and non-dijoint indxing for vtor data) and by Linar Rgion Quadtr bad on B + -tr (hirarhial and dijoint indxing for ratr data), th only rarh work that propo join algorithm btwn th diffrnt data format i [3]. Th author propod vral algorithm to prform thi patial join, and th mot novl u a omplx buffring ytm and th FD-ordr [6] to rdu th I/O ot, whil arhing in th B + -tr for th FD-od that ovrlap with a point in th R-tr.

4 From th dynami point of viw, th mot rprntativ join on moving objt hav bn propod vry rntly. In [8], th author prnt a t of patio-tmporal quri o-alld tim-paramtrizd quri, inluding th tim-paramtrizd join qury, whih w will adapt latr to our problm tting. In [5], qury maintnan algorithm for patial join on ontinuouly moving point that upport updat wr prntd. And finally, in [2], th problm of proing ontinuou intrtion join ovr moving objt, uing TPR*-tr, ha bn addrd. In pratial appliation, th nd for patio-tmporal prditiv joining btwn diffrnt data format i ommon. For xampl, onidr Figur, whr fiv boat (hap A-E), along with thir moving vtor (arrow) and a torm (gray rgion) ar dpitd. At th tim intant of Figur.a (tim I = now), only boat B i in th torm. At th tim intant of Figur.b (tim II = now+ minut), boat B ha jut xitd th torm, whil boat C and D hav jut ntrd th torm. On poibl qury i: giv m all th boat that will b undr th torm for th nxt 9 minut and 59 ond (thi i an xampl of a futurtim-intrval join and th rult i: boat B). Not that w aumd that th rolution of tim i on ond. Anothr poibl qury i: giv m all th boat that ar undr th torm now, th tim point whn thi ituation will hang and th vnt that will au th hang of th ituation (thi i an xampl of a futur-tim-paramtrizd join and th rult i: boat B i undr th torm now, th ituation will hang in minut, bau boat B will xit and boat C and D will ntr th torm). It hould b notd that both quri ar prditiv, in thy rfr to th futur and thy ar bad on th aumption that th moving vtor of a boat do not hang (at lat ignifiantly) btwn ubqunt updat of th poition of a moving objt. Howvr, dpnding on th appliation, th frquny of updat of objt poition and th poibility of uddn hang of th movmnt vtor, th rult uh quri may b nough aurat. Morovr, it hould b notd that th torm data ar onidrd tati, at lat for tim priod quit larg in omparion to th updat frquny of th objt poition. Although, th quri dribd abov ari naturally in pratial appliation, th litratur (to th bt of our knowldg) do not inlud any thniqu for proing thm, prhap du to th diffrnt natur of rgional and vtor data and th diffrnt mthod ud for toring and indxing thm. In thi papr, w onidr that th rgional data (.g. th torm) ar tord and indxd uing Linar Rgion Quadtr (a ommon hoi in GIS ytm). Th od of th quad blok ar tord ithr in B + -tr [6], or in R*-tr (Oral, in gnral, rommnd uing R-tr ovr quadtr [8]), for omparion purpo btwn th two diffrnt altrnativ. At tim priod larg nough for ignifiant hang of th rgional data, th whol torag and indxing trutur i rbuilt. In thi papr, w tudy th ituation within on uh tim priod, during whih th rgional data ar onidrd tati. Morovr, w onidr that th hanging vtor data (.g. moving boat) ar tord and indxd uing TPR*-tr [9] (th mot widly-ud indx trutur for prditing th futur poition of

