Processing Moving Queries over Moving Objects Using Motion Adaptive Indexes

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1 Prcessing Mving Queries ver Mving Objects Using Mtin Adaptive Indexes Cllege f Cmputing Gergia Institute f Technlgy bgedik@cc.gatech.edu Kun-Lung Wu, Philip S. Yu T.J. Watsn Research Center IBM Research {klwu, psyu}@us.ibm.cm Ling Liu Cllege f Cmputing Gergia Institute f Technlgy lingliu@cc.gatech.edu Abstract This paper describes a mtin adaptive indexing scheme fr efficient evaluatin f mving ueries (MQs) ver mving bjects. It uses the cncept f mtin-sensitive bunding bxes t mdel the dynamic behavir f bth mving bjects and mving ueries. Instead f indexing freuently changing bject psitins, we index less freuently changing mtin sensitive bunding bxes tgether with the mtin functins f the bjects. This significantly decreases the number f update peratins perfrmed n the indexes. We use predictive uery results t ptimistically precalculate uery results, thus decreasing the number f search peratins perfrmed n the indexes. Mre imprtantly, we prpse a mtin adaptive indexing methd. Instead f using fixed parameters fr mtin sensitive bunding bxes, we autmatically adapt the sizes f the mtin sensitive bunding bxes t the dynamic mtin behavirs f the crrespnding individual bjects. As a result, the mving ueries can be evaluated faster by perfrming fewer IOs. Furthermre, we intrduce the cncept f guaranteed safe radius and ptimistic safe radius t extend ur mtin adaptive indexing scheme t evaluating mving cntinual k-nearest neighbr (knn) ueries. Our experiments shw that the prpsed mtin adaptive indexing scheme is efficient fr evaluatin f bth mving cntinual range ueries and mving cntinual knn ueries. 1 Intrductin With the emergence f psitining technlgies like GPS [16], and the cntinued advances f mbile cmputing industry, lcatin management has becme an active area f research. Several research effrts have been made t address the prblem f indexing mving bjects r mving bject trajectries t supprt efficient evaluatin f cntinual spatial ueries. Mving ueries ver mving bjects (MQs fr shrt) are mving cntinual spatial ueries ver mving bject psitins. Efficient evaluatin f mving ueries ver mving bjects is an imprtant issue in bth mbile systems and mving bject tracking systems. There are tw majr types f MQs mving range ueries and mving k-nearest Neighbr ueries (knn). Research n evaluating range ueries ver mving bject psitins has s far fcused n static cntinual range ueries [18, 9, 19, 3]. A static cntinual range uery specifies a spatial range tgether with a time interval and tracks the set f bjects that lcate within this spatial regin ver the given time perid. The result f the uery changes as the bjects being ueried mve ver time. Althugh similar, a mving range uery exhibits sme fundamental difference when cmpared t a static range uery. A mving range uery has an assciated mving bject, called the fcal bject f the uery [8]; the spatial regin f the uery mves cntinuusly as the uery s fcal bject mves. Mving ueries intrduce a new challenge in indexing, mainly due t the highly dynamic nature f the system elements. MQs have different applicatins such as envirnmental awareness, bject tracking and mnitring, lcatin-based services, virtual envirnments and cmputer games t name a few. Here is an example mving uery MQ 1: Give me the psitins f thse custmers wh are lking fr taxi and are within 5 miles (f my lcatin at each instance f time r at an interval f every minute) during the next 2 minutes, psted by a taxi driver marching n the rad. The fcal bject f MQ 1 is the taxi n the rad. Different specializatins f MQs can result in interesting and useful classes f MQs. One is called mving ueries ver static bjects, where the target bjects are still bjects in the uery regin. An example f such a uery is MQ 2: Give me the lcatins and names f the gas statins ffering gasline fr less than $1.2 per galln within 1 miles, during the next half an hur psted by a driver f a mving car, where the fcal bject f the uery is the car n the mve and the target bjects are buildings within 1 miles with respect t the lcatin f the car n the mve. Anther interesting specializatin is s called static uery ver mving bjects, where the ueries are psed with static fcal bjects r withut fcal bjects. An example uery is MQ 3: Give me the list f AAA vehicles that are currently n service call in dwntwn Atlanta (r 5 miles frm my ffice lcatin), during the next hur. Traditinal indexing mechanisms d nt wrk well fr indexing mving bject psitins due t the freuent updates reuired n the index structures [18, 9]. In rder t tackle this prblem, recently several researchers have intrduced indexing mechanisms based n indexing the parameters f the mtin functins f the mving bjects [11, 19, 1]. This apprach alleviates the prblem f freuent updates t the index, as the index needs t be updated nly when the mtin functin f an bject changes. Hwever this kind f indexing, mstly based n R-tree like structures, prduces time parameterized minimum bunding rectangles that enlarge cntinuusly and result in the deteriratin f the search perfrmance ver time. As a result, mst f these index structures are used fr a certain interval after which they have t be recnstructed [18, 19]. In this paper, we describe a mtin-adaptive indexing scheme fr efficient prcessing f mving ueries ver mving bjects. It uses the cncept f mtin-sensitive bunding bxes t mdel the dynamic behavir f bth mving bjects and mving ueries. Instead f indexing freuently changing bject psitins, we index less freuently changing mtin sensitive bunding bxes tgether with the mtin functins f the bjects. This significantly decreases the number f update peratins perfrmed n the indexes. We prvide tw techniues t address the prblem f increased 1

2 search cst n such indexes. First, we ptimistically precalculate uery results and maintain such predictive uery results under the presence f bject mtin changes. Secnd, we supprt mtin adaptive indexing. Instead f using fixed size f mtin sensitive bunding bxes, we autmatically adapt the sizes f the mtin sensitive bunding bxes t the changing mvement behavirs f the crrespnding individual bjects. By adapting t mving bject behavir at the granularity f individual bjects, the mving ueries can be evaluated faster by perfrming fewer IOs. Furthermre, we intrduce the cncept f guaranteed safe radius and ptimistic safe radius t extend ur mtin adaptive indexing scheme t the evaluatin f mving cntinual k-nearest neighbr ueries. Our experimental results shw that the mtin adaptive indexing scheme is efficient fr evaluatin f bth mving cntinual range ueries and mving cntinual k-nearest neighbr (knn) ueries. The prpsed mtin-adaptive indexing scheme is independent f the underlying spatial index structures by design. In the experiments reprted in this paper, we use bth R -trees and statically partitined grids fr measuring the perfrmance f ur indexing scheme. We reprt a series f experimental perfrmance results fr different wrklads including scenaris based n skewed bject and uery distributin, and demnstrate the effectiveness f ur mtin adaptive indexing scheme thrugh cmparisns with ther alternative indexing mechanisms. It shuld be mentined that the mving ueries, as we frmalize them in this paper, are different frm mving ueries with a predefined path [19]. T sme extent, ur mving uery definitin is clse t dynamic ueries in [12]. There are, hwever, sme imprtant differences. Dynamic ueries are defined as temprally rdered set f snapsht ueries. Instead f using a prcedural level definitin, the cncept f mving ueries is defined declaratively frm a users perspective [8]. The rest f the paper is rganized as fllws. Sectin 2 gives an verview f the basic cncepts and the system mdel. Sectin 3 describes the mtin-adaptive indexing scheme fr efficient evaluatin f mving range ueries. Sectin 4 extends the slutin t efficient evaluatin f mving knn ueries. Sectin 5 reprts varius perfrmance results t illustrate the effectiveness f the prpsed apprach. We discuss the previus wrk in the literature related t uerying and indexing mving bject psitins in Sectin 6, and cnclude with a summary in Sectin 7. 