5 A A C B B C E D E D (a) Initial intant (b) minut latr Fig.. Fiv moving boat and a torm at a tim intant (a) and minut latr (b). moving point). Thu, w prnt and tudy th firt algorithm for proing futur-tim-intrval and futur-tim-paramtrizd join quri, btwn vtor and ratr data. 3 Th two a mthod 3. Th TPR*-tr W aum that th radr i alrady familiar with th R*-tr []. Th TPRtr [5] xtnd th R*-tr, prdit th futur loation of moving objt by toring th loation and th vloity of ah objt at a givn tim point. Th loation of moving objt ar indxd uing CBR (Conrvativ Bounding Rtangl) intad of MBR (Minimum Bounding Rtangl). A CBR i ompod of an MBR, rprnting th rgion that ovr a t of moving objt at a pifi tim point, and th maximum and minimum moving vloiti of th objt within an MBR at ah axi (vloity bounding rtangl, VBR). Th loation of a moving objt at any futur-tim point an b aily prditd with th loation and moving vloity tord in a CBR. Th prditd rgion of a nod omputd by uing a urrnt loation of an MBR and it maximum and minimum vloiti at ah axi i dfind a a bounding rtangl. Aording to [9], a moving objt o i rprntd with () an MBR o R that dnot it xtnt at rfrn tim, and (2) a vloity bounding rtangl (VBR) o V = {o V, o V +, o V 2, o V 2+ } whr o V i (o V i+ ) drib th vloity of th lowr (uppr) boundary of o R along th i-th dimnion ( i 2). Figur 2.a how th MBR and VBR of 4 objt a, b,, d. Th arrow (numbr) dnot th dirtion (valu) of thir vloiti. For xampl, th VBR of i V = { 2,,, 2}, whr th firt two numbr ar for th X-axi. A non-laf ntry i alo rprntd with an MBR and a VBR. Spifially, th MBR (VBR) tightly bound th MBR (VBR) of th ntri in it hild nod.

6 In Figur 2.a, th objt ar lutrd into two laf nod N and N2, whih VBR ar N V = { 2,, 2, } and N2 V = { 2,,, 2}. Figur 2.b how th MBR at timtamp (noti that ah dg mov aording to it vloity). Th MBR of a non-laf ntry alway nlo tho of th objt in it ubtr, but it i not narily tight. Y Y N N -2-2 a -2 b d -2 2 N N a b d X (a) MBR and VBR at tim (b) MBR and VBR at tim X Fig. 2. Entry rprntation in a TPR*-tr. Th TPR*-tr [9] u a t of improvd algorithm to build th TPR-tr and ahiv an almot optimal tr. In gnral, th TPR*-tr baially u th am trutur a th TPR-tr. During th updat opration, howvr, th TPR-tr mploy th inrtion and th dltion algorithm of th R*-tr a thy ar, whil th TPR*-tr mploy modifid vrion that rflt objt mobility. Thi mak it poibl to improv th prforman of updat and rtrival in th TPR*-tr ovr th TPR-tr. Sin th TPR-tr onidr th ara, irumfrn, ovrlapping, and ditan of an MBR only at th tim of updat of moving objt, it annot rflt th proprty that objt mov with tim. On th othr hand, th TPR*-tr prform updat in uh a way that it minimiz th ara of a wping rgion, whih i an xtnion of th rtangl that orrpond to a nod with tim aftr th updat of moving objt. Taking into aount th inrtion tratgy, th TPR-tr inrt a moving objt into uh a nod who MBR xtnion rquird i minimum at th tim of th inrtion. On th othr hand, th TPR*-tr inrt a moving objt into uh a nod with a minimum xtnion of th bounding rtangl aftr th inrtion. Travring from th root to lowr-lvl nod, thir rtangl xtnion rquird for th inrtion ar omputd, and alo ar tord into a priority quu. And finally, th optimal nod for th inrtion i th on having th mallt valu. With thi tratgy, th TPR*-tr rquir a ot highr than th TPR-tr for updat. Howvr, it ompatn of bounding rtangl, it gratly improv th ovrall qury prforman.

7 3.2 Rgion Quadtr Th Rgion Quadtr i th mot popular mmbr in th family of quadtrbad a mthod. It i ud for th rprntation of binary imag, that i 2 n 2 n binary array (for a poitiv intgr n), whr a () ntry tand for a blak (whit) pitur lmnt. Mor prily, it i a dgr four tr with hight n, at mot. Eah nod orrpond to a quar array of pixl (th root orrpond to th whol imag). If all of thm hav th am olor (blak or whit) th nod i a laf of that olor. Othrwi, th nod i olord gray and ha four hildrn. Eah of th hildrn orrpond to on of th four quar ub-array to whih th array of that nod i partitiond. W aum hr, that th firt (lftmot) hild orrpond to th NW ub-array, th ond to th NE ub-array, th third to th SW ub-array and th fourth (rightmot) hild to th SE ub-array. For mor dtail rgarding Quadtr [6]. Rgion Quadtr, a prntd abov, an b implmntd a main mmory tr trutur. Variation of Rgion Quadtr hav bn dvlopd for ondary mmory. Linar Rgion Quadtr ar th on ud mot xtnivly. A Linar Quadtr rprntation onit of a lit of valu, whr thr i on valu for ah blak nod of th pointr-bad Quadtr. Th valu a nod i an addr dribing th poition and iz of th orrponding blok in th imag. Th addr an b tord in a ffiint trutur for ondary mmory (uh a an B-tr or on of it variation). Thr ar alo variation of thi rprntation whr whit nod ar tord too, or variation whih ar uitabl for multiolor imag. Evidntly, thi rprntation i vry pa ffiint, although it i not uitd too many uful algorithm that ar dignd for pointr-bad Quadtr. Th mot popular linar implmntation ar th FL (Fixd Lngth), th FD (Fixd Lngth-Dpth) and th VL (Variabl Lngth) linar implmntation. For mor dtail rgarding FL and VL implmntation [6]. In th rt if thi papr, lik in [3], w aum that Linar Quadtr ar prntd with FD-od tord in a B + -tr or in an R*-tr. Th hoi of FD linar rprntation i not aidntal, in it i mad of ba 4 digit and i thu aily handld uing two bit for ah digit. Bid, th ortd qun of FD-od i a Dpth-Firt travral of th tr. Sin intrnal and whit nod ar omittd, ibling blak nod ar tord onutivly in th B + -tr or, in gnral, nod that ar lo in pa ar likly to b tord in th am or onutiv B + -tr lav. Thi proprty hlp at rduing th I/O ot of join proing. Sin in th am quadtr two blak nod that ar antor and dndant annot o-xit, two FD-od that oinid at all th dirtional digit annot xit nithr. Thi man that th dirtional part of th FD-od i uffiint for building B + -tr at all th lvl. At th laf-lvl, th dpth of ah blak nod hould alo b tord o that imag ar auratly rprntd [3]. Sin th FD-od an b tranformd in dijoint MBR w an tor th qun of FD-od in an R*-tr, and apply of xiting qury algorithm ovr thi popular patial indx.

8 4 Futur-tim Join Algorithm Bfor join proing, w mut rat th indx that tor th tati rgion (linar rgion quadtr tord in a B + -tr, or an R*-tr) and moving point (TPR*-tr). W hav didd to tor th qun of FD-od in an R*-tr, a an altrnativ to a B + -tr, bau Oral, in gnral, rommnd uing R- tr ovr quadtr (du to highr tiling lvl in th quadtr that au vry xpniv prproing and torag ot) [8]. Morovr, th orrpondn of th pa ovrd by th two trutur ha bn tablihd in [3]. Finally, w hav to tak into aount th two typ of futur-tim join quri that w will tudy in thi papr: futur-tim-intrval join and futur-tim-paramtrizd join. Joining th two trutur an b arrid out following two join proing thniqu: () multipl quri and (2) ynhronizd tr travral. Multipl quri