2 The System Mdel The basic elements f ur system mdel are a set f mving r still bjects and a set f mving r static cntinual (range r knn) ueries. A fundamental challenge we address in this paper is t study what kind f indexing scheme can efficiently answer the mving ueries. Fast evaluatin is critical fr prcessing mving ueries, as it nt nly imprves the freshness f the uery results by enabling mre freuent re-evaluatin, but als increases the scalability f the system by enabling timely evaluatin f large number f mving ueries ver large number f bjects. 2.1 Basic Cncepts and Prblem Statement We dente the set f mving r still bjects as O, where O = O m O s and O m O s =. O m dentes the set f mving bjects and O s dentes the set f still bjects. We dente the set f mving r static ueries as Q, where Q = Q m Q s and Q m Q s =. Q m dentes the set f mving range ueries and Q s dentes the set f static range ueries. Mving Objects. We describe a mving bject m O m by a uadruple: id, ps, vel, {prps}. Here, id is the uniue bject identifier, ps is the current psitin f the mving bject, vel =(velx, vely) is the current velcity vectr f the bject, where velx is its velcity in the x-dimensin and vely is its velcity in the y-dimensin, and {prps} is a set f prperties abut the bject. A still bject can be mdeled as a special case f mving bjects where the velcity vectr is set t zer, s O s, s.vel =(, ). Mving Queries. We describe a mving uery m Q m by a uadruple: id, id, regin, filter. Here, id is the uniue uery identifier, id is the bject identifier f the fcal bject f the uery, regin defines the shape f the spatial uery regin bund t the fcal bject f the uery, and filter is a Blean predicate defined ver the prperties ({prps}) f the target bjects f the uery. Nte that, regin can be described by a clsed shape descriptin such as a rectangle r a circle. This clsed shape descriptin als specifies a binding pint, thrugh which it is bund t the fcal bject f the uery. In the rest f the paper we assume that a mving cntinual uery specifies a circle as its range with its center serving as the binding pint. A static spatial cntinual range uery can be described as a special case where the ueries either have n fcal bjects r the fcal bject is a still bject. I.e., s Q s, s.id = null s.id O s. Befre we describe the mtin mdeling basics, we first review three basic types f indexing techniues fr evaluating range ueries ver mving bjects and discuss their advantages and inherent weaknesses. The three indexing mechanisms we cnsider are Object-nly Indexing (OI), Query-nly Indexing (QI) and Object and Query Indexing (OQI). T simplify the discussin withut lss f generality, we cnsider these indexes in their simplest frms and we assume that we are fed with a stream f bject psitin updates, where an update is received at each time step frm every bject that has mved since the last time step. We cmment n whether a particular indexing apprach is pen t ptimizatins but will nt cnsider specific ptimizatins in the discussin. Object-nly Indexing (OI). In the bject-nly indexing apprach a spatial index is built n the bject psitins. Each time a new bject psitin is received, the bject index is updated. At each uery evaluatin step, all ueries are evaluated against the bject index. An inherent drawback f the basic bject-nly indexing apprach is the re-evaluatin f all ueries against the bject index regardless f whether we have a static r mving uery and whether the bject psitin changes are f interest t the uery r nt. Object-nly indexing is pen t ptimizatins that can decrease the number r cst f the updates n the bject index (see velcity cnstrained indexing in [18] and time parameterized R-trees in [19]). Query-nly Indexing (QI). In the uery-nly indexing apprach a spatial index is built n the spatial regins f the ueries. Each time a new uery psitin (the psitin f the uery s fcal bject) is received, the uery index is updated. At each uery evaluatin step, each bject psitin is evaluated against the uery index and the ueries that cntain the bject s psitin are determined. Nte that this has t be dne fr every bject as ppsed t ding it nly fr bjects that have mved since the last uery evaluatin 2

3 step. This is due t the fact that underlying ueries are ptentially mving. This significantly decreases the effectiveness f uerynly indexing apprach, althugh in the cntext f static cntinual range ueries it has been shwn that a uery index may imprve perfrmance significantly [18]. Object and Query Indexing (OQI). In the bject and uery indexing apprach tw spatial indexes are built, ne fr the bject psitins and anther fr the spatial regins f the ueries. Each time an bject psitin is received, the bject index is updated. Similarly, each time a new uery psitin (the psitin f a uery s fcal bject) is received, the uery index is updated. At each uery evaluatin step, each new bject psitin is evaluated against the uery index and the ueries that cntain the bject s psitin are determined. Then the uery results are updated differentially. Similarly at each uery evaluatin step, each new uery psitin is evaluated against the bject index and the new result f the uery is determined. The bject and uery indexing apprach evaluates bject psitins against the uery index nly fr the bjects that have changed their psitins since the last uery evaluatin step as ppsed t uery-nly indexing which has t d it fr all bjects. It als evaluates the ueries against the bject index nly fr ueries that have mved since the last uery evaluatin step, as ppsed t bject-nly indexing which has t d it fr all ueries. Althugh the bject and uery indexing apprach incurs higher cst due t maintenance f an additinal index structure, it is pssible t emply a larger range f ptimizatins t reduce the additinal cst incurred and it des nt have certain restrictins f bject-nly indexing r uery-nly indexing. 