thniqu prform a window qury on th TPR*-tr for ah FD-od indxd in th B + -tr, or in th R*-tr. And th ynhronizd tr travral thniqu follow a Dpth-Firt or Bradth-Firt ordr to travr both trutur during th qury proing. Mor pifially, w hav dignd and implmntd th following fiv algorithm for futur-tim-intrval join and on algorithm for futur-tim-paramtrizd join btwn rgion and moving objt. 4. Futur-tim-intrval join Th futur-tim-intrval join riv th tim-intrval of intrt ([T t, T d ]) and rturn th rult, valid only during uh a tim-intrval. B + to TPR*-tr join (B-TPR) Thi algorithm follow th multipl quri thniqu and it dnd th B + -tr from th root to it lftmot laf. It a quntially th FD-od prnt in thi laf, and for ah FD-od it prform a prditiv window (MBR of FD-od and VBR, whih i, in th rgion i tati) qury in th TPR*-tr (rporting intrtion of thi FD-od and lmnt in th TPR*-tr lav within [T t, T d ]. By making u of th horizontal intr-laf pointr, it a th nxt B + -tr laf and rpat th prviou tp. Of our, th rvr altrnativ (from TPR* to B +, i.. to an th ntri of th TPR*-tr and prform window quri in th B + -tr) an b aily implmntd, and w xpt that th rult will b vry imilar. R* to TPR*-tr join (R-TPR) Auming that w tor th FD-od in an R*-tr, thi algorithm follow th multipl quri thniqu a wll, and it travr rurivly th R*-tr, aing th MBR in ah nod in ordr of apparan within th nod. For th MBR (FD-od) of ah laf ad, it prform a prditiv window (MBR and VBR (it i in th rgion i tati)) qury in th TPR*-tr, rporting intrtion of thi MBR and lmnt in th TPR*-tr lav within [T t, T d ]. Of our, th rvr altrnativ (from TPR*

9 to R*, i.. to an th ntri of th TPR*-tr and prform window quri in th R*-tr) an b aily implmntd, and w xpt that th rult will b vry imilar. Dpth-Firt Travral join (R-TPR-DFJ) Thi algorithm follow th ynhronizd tr travral thniqu, uing a Dpth-Firt ordr of both tr for ovrlap join [2]. It i bad on th nlour proprty of R-tr nod: if th MBR of two intrnal nod do not ovrlap, thn thr an not b any MBR blow thm that ovrlap. In thi a, to apply thi thniqu on TPR*-tr w nd an additional funtion (omput intrtion priod) to hk if two dynami ntri (MBR and VBR) ovrlap in th rquird tim-intrval. R-TPR-DFJ(nodR, nodtpr, Tt, Td) If nodr and nodtpr ar lav For ah pair of ntri (Rntry, TPRntry) Sav Rntry in RnTPRntry If omput intrtion priod(rntprntry, TPRntry, Tt, Td) Add TPRntry to th rult El For ah pair of ntri (Rntry, TPRntry) Sav Rntry in RnTPRntry If omput intrtion priod(rntprntry, TPRntry, Tt, Td) nodraux = RadNodR(Rntry.p) nodtpraux = RadNodTPR(TPRntry.p) R-TPR-DFJ(nodRaux, nodtpraux, Tt, Td) An advand variant of th algorithm appli a loal optimization (bau it improv th ovrlap omputation with ah nod-pair join proing) in ordr to rdu CPU ot. In partiular, whn joining two nod, th ovrlapping of ntri i omputd uing a plan-wp thniqu [2] intad of brut-for ntd loop algorithm. In gnral, th MBR of ah nod ar ortd on th x- axi, and a mrg-lik algorithm i arrid out, rduing ignifiantly th numbr of intrtion tt. W will alld to thi variant, R-TPR-DFJ-PS. Bradth-Firt Travral join (R-TPR-BFJ) Thi algorithm follow th ynhronizd tr travral thniqu, uing a Bradth-Firt ordr of both tr [4]. Th algorithm travr down th two tr ynhronouly lvl by lvl. At ah lvl, th algorithm rat an intrmdiat join indx (IJI) and dploy global optimization thniqu (.g. ordring) to improv th join omputation at th nxt lvl. It trminat whn th IJI i ratd by joining th laf ntri in th R*-tr with th dynami laf ntri in th TPR*-tr. Again, w nd th funtion (omput intrtion priod) to hk if two dynami ntri (MBR

10 and VBR) ovrlap in th rquird tim-intrval. Aording to [4], w hav implmntd two ordring of intrmdiat indx join a global optimization: ordring by th um of th ntr (OrdSum) and ordring by ntr point (OrdCn). In thi a, th MBR of th TPR*-tr i th bounding rtangl that ovr th VBR in th rquird tim-intrval. B-TPR FD-buffr join (B-TPR-FD) Thi algorithm follow a partiular thniqu for join proing. It u a omplx buffring ytm and th FDordr in th B + -tr to rdu th I/O ot for join proing btwn an R*-tr (TPR*-tr) and a FD-linar quadtr tord in a B + -tr [3]. In gnral, th algorithm work a follow: () pro ah ntry of th TPR*-tr root in FDordr; (2) rad a many FD-od a poibl for th urrnt ntry and tor thm in th FD-buffr, (3) all rurivly th join routin for thi ntry; (4) whn th join routin rturn, mpty th FD-buffr and rpat th prviou two tp until th urrnt ntry ha bn ompltly hkd; (5) rpat for th nxt ntry of th TPR*-tr root. On th othr hand, Th join routin for a TPR*-tr nod and th rquird tim-intrval work a follow: () if th nod i a laf, hk intrtion at th rquird tim-intrval uing th omput intrtion priod funtion and rturn; (2) If not (non-laf nod), for ah hild of th nod that ha not bn xamind in rlation to th FD-od in FD-buffr, all th join routin rurivly. 4.2 Futur-tim-paramtrizd join (TP-Join) In gnral, thi typ of futur-tim join do not riv any paramtr and it rturn () th atual rult at th tim that th qury (join) i mittd, (2) th xpiry tim of th rult givn in; and (3) th hang that au th invalidation of th rult. That i, th anwr ar in format of triplt (R, T, C) [8]. A tati patial join rturn all pair of objt from two datat that atify om patial prdiat (uually ovrlap). Th join rult hang in th futur whn: () a pair of objt, in th urrnt rult, a to atify th join ondition, or (2) a pair not in th rult tart to atify th ondition. In gnral, w dnot th influn tim of a pair of objt (o, o 2 ) a T INF (o, o 2 ), and it th nxt timtamp that will hang th rult [8]. Th influn tim i, if a pair will nvr hang th join rult, and th xpiry tim i th minimum influn tim. Th influn tim of two non-laf ntri, T INF (E, E 2 ), hould b a lowr bound of th T INF (o, o 2 ) of any two objt o and o 2 in th ubtr of th non-laf ntri E and E 2, rptivly. In gnral, our join algorithm work a follow: it travr, in Dpth-Firt ordr, th two tr (R*-tr and TPR*-tr) imultanouly tarting from th two root. Suppo E and E 2 to b two ntri in non-laf nod, on from th R*-tr and th othr from th TPR*-tr. Th travral go down th ubtr pointd by E and E 2 if on of th following ondition hold: () th MBR of E and E 2 ovrlap, or (2) T INF (E, E 2 ) i l than or qual to th minimum influn tim of all objt pair n o far (in thi a thir ubtr may