2.2 Overview f the Slutin Apprach Bearing in mind the prs and cns f the abve three basic indexing schemes, we prpse a mtin-adaptive indexing scheme fr efficient prcessing f mving ueries ver mving bjects. A mtin adaptive index is defined as an index f parameterized mtin-sensitive bunding bxes. We use the cncept f mtinsensitive bunding bxes t mdel the dynamic behavir f bth mving bjects and mving ueries. Such bunding bxes are nt updated unless the psitin f a mving bject r the spatial regin f a mving uery exceeds the brders f its bunding bx. Instead f indexing freuently changing bject psitins r spatial regins f mving ueries, we index less freuently changing mtin sensitive bunding bxes tgether with the mtin functins f the bjects. This significantly decreases the number f update peratins perfrmed n the indexes. Our indexing scheme maintains bth an index f bject-based mtin sensitive bunding bxes (dented as Index msb ) and an index f uery-based mtin sensitive bunding bxes (dented as Index msb ). T address the prblem f increased search cst due t uerying bth Index msb and Index msb, we emply tw ptimizatin techniues: the predictive uery results, which ptimistically precmputes the uery results in the presence f bject mtin changes, and the mtin adaptive indexing, which dynamically adapts the sizes f the mtin sensitive bunding bxes t the changing mtin behavirs at the granularity f individual bjects, allwing mving ueries t be evaluated faster by perfrming fewer IOs. In the rest f this sectin we describe the mtin mdeling and mtin update generatin, which prvides the fundatin fr mtin sensitive bunding bxes and predictive uery results. Mtin Mdeling Mdeling mtins f the mving bjects fr predicting their psitins is a cmmnly used methd in mving bject indexing [26, 11]. In reality a mving bject mves and changes its velcity vectr cntinuusly. Mtin mdeling uses apprximatin fr predictin. Cncretely, instead f reprting their psitin updates each time they mve, mving bjects reprt their velcity vectr and psitin updates nly when their velcity vectrs change and this change is significant enugh 1. In rder t evaluate mving ueries in between the last update reprting and the next update reprting, the psitins f the mving bjects are predicted using a simple linear functin f time. Given that the last received velcity vectr f an bject is vel, its psitin is ps and the time its velcity update was recrded is t, the future psitin f the bject at time t + t can be predicted as ps + t vel. We use a linear mtin functin in this paper, since it is the cmmnly used mdel in mving bject databases. We refer readers t [1] fr a study f nn-linear mtin mdeling fr mving bject indexing. Predictin-based mtin mdeling decreases the amunt f infrmatin sent t the uery prcessing engine by reducing the freuency f psitin reprting frm each mving bject. Furthermre, it allws the system t ptimistically precmpute future uery results. We belw briefly describe hw the mving bjects generate and send their mtin updates t the server where the uery evaluatin is perfrmed. Mtin Update Generatin In rder fr the mving bjects t decide when t reprt their velcity vectr and psitin updates, they need t peridically cmpute if their velcity vectrs have changed significantly. Cncretely, at each time step a mving bject samples its current psitin and calculates the difference between its current psitin and its psitin as predicted based n the last mtin update it reprted t the server. In case this difference is larger than a specified threshld, say D, the new mtin functin parameters are relayed t the server. 3 Efficient Evaluatin f Mving Range Queries 3.1 Mtin Sensitive Bunding Bxes Mtin sensitive bunding bxes (MSBs) can be defined fr bth mving ueries and mving bjects. Given a mving bject m, its assciated MSB is calculated by extending the psitin f the bject alng each dimensin by α( m) times the velcity f the bject in that directin. Given a mving uery m, MSB f the mving uery is calculated by extending the minimum bunding bx f the uery alng each dimensin by β( m) times the velcity f the fcal bject f the uery in that directin (See Figure 1 fr illustratins). Let Rect(l, m) dente a rectangle with l and m as any tw end pints f the rectangle that are n the same diagnal. Let sign(x) dente a functin ver a vectr x, which replaces each entry in x with its sign (+1 r -1). Then we define the MSB fr a mving bject and the MSB fr a mving uery with fcal bject f as fllws: 1 This techniue is knwn as dead reckning [6]. 3

4 .vely * α().velx * α() f.vely * β() + 2 *.radius f f.velx * β()+ 2 *.radius Figure 1: Mtin sensitive bunding bxes O m,msb() =Rect(.ps,.ps + α().vel) Q m,msb() =Rect( f.ps.radius sign(.vel), f.ps + β().vel +.radius sign(.vel)) Fr each mving uery, its MSB is calculated and used in place f the uery s spatial regin in the uery-based MSB index, referred t as the Index msb. Similarly, fr each mving bject, its MSB is calculated and used in place f the bject s psitin and we refer t such an bject-based MSB index as the Index msb. An imprtant feature f indexing mtin sensitive bxes f mving bjects and mving ueries is the fact that an MSB is nt updated unless the uery s spatial regin r the bject s psitin exceeds the brders f its mtin sensitive bunding bx. When this happens, we need t invalidate the MSB. As a result, a new MSB is calculated and the Index msb r the Index msb is updated. This apprach significantly reduces the number f update peratins perfrmed n the spatial indexes and thus decreases the verall cst f updating the spatial indexes (Index msb and Index msb ). Althugh maintaining MSB indexes increase the cst f searching the index due t higher verlap f spatial bjects (MBRs) being indexed, it is imprtant t nte that fr apprpriate values f the α and β parameters, the gain f using MSB indexes is significant. Therefre, we need nt nly mechanisms fr reducing the cst f search peratins in the MSB indexes, but als mechanisms fr dynamically determining the mst apprpriate values f the α and β parameters based n the mtin behavir f mving bjects and mving ueries. It is als crucial t nte that, using MSBs des nt intrduce any inaccuracy in the uery results, as we stre the mtin functin f the bject r the uery tgether with its MSB inside the spatial index 2. Furthermre, MSBs prvide the fllwing tw advantages: (1) As ppsed t appraches that alter the implementatin f traditinal spatial indexes fr decreasing the update cst [19, 18], mtin sensitive bunding bxes reuire almst n significant change t the underlying spatial index implementatin. (2) Mtin sensitive bunding bxes perfrm size adaptatin at the granularity f individual bjects, leading t significant reductin f IO cst (See Sectin 3.5 fr further detail). 3.2 Predictive Query Results n Per Object Base It is well knwn that ne way f saving IO and imprving efficiency f evaluating mving ueries is t precalculate future results f the cntinual ueries. This apprach has been successfully used 2 The inaccuracy due t mtin mdeling is nt cnsidered here. See [27] fr a discussin f mtin update plicies and their tradeffs. time t +b time t +a time t 1 static uery mving bject 1 static bject mving uery mving uery mving bject case I case II case III Figure 2: Calculating Intervals in the cntext f cntinuus mving knn ueries ver static bjects [21]. Mst f existing appraches t precaculating uery results assciate a time interval t each uery that specifies the valid time fr the precalculated results. One prblem with per uery based predictin in the cntext f mving ueries ver mving bjects is the fact that a change n the mtin functin f anyne f the mving bjects may cause the invalidatin f sme f the precalculated results. This mtivates us t intrduce predictive uery results where the predictin is cnducted n per-bject base. Given a uery, its predictive uery result differs frm a regular uery result in the sense that each bject in the predictive uery result has an assciated time interval indicating the time perid in which the bject is expected t be included in the uery result. We dente the predictive uery result f uery Q by PQR(). Each entry in a predictive uery result takes the frm, [t s,t e]. We call the entry assciated with bject O in PQR() the predictive uery result entry f bject with regard t uery, and the interval [t s,t e] assciated with bject the valid predictin time interval f the predictive uery result entry. Calculating the valid predictin time intervals is dne as fllws. Given a static cntinual range uery and a mving bject with its mtin functin, it is straight frward t calculate the intersectin pints f the uery s spatial regin and the ray frmed by the mving bject s trajectry (See case I in Figure 2). Similarly, t calculate the intersectin pint f a mving uery and a mving r nn-mving bject (assuming that we nly cnsider mving ueries with circle shaped spatial regins), we need t slve a uadratic functin f time. Frmally, let Q be a uery with fcal bject f O m, and O be an bject, and let Dist(a, b) dente the Euclidian distance between the tw pints a and b. We can calculate the time interval in which the bject is expected t be in the result set f uery by slving the frmula: Dist( f.ps + t f.vel,.ps + t.vel) m.radius. Figure 2 illustrates three different cases that arise in the calculatin f predictin time interval fr each per-bject based predictive uery result entry. The predictive uery results are precalculated n per bject base and predictive uery result entries are crrect unless the mtin functin f the fcal bject f a uery r the mtin functin f the mving bject assciated with the uery result entry have changed within the valid predictin time interval. As a result, there are tw key uestins t answer in rder t effectively use the predictive uery results in evaluating mving ueries: Predictin Fr each mving uery, shuld we perfrm predictin n all mving bjects? If nt, hw t determine fr which bjects we shuld d predictin? Obviusly we shuld nt perfrm predictin fr bjects that are far away frm the spatial regin f the uery within a perid f time, 4

5 at time t 2 and 3 are subject t Res( 1 ) PQR( 1 ) = {( 2,[t +a, t +b])} at time t 1 2 and 4 are subject t Res( 1 ) PQR( 1 ) = {( 2,[t +a, t +b]), ( 4,[t 1 +c, t 1 +d])} Figure 3: An illustratin f tempral uery results and hw it integrates with mtin sensitive bunding bxes as the predicted results are less likely t hld until thse bjects reach t the prximity f the uery, e.g., entering the mtin sensitive bunding bx f the uery. Invalidatin When and hw t update the predictive results? This can be referred t as the invalidatin plicy fr per-bject based predictin. The predictive uery results may be invalid and thus need t be updated when the mtin functin f a mving uery r the mtin functin f a mving bject changes. In additin, the predictive results may reuire t be refreshed when the bjects in the predictive uery results have mved away frm the prximity f the uery r when the bjects that did nt participate in the predictin have entered the prximity f the uery. 3.3 Determining Predictive Query Results Using MSBs In additin t serving as an ptimizatin techniue t minimize the update cst n the spatial indexes, MSBs can be used t effectively determine fr which bjects we shuld perfrm result predictin with respect t a uery (answering the first uestin listed in Sectin 3.2). Cncretely, fr a given uery, bjects whse MSBs intersect with the uery s MSB are cnsidered as ptential candidates f the uery s predictive result. Figure 3 gives an illustratin f hw predictive uery results integrate with mtin sensitive bunding bxes. Cnsider the mving uery 1 with its uery MSB and fur mving bjects 1, 2, 3 and 4 as shwn in Figure 3. In the figure, 1 is the fcal bject f uery 1 and the ther three mving bjects 2, 3 and 4 are assciated with their bject MSBs. At time t nly bjects 2 and 3 are subject t uery 1 s predictive uery result, as their MSBs intersect with the uery s MSB. Hwever the valid predictin time interval f bject 3 with regard t uery 1 is empty because there is n such time interval during which bject 3 is expected t be inside the uery result f uery 1. Thus bject 3 shuld nt be included in the predictive result f uery 1. At sme later time t 1, bject 2 and uery 1 remain inside their MSBs. Hwever bjects 3 and 4 have changed their MSBs. As a result, bjects 2 and 4 becme ptential candidates f uery 1 s predictive result at time t 1. Since 2 has nt changed its MSB, it remains included in uery 1 s predictive result. By applying the valid predictin time interval test n 4, we btain a nn-empty time interval with respect t uery 1, during which 4 is expected t be included in the uery result. Thus 4 is added int the predictive result f uery Setting α and β Values The α and β parameters used fr calculating MSBs can be set based n the mtin behavir f the bjects, in rder t achieve mre efficient uery evaluatin. There are tw imprtant characteristics f bject mtins: (a) the speed f the bject and (b) the perid f cnstant mtin f the bject (i.e. the length f the time perid it takes fr the mtin functin t change). Fr instance, fr a uery whse fcal bject changes its mtin functin freuently, it may nt be a gd idea t perfrm t much predictin, thus β value fr this uery s MSB shuld be kept smaller. Hwever, fr an bject with high speed, a small α value may nt be apprpriate, as it may cause freuent MSB invalidatins. In rder t chse apprpriate α and β values fr each MSB, it is imprtant t design a mtin-adaptive methd that can set the values f α and β parameters adaptively. A cmmn apprach t runtime parameter settings is t develp an analytical mdel and use the analytical results t guide the runtime selectin f the best parameter settings. We develp an analytical mdel fr estimating the IO cst f perfrming uery evaluatin (see Appendix fr details). This mdel is used as the guide t build an ff-line cmputed αβtable, giving the best α and β values fr different value pairs f speed and perid f cnstant mvement f a mving bject. We will discuss details f α and β value selectin in Sectin Mtin Adaptive Indexing We have described the main ideas and mechanisms used in ur mtin-adaptive indexing scheme. In this subsectin we describe mtin-adaptive indexing as a uery evaluatin techniue that integrates the ideas and mechanisms presented s far fr efficient prcessing f mving ueries ver mving bjects Prcessing Mving Queries: An Overview The evaluatin f mving ueries is perfrmed thrugh multiple uery evaluatin steps executed peridically with regular time intervals f P s (scan perid) secnds. We build tw spatial MSB indexes, Index msb fr the bjects and Index msb fr the ueries. Index msb stres MSBs f the bjects accmpanied by the assciated mtin functins as data. Static bjects have pint MSBs. Similarly, Index msb stres the MSBs f the ueries accmpanied by the assciated mtin functins f the fcal bjects f the ueries and their radiuses as data. Static ueries have MSBs eual t their minimum bunding rectangles and they d nt have assciated mtin functins. We create and maintain tw tables, a mving bject table and a mving uery table. They stre infrmatin regarding the mving bjects and mving ueries. The static ueries and static bjects are included in the spatial MSB indexes but nt in the tw tables. The peridic evaluatin is perfrmed by scanning these tables at each uery evaluatin step and perfrming updates and searches n the spatial indexes as needed in rder t maintain the uery results as bjects and the spatial regins f the ueries mve. The detailed descriptin f the tw tables is given belw: Mving Object Table (MOT): An MOT entry is described as (id, id, ps, vel, time, bx, P cm,v ch ) and stres infrmatin regarding a mving bject. Here, id is the mving bject identifier, id is the uery identifier f the mving uery whse fcal bject s identifier is id, id is null if n such mving uery exists, ps is the last received psitin, vel is the last received velcity vectr f the mving bject, time is the timestamp f the mtin updates (ps and vel) received frm the mving bject, bx is the MSB f the mving bject, P cm is an estimate n the 5