11 ontain objt pair that au th nxt rult hang). Condition () find th urrnt join pair and ondition (2) idntifi th nxt timtamp. Th travral top whn laf lvl ar rahd for both tr. Noti that to omput T INF w u th omput intrtion priod funtion that rturn th tim-intrval in whih th two ntri ovrlap for a givn tim-intrval. TP-Join(nodR, nodtpr) If nodr and nodtpr ar lav For ah pair of ntri (Rntry, TPRntry) If T INF (Rntry, TPRntry) < T C = (Rntry, TPRntry); T = T INF (Rntry,TPRntry); El if T INF (Rntry, TPRntry) == T C = C (Rntry, TPRntry) If Rntry ovrlap TPRntry R = R (Rntry, TPRntry) El For ah pair of ntri (Rntry, TPRntry) If (T INF (Rntry, TPRntry) T) or (Rntry ovrlap TPRntry) nodraux = RadNodR(Rntry.p) nodtpraux = RadNodTPR(TPRntry.p) TP-Join(nodRaux, nodtpraux) Not that all th abov algorithm ar ignifiantly diffrnt from xiting R-tr bad join algorithm. Th pial proprti of TPR*-tr for qury proing hav bn utilizd, a wll a, th funtion omput intrtion priod() ha bn ud. Bid, prviou algorithm that ombin FD-Linar-Quadtr tord in a B + -tr and R*-tr hav bn adaptd for u with TPR*-tr (not a trivial tak). 5 Exprimntal Rult In thi tion, w hav valuatd th prforman of our prditiv join algorithm ovr rgional data (blak-whit imag of 2 2 pixl) and moving point uing ynthti (uniform ditribution) and ral data (24493 populatd pla of north-amria). Th rgional data orrpond to viibl ptrum of ara of California (Squoia data). Noti that, in n =, an FD-od for uh an imag rquir 2 + log 2 ( + ) = 26 bit. For th moving objt, ah objt i aoiatd with a VBR uh that on ah dimnion, th vloity valu ditribution i uniform in th rang [,5]. All xprimnt wr prformd on an Intl/Linux worktation with a Pntium IV 2.5 GHz proor, GByt of main mmory, and vral GByt of ondary torag, uing th g ompilr. Th nod iz for th tr trutur (B + -tr, R*-tr and TPR*-tr) i

12 KByt, aording to [9]. Th prforman maurmnt ar: () th numbr of pag a and (2) th rpon tim (lapd tim) rportd in ond. Our firt xprimnt k th mot appropriat LRU buffr iz (nod) for our prditiv join algorithm that will b ud in th nxt xprimnt. W hav onidrd th following onfiguration: th numbr of moving objt (ynthti-uniform) i, th qury tim-intrval i [, 5], and th LRU buffr iz i variabl (8, 32, 64, 28 and 52 Kbyt, or nod). In Figur 3, th rult of 6 algorithm ar hown. W hav to highlight that R-TPR-DFJ-PS wa vry imilar to R-TPR-DFJ; and R-TPR-BFJ (OrdSum) obtaind vry imilar rult to R-TPR-BFJ2 (OrdCn), for thi raon w only how th rult of R-TPR-DFJ-PS (R-TPR-DFJ nhand with th plan-wp thniqu) and R- TPR-BFJ. B-TPR and R-TPR ar th mot I/O-onuming whn th buffr iz ar mall, but whn thy ar larg nough ( 32); th algorithm obtain th bt bhavior (th bt i B-TPR, 32 nod a for 52 nod in th LRU buffr). Th raon of thi xllnt I/O bhavior i du to th good patial loality of th TPR*-tr, thr i a high probability that in th nxt prditiv window qury ovr th TPR*-tr, a grat part of thi indx rmain in th LRU buffr. On th othr hand, th algorithm ar th mot tim-onuming, du to th join proing thniqu, i.. multipl quri. R-TPR-DFJ-PS how an xllnt bhavior with rpt to th I/O ot and rpon tim, mainly du to th u of ynhronizd tr travral a a join proing thniqu. R-TPR- BFJ do not improv th prviou algorithm, du to th high ot of managing th global IJI, although for big LRU buffr iz th I/O ativity i aptabl. R-TPR-FD i an algorithm dignd for rduing th numbr of nod a, and for thi raon it gt good bhavior for thi prforman maurmnt, but it onum a lot of tim to rturn th final rult, du to th ontinuou arh th FD-od in th B + -tr and th managmnt of th FD-buffr. Finally, th TP-join qury gt a imilar bhavior to th R-TPR-DFJ-PS for I/O ativity, but th rpon tim i lightly largr, in thi algorithm ha to rport thr anwr (R, T, C) and th influn tim [8]. ) 3 ^ 8 ( a d o N ) 4 ( 35 tim n 3 o p 25 R LRU buffr iz (nod) LRU buffr iz (nod) Fig. 3. Prforman omparion with rpt to th LRU buffr iz.

13 In th ond xprimnt, w hav tudid th bhavior of th prditiv join algorithm whn th ardinality of th moving objt datat vari. W hav th following onfiguration: LRU buffr iz i 28 nod, th qury tim-intrval [, 5], and th ardinality of th datat i variabl (,, 5 and ). Figur 4 how B-TPR and R-TPR gt th bt rult for thi LRU buffr iz, although thy ar tim-onuming. B-TPR-FD obtain alo good bhavior for I/O ativity until th numbr of moving point i. R-TPR- BFJ nd many nod a, although it nd an aptabl rpon tim. R-TPR-DFJ-PS i th fatt, although it nd mor dik a than th join algorithm that u multipl quri a th thniqu for th join proing. TPjoin ha imilar bhavior than R-TPR-DFJ-PS, in thy follow th am join proing thniqu, although th rpon tim i lightly largr. 5 ) 3 ^ ( a d o N ) 2 ( 9 t im n 6 o p 3 R 5 Numbr of moving objt 5 Numbr of moving objt Fig. 4. Prforman omparion with rpt to th moving objt datat iz. In th third xprimnt, w hav ompard th bhavior of th prditiv join algorithm, varying th qury tim-intrval. W hav th following onfiguration: LRU buffr iz i 28 nod, th ardinality of th moving objt datat i, and th qury tim-intrval ar [, ], [, 5], [, ] and [, 2]. Sin TP-Join do not riv any qury tim-intrval, th rult i not rportd. W hav to highlight that whn th tim-intrval nlarg, th moving objt ovr mor pa along thir movmnt and th MBR that ovr thm grow a wll. Thi fat gnrat mor ovrlap btwn MBR, and it inrmnt th numbr of opration nary to olv th join. Figur 5 how again that B-TPR and R-TPR gt th bt rult for th numbr of nod a (B-TPR nd l nod a than R-TRP to prform th am qury), although thy ar tim-onuming. Morovr, B-TPR-FD gt intrting rult for mall qury tim-intrval, but for largr on it nd mor nod a. Th bt prforman balan orrpond to R-TPR-DFJ- PS, whih onum a raonabl quantity of nod a, but it i th fatt for mall qury tim-intrval ([, ] and [, 5]), but for [, ] and [, 2] iz it i lightly lowr than R-TPR-BFJ. Th join algorithm whih u a Bradth-

14 8 9 ) 3 ^ 64 ( a d o 6 N ) 72 ( 54 t im n 36 o p 8 R [,] [,5] [,] [,2] [,] [,5] [,] [,2] Qury tim-intrval Qury tim-intrval Fig. 5. Prforman omparion with rpt to th qury tim-intrval iz. Firt travral ordr of both tr ha a urpriing bhavior; it i I/O-onuming, but it i th fatt for larg qury tim-intrval. It i mainly du to th applid qury proing thniqu, in it only onidr th ovrlappd lmnt lvl by lvl (thr i no baktraking) and whn th laf nod ar rahd, th rult i rportd. Of our, it alo nd additional main mmory rour to tor th IJI. Th lat rult rportd hr ar a ummary (rprntativ t) of xprimnt with ral moving objt datat. In gnral, th tndni ar vry imilar to th ynthti uniform data. For xampl, th lft hart of Figur 6 i vry imilar to th lft hart of Figur 3, xpt for th LRU buffr iz for B- TPR and R-TPR, tarting from whih thy bom th bt ( 28). Morovr, obrv that in th right hart of Figur 5 and 6, th trnd ar quit imilar, whr R-TPR-DFJ-PS i th fatt for mall qury tim-intrval and for largr on R-TPR-BFJ onum lightly l tim to rport th final rult; and B-TPR and R-TPR ar th mot xpniv altrnativ for rpon tim onumd (R-TPR i lightly fatr than B-TPR for th am qury). 6 4 ) 3 ^ ( a d o 2 N ) 2 ( 8 t im n o p R [,] [,5] [,] [,2] LRU buffr iz (nod) Qury tim-intrval Fig. 6. Prforman for ral data, for LRU buffr and qury tim-intrval iz.