6 Algrithm 1: Mtin Update Received r = id, ps, vel, time e = id, id, ps, vel, time, bx, P cm,v ch MOT where e.id = r.id e.p cm γ (r.time e.time) +(1 γ) e.p cm e.ps r.ps e.vel r.vel e.time r.time e.v ch true if e.id null then e = id, ps, vel, radius, time, bx, P cm,v ch MQT where e.id = e.id e.p cm = γ (r.time e.time) +(1 γ) e.p cm e.ps r.ps e.vel r.vel e.time r.time e.v ch true end if perid f cnstant mtin f the bject and V ch is a Blean variable indicating whether the mving bject has changed its mtin functin since the last uery evaluatin step. Mving Query Table (MQT): An MQT entry is described as (id, ps, vel, radius, time, bx, P cm,v ch ) and stres infrmatin regarding a mving uery. Here, id is the mving uery identifier, ps and vel are the last received psitin and the last received velcity vectr f the uery s fcal bject respectively, time is the timestamp f the mtin updates (ps and vel) received frm the fcal bject, radius is the radius f the mving uery s spatial regin, bx is the MSB f the mving uery, P cm is an estimate n the perid f cnstant mtin f the bject and V ch is a Blean variable indicating whether the fcal bject has changed its mtin functin since the last uery evaluatin step r nt. Nte that the infrmatin in MQT is t sme extent redundant with respect t MOT. Hwever the redundant infrmatin is reuired during the mving uery table scan. Withut redundancy we will need t lk them up frm the mving bject table, which is uite cstly. The MOT and MQT table entries are updated whenever new mtin updates are received frm the mving bjects. The P cm entries are updated using a simple weighted running average. The details are given in Algrithm 1. The effect f a mtin update is reflected n the uery results when the next peridic uery evaluatin step is perfrmed. Assuming that mving bjects decide whether they shuld send new mtin updates r nt at every P mu secnds (called the mtin update time perid), ne f ur aims is t perfrm a single uery evaluatin step in less that P mu secnds in rder t prvide fresh uery results, i.e. having P s P mu. At each uery evaluatin step, we need t perfrm uery table scan and bject table scan. The scan algrithms presented in the next subsectin describe hw these tw tasks are perfrmed The Scan Algrithms At each uery evaluatin step, tw scans are perfrmed. The first scan is n the mving bject table, MOT, and the secnd scan is n the mving uery table, MQT. The aim f the MOT scan is t update the Index msb and t differentially update sme f the uery results by perfrming searches n the Index msb. The aim f the MQT scan is t update the Index msb and t recalculate sme f the uery results by perfrming searches n the Index msb. MOT Scan. During the MOT scan, when prcessing an entry we first check whether the assciated bject f the entry has invalidated its MSB r changed its mtin functin since the last uery evaluatin perid. If nne f these has happened, we prceed t the next entry withut perfrming any peratin n the spatial MSB indexes. Otherwise we first update the Index msb. In case there is an MSB invalidatin, a new MSB is calculated fr the bject and the Index msb is updated. The α value used fr calculating the new MSB is selected adaptively (See Sectin 3.6 fr further details). If there has been a mtin functin change, the data assciated with the entry f the bject s MSB in the Index msb is als updated. Once the Index msb is updated, tw searches are perfrmed n the Index msb. First, using the ld MSB f the bject, the Index msb is searched and all the ueries whse MSBs intersect with the ld MSB f the bject are retrieved. The bject is then remved frm the results f thse ueries (if it is already in). Then a secnd search is perfrmed with the newly calculated MSB f the bject and all ueries whse MSBs intersect with the new MSB f the bject are retrieved. Fr all thse ueries, result predictin is perfrmed against the bject. Lastly, the uery result entries btained frm the predictin with nn-empty time intervals are added int their assciated uery results. Algrithm 2 gives the pseud cde fr the MOT scan. MQT Scan. During the MQT scan, when prcessing a uery entry we first check whether the assciated uery f the entry has invalidated its MSB r its fcal bject has changed its mtin functin since the last uery evaluatin step. If nne f these has happened, we prceed t the next entry withut perfrming any peratin n the spatial indexes. Otherwise we first update the Index msb. In case there is an MSB invalidatin, a new MSB is calculated fr the uery and the Index msb is updated. The β value used fr calculating the new MSB is selected adaptively (See Sectin 3.6 fr details). If there has been a mtin functin change, the data assciated with the entry f the uery s MSB in the Index msb is als updated. Once the Index msb is updated, a single search is perfrmed n the Index msb with the newly calculated MSB f the uery. All bjects whse MSBs intersect with the new uery MSB are retrieved. Fr all thse bjects, result predictin is perfrmed against the uery. The predictive uery result entries with nn-empty time intervals are added int the uery result and all ld uery results are remved. Algrithm 3 gives the pseud cde fr the MQT scan. Nte that after the MOT scan all results are crrect fr the ueries whse MSBs are nt invalidated and their fcal bjects have nt changed their mtin functin. Fr ueries that have invalidated their MSBs r whse fcal bjects have changed their mtin functins, the uery results are recalculated during the MQT scan. Therefre, all f the uery results are up t date after the MQT scan, given that MOT scan is perfrmed first. The rder f the scans can be reversed with sme minr mdificatins. 3.6 αβt able and Adaptive Parameter Selectin The cst functin develped in Appendix has a glbal minimum that ptimizes the IO cst f the uery evaluatin. We build an ff-line cmputed αβt able, which gives the ptimal α and β values fr different value pairs f bject speed and perid f cnstant mtin, calculated using the cst functin we have develped. We implement the αβt able as a 2D matrix, whse rws crrespnd t different bject speeds and clumns crrespnd t different perids f cnstant mtin and the entries are ptimal (α, β) pairs. Recall Sectin 3.5, when we calculate the MSBs f mving bjects and mving ueries, we already have the estimates n perids f cnstant mtin and speeds f all mving bjects including the 6

7 Algrithm 2: Mving Object Table Scan 1: fr all e = id, id, ps, vel, time, bx, P cm,v ch MOT d 2: ctime current time 3: e.ps e.ps +(ctime e.time) e.vel 4: e.time ctime 5: bxinvalid cps e.bx 6: if bxinvalid e.v ch then 7: cntinue 8: end if 9: ldbx e.bx 1: if e.v ch then 11: e.v ch false 12: if bxinvalid then 13: {Update the data assciated with e.id in the Index msb } 14: idx.updatedata(id, e.ps, e.vel, e.time ) 15: end if 16: end if 17: if bxinvalid then 18: α αβt able.lkup( e.vel,e.p cm) 19: e.bx Rect(e.ps, e.ps + α e.vel) 2: {Update the entry assciated with e.id in the Index msb } 21: idx.update(id,e.bx, e.ps, e.vel, e.time ) 22: end if 23: ldqids idx.uery(ldbx) 24: newresults idx.uery(e.bx, e.ps, e.vel, e.time) 25: fr all r = id, itv = itvs, itve newresults d 26: result results.get(r.id) 27: result.put(e.id, itv) 28: ldqids.remve(r.id) 29: end fr 3: fr all id ldqids d 31: result results.get(r.id) 32: result.remve(e.id) 33: end fr 34: end fr Algrithm 3: Mving Query Table Scan 1: fr all e = id, ps, vel, radius, time, bx, P cm,v ch MQT d 2: ctime current time 3: e.ps e.ps +(ctime e.time) e.vel 4: e.time ctime 5: bxinvalid cps e.bx 6: if bxinvalid e.v ch then 7: cntinue 8: end if 9: if e.v ch then 1: e.v ch false 11: if bxinvalid then 12: {Update the data assciated with e.id in the Index msb } 13: idx.updatedata(id, e.ps, e.vel, e.radius, e.time ) 14: end if 15: end if 16: if bxinvalid then 17: β αβt able.lkup( e.vel,e.p cm) 18: fps e.ps + β e.vel 19: e.bx Rect(e.ps sign(e.vel) e.radius, fps + sign(e.vel) e.radius) 2: {Update the entry assciated with e.id in the Index msb } 21: idx.update(id,e.bx, e.ps, e.vel, e.radius, e.time ) 22: end if 23: result results.get(e.id) 24: result.remveall() 25: newresults idx.uery(e.bx, e.ps, e.vel, e.radius, e.time) 26: fr all r = id, itv = itvs, itve newresults d 27: result.put(r.id, itv) 28: end fr 29: end fr uery evaluatin time (secs) Emprical uery evaluatin time measurement 4 β 2 2 α 4 6 nde IO cunt x β Analytical nde IO cst estimate Figure 4: Analytical nde IO estimate and experimental uery evaluatin time fcal bjects f the mving ueries. We can decide the best α and β values t use during MSB calculatin by perfrming a single lkup frm the ff-line cmputed αβt able. The graph n the left in Figure 4 plts the average time it takes t perfrm ne cmplete uery evaluatin step (labelled as ttal uery evaluatin time) as a functin f α and β. These values are frm the actual implementatin f mtin adaptive indexing. The graph n the right in Figure 4 plts the analytical nde IO cunt estimate f perfrming ne uery evaluatin step as a functin f α and β. Tw imprtant bservatins can be btained by cmparing these graphs. First, it shws that the IO cst is dminant n the time it takes t perfrm uery evaluatin, as the nde IO cunt graph highly determines the shape f the uery evaluatin time graph. Secnd, the ptimal values f α and β calculated using the analytical cst functin indeed results in faster uery evaluatin. The graph n the left in Figure 5 plts the ptimal α and β values calculated by the analytical cst estimate (using the right y-axis), as a functin f perid f cnstant mtin. The nde IO cunt is als shwn in the graph as an area chart (using the left y-axis). Nte that as the perid f cnstant mtin increases, the bject mvements are mre predictable. Fr small values f the perid f cnstant mtin, the ptimal β value turns ut t be small. This is because large β values will result in mre predictin, which is nt 2 2 α 4 6 analytical IO cunt estimate 2.5 x alpha beta ptimal alpha and beta analytical IO cunt estimate x perid f cnstant mtin (minutes) mving bject speed (m/h) Figure 5: Optimal α and β values desirable when the perid f cnstant mtin is small (predictability is pr). The graph n the right in Figure 5 plts the ptimal α and β values calculated by the analytical cst estimate (using the right y- axis), as a functin f bject speeds. The nde IO is als shwn in the graph as an area chart (using the left y-axis). The graph shws that fr high speeds the α and β parameters shuld be kept small, in rder t avid large MSBs which will cause high verlap and increase the cst f spatial index peratins. Mre experiments n the effect f adaptive parameter selectin will be prvided in Sectin 5. 4 Evaluating Mving knn Queries Mving cntinual k-nearest neighbr (knn) ueries ver mving bjects can be evaluated using the main mechanisms emplyed fr mving range uery evaluatin. A mving knn uery is defined similar t a mving range uery, except that instead f a range, the parameter k is specified fr retrieving the k nearest neighbrs f the fcal bject f the uery. A uniue feature f ur mtin adaptive indexing scheme is its ability t efficiently prcess bth cntinual mving range ueries and cntinual mving knn ueries. In rder t extend the mtin adaptive indexing develped fr alpha beta ptimal alpha and beta 7

8 evaluating mving range ueries t the evaluatin f mving knn ueries, we intrduce the cncept f safe radius and tw mechanisms guaranteed safe radius and ptimistic safe radius. T evaluate knn ueries with the use f safe radiuses, we need t make the fllwing three changes: a. Instead f string time intervals in uery result entries, we stre the distance f the bjects frm the fcal bject f the uery as a functin f time. b. During the MQT table scan, when a uery invalidates its MSB r changes its mtin functin, we calculate a safe radius which is guaranteed t cntain at least k mving bjects until the next time the safe radius is calculated (β is an upper bund fr this time). Then the knn uery is installed as a standard MQ with its range eual t the safe radius. c. At the end f each uery evaluatin step, results are srted based n their distances t their assciated fcal bjects by using the distance functins stred within the uery result entries. The tp k result entries are then marked as the current results. The imprtant step here is t calculate a safe radius, that will make sure that at least k bjects will be cntained within the safe radius during the next t time units. We prpse tw different appraches t tackle this prblem: the guaranteed safe radius (GSR) and the ptimistic safe radius (OSR). The guaranteed safe radius apprach retrieves the current k nearest neighbrs, and fr each bject in the list calculates the maximum pssible distance between the bject and the fcal bject f the uery at the end f the t time units. This can be calculated using the fcal bject s mtin functin and the upper bunds n the maximum speeds f these k nearest neighbr bjects. The maximum f these k calculated distances will give the safe radius. Hwever, there are tw prblems. First, it reuires us t knw the upper bunds n the speeds f mving bjects. Secnd, the calculated safe radius may becme t large. The ptimistic safe radius apprach retrieves the current k nearest neighbrs, and fr each bject in the list calculates the maximum distance between the bject and the fcal bject f the uery thrughut the next t time units. Fr each f the k bjects, this calculatin can be dne using the current mtin functin f the bject and the mtin functin f the uery s fcal bject. The maximum f these k calculated distances will give the safe radius. This apprach guarantees that k bjects will be cntained within the safe radius during the next t time units under the assumptin that the initial set f k nearest neighbrs d nt change their mtin functins during this perid. When using this apprach, if the number f bjects in the result f a knn uery turns ut t be smaller than k, we fall back t the traditinal spatial index knn search plan fr that uery until the next time a new safe radius is calculated. 