15 From th prviou prforman omparion (for ynthti and ral data), th mot important onluion ar th following: () B-TPR and R-TPR ar appropriat whn w hav availabl nough rour for buffring. (2) Th prditiv join algorithm whih u a Bradth-Firt travral ordr of both tr (R-TPR-BFJ) hav a good bhavior for larg qury tim-intrval and buffr iz, obtaining th bt rpon tim. (3) B-TPR-FD rport intrting rult with rpt to th I/O ativity, but it i tim-onuming du to th high omputational ot of managing th FD-buffr. (4) Finally, th prditiv join algorithm whih u a ynhronou Dpth-Firt travral ordr (R-TPR-DFJ- PS and TP-join) hav th bt prforman balan (on avrag) in all xutd xprimnt, taking into aount th I/O ativity and rpon tim. 6 Conluion and Futur Work Th ontribution of thi papr fall within in th tudy of a patio-tmporal problm that appar in ral-world appliation: proing of prditiv join btwn moving objt and rgion. To th bt of our knowldg, thi i th firt tudy of it kind. For thi purpo: () W hav onidrd two typ of futur-tim join quri: futur-tim-intrval join and futur-tim-paramtrizd join. To olv th quri, w hav ud th TPR*-tr, whih i th mot widly-ud indx trutur for prditing th futur poition of moving point, and th Linar Rgion Quadtr (FD Linar Quadtr, a pointrl rprntation) tord in a B + -tr, or in R*-tr. (2) W hav propod vral join algorithm btwn th two indx, following two join proing thniqu: multipl quri and ynhronizd tr travral, to olv uh futur-tim join quri. (3) By xtniv xprimntal rult, w hav hown that th u of a ynhronou Dpth-Firt travral ordr (R-TPR-DFJ-PS and TP-join) ha th bt prforman balan (on avrag), onidring th I/O ativity and rpon tim. Futur rarh may inlud th xtnion of our algorithm to prform ontinuou intrtion join [2], and th u of moving and/or hanging, intad of tati, rgion. Rfrn. N. Bkmann, H.P. Krigl, R. Shnidr and B. Sgr. Th R*-tr: an Effiint and Robut A Mthod for Point and Rtangl. SIGMOD Confrn, pp , T. Brinkhoff, H.P. Krigl and B. Sgr, B Effiint Proing of Spatial Join Uing R-tr. SIGMOD Confrn, p.p , A. Corral, M. Vailakopoulo and Y. Manolopoulo. Algorithm for Joining R- tr and Linar Rgion Quadtr. SSD Confrn. LNCS Vol. 65, p.p , Y.M. Huang, N. Jing, and E. Rundntinr. Spatial Join Uing R-tr: Bradth- Firt Travral with Global Optimization. VLDB Confrn, p.p , 997.

16 5. G.S. Iwrk, H. Samt and K.P. Smith. Maintnan of K-nn and Spatial Join Quri on Continuouly Moving Point. TODS 3(2), p.p , E.H. Jaox and H. Samt. Itrativ Spatial Join.TODS 28(3), p.p , E.H. Jaox and H. Samt. Spatial Join Thniqu. TODS 32(), artil 7, p.p. -44, R.K. Kothuri, S. Ravada and D. Abugov. Quadtr and R-tr Indx in Oral Spatial: A Comparion uing GIS Data. SIGMOD Confrn, p.p , M.L. Lo and C.V. Ravihankar. Spatial Join Uing Sdd Tr. SIGMOD Confrn, p.p , M.L. Lo and C.V. Ravihankar. Spatial Hah-Join. SIGMOD Confrn, p.p , N. Mamouli and D. Papadia. Slot Indx Spatial Join. TKDE 5(), p.p. 2-23, Y. Manolopoulo, A. Nanopoulo, A.N. Papadopoulo and Y. Thodoridi. R-Tr: Thory and Appliation. Springr, A.N. Papadopoulo, P. Rigaux and M. Sholl. A Prforman Evaluation of Spatial Join Proing Stratgi. SSD Confrn. LNCS, Vol. 65, p.p , J.M. Patl and D.J. Dwitt. Partition Bad Spatial-Mrg Join. SIGMOD Confrn, p.p , S. Saltni, C. S. Jnn, S. T. Lutnggr and M. A. Lopz. Indxing th Poition of Continouly Moving Objt. SIGMOD Confrn, p.p , H. Samt. Appliation of Spatial Data Strutur. Addion-Wly, Rading MA, A.P. Sitla, O. Wolfon, S. Chambrlain and S. Dao. Modling and Qurying Moving Objt. ICDE Confrn, pp , Y. Tao and D. Papadia. Tim-Paramtrizd Quri in Spatio-Tmporal Databa. SIGMOD Confrn, pp , Y. Tao, D. Papadia and J. Sun. Th TPR*-tr: An Optimizd Spatio-Tmporal A Mthod for Prditiv Quri. VLDB Confrn, p.p. 79-8, R. Zhang, D. Lin, K. Ramamohanarao and E. Brtino. Continuou Intrtion Join Ovr Moving Objt. ICDE Confrn, pp , 28.