5 Experimental Results This sectin describes five sets f implementatin based experiments, which are used t evaluate ur slutin. The first set f experiments cmpares the perfrmance f mtin adaptive indexing against varius existing appraches. The secnd set f experiments illustrates the advantages f adaptive parameter selectin Parameter Default value area f the regin f interest 5 suare miles number f bjects 5 percentage f mving bjects 5% number f ueries 5 percentage f mving ueries 5% mving uery range distributin {5, 4, 3, 2, 1} miles with Zipf param.6 static uery side range distributin {8, 7, 5, 4, 2} miles with Zipf param.6 perid f cnstant mtin mean 1 minutes gemetrically distributed mving bject speed between -16 miles/hur unifrmly randm scan perid 3 secnds mtin update time perid 3 secnds Table 1: System Parameters ver fixed parameter setting. The third set f experiments studies the effect f skewed data and uery distributin n uery evaluatin perfrmance. The furth set f experiments analyzes the scalability f the prpsed apprach with respect t ueries with varying sizes f spatial regins, varying percentages f mving ueries, and varying number f bjects. Finally the fifth set f experiments present the effectiveness f the mtin adaptive apprach t evaluating mving cntinual knn ueries ver mving bjects. 5.1 System Parameters and Setup We list the set f parameters used in the experiment in Table 1. In all f the experiments presented in the rest f the paper, the parameters take their default values if nt specified therwise. The default bject density is taken in accrdance with previus wrk [18, 19]. Objects and ueries are randmly distributed in the area f interest, except in Sectin 5.4 where we cnsider skewed distributins. Objects that belng t different classes with strictly varying mvement behavirs are cnsidered in Sectin 5.3. The paths fllwed by the bjects are randm, i.e. each time a mtin functin update ccurs, a randm directin and a randm speed are chsen. Fr R -trees a 11 nde LRU buffer is used with 4KB page size. Branching factr f the internal tree ndes is 1 and the fill factr is.5. All experiments are perfrmed using R -trees, except that in Sectin 5.4 a static grid based spatial index implementatin is used fr cmparisn purpses. 5.2 Perfrmance Cmparisn We cmpare the perfrmance f mtin adaptive indexing against varius existing appraches, in terms f uery evaluatin time and nde IO cunts. The appraches used fr cmparisn are: Brute Frce (BF), Object-nly Indexing (OI), Query-nly Indexing (QI), Object and Query Indexing (OQI), Mtin Adaptive Indexing (MAI), and Object Indexing with MSBs (OIB). The Brute Frce calculatin is perfrmed by scanning thrugh the bjects. During the scan, all ueries are cnsidered against each bject in rder t calculate the results. Object Indexing with MSBs is similar t pure bject-nly indexing, except that the mtin sensitive bxes are used in the place f bjects in the spatial index (withut the predictive uery results). Figure 6 plts the ttal uery evaluatin time fr fixed number f bjects (5K) with varying number f ueries (2.5K t 2K). The hrizntal line in the figure represents the scan perid. We cnsider a uery evaluatin scheme as acceptable when the ttal uery evaluatin time is less than the scan perid. Nte that the scan perid, P s, is set t be eual t the mtin update time perid P mu in this set f experiments. Figure 7 plts the uery evaluatin nde IO cunt fr the same setup. The nde IO is divided int fur different cmpnents. These are: (a) nde IO due t bject index 8

9 x bject table scan time uery table scan time bject index update is bject index search is uery index update is uery index search is scan time (s) nde IO MAIOIB OI QI OQI BF MAIOIB OI QI OQI BF MAIOIB OI QI OQI BF MAIOIB OI QI OQI BF 2.5K 5K 1K 2K number f ueries Figure 6: Query evaluatin time update, (b) nde IO due t bject index search, (c) nde IO due t uery index update and (d) nde IO due t uery index search. Each cmpnent is depicted with a different clr in Figure 7. Several bservatins can be btained frm Figure 6 and Figure 7. First, the appraches with an bject index that is updated fr all mving bjects, d nt perfrm well when the number f ueries is small. This is clear frm the pr perfrmances f OI and OQI fr 2.5K ueries, as shwn in Figure 6. The reasn is straightfrward. The cst f updating the bject index dminates when the number f ueries is small. This can als be bserved by the bject index update cmpnent f the OI in Figure 7. Secnd, the appraches with a uery index that is searched fr large number f bjects, d nt perfrm well fr large number f ueries. This is clear frm the pr perfrmances f QI and OQI fr 2K ueries, as shwn in Figure 6. This is due t the fact that, the cst f searching the uery index dminates when the number f ueries is large. This can als be bserved by the uery index search cmpnent f the QI in Figure 7. Third, the brute frce apprach perfrms relatively gd cmpared t OQI and slightly better cmpared t OI, when the number f ueries is small (2.5K), as shwn in Figure 6. Obviusly BF des nt scale with the increasing number f ueries, since the cmputatinal cmplexity f the brute frce apprach is O(N N ) where N is the ttal number f bjects and N is the ttal number f ueries. Althugh OQI seems t be a cnsistent lser when cmpared t ther indexing appraches, it is interesting t nte that the mtin adaptive indexing is built n tp f it and perfrms better than all ther appraches. Finally, it is wrth nting that nly MAI manages t prvide gd enugh perfrmance t satisfy P s P mu under all cnditins. MAI prvides arund 75-8% savings in uery evaluatin time under all cases when cmpared t the best cmpeting apprach except OIB. OIB perfrms reasnably well, but fails t scale well with increasing number f ueries when cmpared t the prpsed MAI apprach. 5.3 Effect f Adaptive Parameter Selectin In rder t illustrate the advantage f adaptive parameter selectin, we cmpare mtin adaptive indexing against itself with static parameter selectin. Fr the purpse f this experiment, we intrduce three different classes f mving bjects with strictly different mvement behavirs. The first class f mving bjects change their mtin functins freuently (avg. perid f cnstant mtin MAIOIB OI QI OQI MAIOIB OI QI OQI MAIOIB OI QI OQI MAIOIB OI QI OQI 2.5K 5K 1K 2K number f ueries Figure 7: Query evaluatin nde IO 1 minute) and mve slw (max. speed 2 miles/hur). The secnd class f mving bjects pssess the default prperties described in Sectin 5.1. The third class f mving bjects seldm change their mtin functins (avg. perid f cnstant mtin 1 hur) and mve fast (max. speed 5 miles/hur). In rder t bserve the gain frm adaptive parameter selectin, we set the α and β parameters t the ptimal values btained fr mving bjects f the secnd class fr the nn-adaptive case. Figure 8 plts the time and IO cst f uery evaluatin fr MAI and static parameter setting versin f MAI. The x-axis represents the bject class distributins. Hence, 1:1:1 represents the case where the number f bjects belnging t different classes are the same. Alng the x-axis we change the number f bjects belnging t the secnd class. 1:.25:1 represents the case where the number f bjects belnging t the first class and the number bjects belnging t the third class are bth 4 times the number f bjects belnging t the secnd class. Dually, 1:4:1 represents the case where the secnd class cardinality is 4 times the ther tw class cardinalities. Ttal uery evaluatin times are depicted as lines in the figure and their crrespnding values are n the left y-axis. The nde IO cunts are depicted as an embedded bar chart and their crrespnding values are n the right y-axis. There are tw imprtant bservatins frm Figure 8. First, we ntice that the adaptive parameter selectin has a clear perfrmance advantage. This is clearly bserved frm Figure 8, which shws significant imprvement prvided by mtin adaptive indexing ver static parameter setting in bth uery evaluatin time and nde IO cunt. Secnd, it is imprtant t nte that the bjects belnging t the first class r the third class cannt be ignred even if their numbers are small. Even fr 1:4:1 distributin, where the secnd class f bjects is dminant, we see a significant imprvement with MAI. Nte that bjects belnging t the first and the third class are expensive t handle. The first class f bjects are expensive, as they cause freuent mtin updates which in turn causes mre prcessing during MOT and MQT scans. The third class f bjects are als expensive, as they cause freuent MSB invalidatin which instigates mre prcessing during MOT and MQT scans. The fact that bth uery evaluatin time and nde IO cunt are declining alng the x-axis shws that it is bviusly mre expensive t handle the first and the third class f bjects. 9

10 ttal uery evaluatin time (s) nn adaptive adaptive uery eval. time nn adaptive adaptive nde IO x nde IO cunt ttal uery evaluatin time (s) R*Tree OMSB index R*Tree QMSB index StaticGrid OMSB index R*Tree QMSB index R*Tree OMSB index StaticGrid QMSB index StaticGrid OMSB index StaticGrid QMSB index ttal uery evaluatin time (s) range factr=.5 range factr=1. range factr=1.5 range factr=2. 6 1:.25:1 1:.5:1 1:1:1 1:2:1 1:4:1 mving bject type distributin Figure 8: Perfrmance gain due t adaptive parameter selectin Figure 11: Query and bject distributin fr N h =5and N h =3 5.4 Effect f Data and Query Skewness Our experiments up t nw have assumed unifrm bject and uery distributin. In this sectin we cnduct experiments with skewed data and uery distributins. We mdel skewness using tw parameters, number f ht spts (N h ) and scatter deviatin (d). We pick N h different psitins within the area f interest randmly, which crrespnd t ht spt regins. When assigning an initial psitin t an bject, we first pick a randm ht spt psitin frm the N h different ht spts and then place the bject arund the ht spt psitin using a nrmally distributed distance functin n bth x and y dimensins with zer mean and d standard deviatin. Scatter deviatin d is set t 25 miles in all experiments and the number f ht spts is varied t experiment with different skewness cnditins. Queries als fllws the same distributin with bjects. Figure 11 shws the bject and uery distributin fr N h =5and N h =3. We als experiment with different spatial indexing mechanisms. We have implemented a static grid based spatial index, backed up byab + -tree with z-rdering [7]. The ptimal cell size f the grid is determined based n the wrklad. The mtivatin fr using a static grid is that, with freuently updated data it may be mre prfitable t use a statically partitined spatial index that can be easily updated. Actually, previus wrk dne fr static range ueries ver mving bjects [9] has shwn that using a static grid utperfrms mst ther well knwn spatial index structures fr in-memry databases. With this experiment we als investigate whether a similar situatin exists in secndary strage based indexing in the cntext f MQs. Figure 9 plts the ttal uery evaluatin time as a functin f number f ht spts fr different spatial index structures used fr Index msb and Index msb. Nte that the smaller the number f ht spts, the mre skewed the distributin is. Figure 9 shws that decreasing the number f ht spts expnentially increases the uery evaluatin scan time. But even fr N h =5, the uery evaluatin time des nt exceed the uery evaluatin perid. Figure 9 als number f htspts Figure 9: Effect f data and uery skewness n perfrmance percentage f mving ueries (%) Figure 1: Effect f uery range & mving uery % n perfrmance shws that R -tree perfrms the best under all cnditins. But during ur experiments we als bserved that withut using the ptimizatin discussed under R*-tree implementatin detail (in the Appendix) fr decreasing the update peratin cst f the R -tree, the results are in favr f the static grid. Put differently, ur experiments cnclude that nly a prperly implemented R -tree with ptimized update peratin utperfrms a static grid based apprach. 5.5 Scalability Study In this sectin we study the scalability f the prpsed slutin with respect t the varying size f uery ranges, the varying percentage f mving ueries ver the ttal number f spatial ueries, and the varying ttal number f bjects. We first measure the impact f the uery range and the mving uery percentage n the uery evaluatin perfrmance. We use the range factr (r f )t experiment with different wrklads in terms f different uery ranges. The uery radius and uery side length parameters given in Sectin 5.1 are multiplied by the range factr r f in rder t alter the size f uery regins. Nte that multiplying the range factr by tw in fact increases the area f the uery range by fur. Figure 1 plts the ttal uery evaluatin scan time as a functin f mving uery percentage fr different range factrs. As shwn in Figure 1, the scalability in terms f mving uery percentage is extremely gd. The slpe f the uery evaluatin time functin shws gd reductin with increasing percentage f mving bjects. On the ther hand, increasing the range factr shws rughly linear increase n the uery evaluatin time. In Figure 12 we study the effect f the number f bjects n the uery evaluatin perfrmance. Figure 12 plts the ttal uery evaluatin time as a functin f number f bjects fr different spatial index structures used fr Index msb and Index msb. The number f ueries is set t its default value f 5K. Frm Figure 12 we bserve a linear increase in scan time with the increasing number f bjects, where the R -tree implementatin f Index msb and Index msb shw better scalability with increasing number f bjects than the static grid implementatin fr the similar reasn discussed befre. 5.6 Perfrmance f Cntinual knn Queries We cmpare the perfrmance f MQ based mving cntinual knn uery evaluatin against the bject-nly indexing apprach. In bject-nly indexing apprach, the bject index is updated and the knn ueries are evaluated against the updated bject index during each uery evaluatin step. In this experiment 1K bjects are used with the same bject density (N /A) specified in Sec- 1