A Trajectory Splitting Model for Efficient Spatio-Temporal Indexing

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1 A Trajectory Splitting Model for Efficient Spatio-Teporal Indexing Slobodan Rasetic Jörg Sander Jaes Elding Mario A. Nasciento Departent of Coputing Science University of Alberta Edonton, Alberta, Canada {rasetic, joerg, elding, Abstract This paper addresses the proble of splitting trajectories ially for the purpose of efficiently supporting spatio-teporal range ueries using index structures (e.g., R-trees) that use iniu bounding hyper-rectangles as trajectory approxiations. We derive a foral cost odel for estiating the nuber of I/Os reuired to evaluate a spatio-teporal range uery with respect to a given uery size and an arbitrary split of a trajectory. Based on the proposed odel, we introduce a dynaic prograing algorith for splitting a set of trajectories that iniizes the nuber of expected disk I/Os with respect to an average uery size. In addition, we develop a linear tie, near ial solution for this proble to be used in a dynaic case where trajectory points are continuously updated. Our experiental evaluation confirs the effectiveness and efficiency of our proposed splitting policies when ebedded into an R-tree structure.. Introduction Producing and collecting large volues of spatioteporal data has becoe ore practical in recent years, leading to increased availability and conseuently the need for efficient anageent of this type of data. For any applications, it is iportant to track, store, and uery data about oving objects, for instance, to deliver real tie services to clients based on spatial and teporal context. Application areas include fleet control, wireless counication networks, and obile coputing. In this paper, we focus on spatio-teporal ueries Perission to copy without fee all or part of this aterial is granted provided that the copies are not ade or distributed for direct coercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by perission of the Very Large Data Base Endowent. To copy otherwise, or to republish, reuires a fee and/or special perission fro the Endowent Proceedings of the 3 st VLDB Conference, Trondhei, Norway, 25 over historical trajectory data. Trajectories are often used to represent the path of oving objects. The authors in [2] distinguish two ain types of spatio-teporal ueries: coordinate-based ueries and trajectory-based ueries. Coordinate-based ueries return only the ids or the count of objects, for instance, the ids of objects whose trajectories intersect a given spatial region during a given tie interval. Trajectory-based ueries reuire the exact inforation about (partial) trajectories to deterine possibly coplex topological relationships (e.g., whether they cross or bypass an area) or navigational inforation (e.g., what was their top speed and direction within a certain area during a given tie interval). To process those ueries, usually, one or ore range ueries are used to extract the relevant trajectory segents fro an index. In order to process the iportant class of trajectorybased ueries efficiently, specialized index structures that support spatio-teporal range ueries are needed. Virtually all spatio-teporal index structures proposed in the literature are derived fro spatial index structures such as R-trees [9]. These approaches are based on the intuition that spatio-teporal data can be viewed as spatial objects in an extended spatial doain, where tie is treated as an additional diension. Trajectories are then represented by iniu bounding rectangles (MBRs). One straightforward solution within an R-tree is to approxiate each trajectory by a single MBR. This approach, however, yields poor approxiations, leading to low uery perforance in general (except possibly for ueries with very large spatial and teporal extent). Another straightforward solution is to approxiate each line segent of a trajectory individually by an MBR. Since each line segent can be oriented in only four different ways within an MBR [2], the orientation inforation and an MBR can be stored within each leaf node entry. In this case, inforation about each trajectory is copletely stored within the R-tree and can be reconstructed without any additional disk I/Os to a separate data level. This type of index is particularly effective for coordinate based ueries. However, the size of such an R-tree is, in general, uch larger than in the first approach, and the disk I/Os at the directory level are ore significant. A ore effective

2 alternative is to split trajectories and approxiate the resulting sub-trajectories independently by MBRs, balancing between the size of the index and the approxiation uality, which ay lead to a better overall perforance. The proble of splitting trajectories ially with the goal of iniizing the expected nuber of I/Os has not yet been treated rigorously. To our best knowledge, the only work that addresses this proble of splitting a set of trajectories to iprove uery perforance is presented in [7]. The authors assue a predeterined total nuber of allowed splits for a static set of trajectories, and propose a solution that distributes the splits aong the given trajectories so that the total volue of the resulting MBRs is iniized. We argue, and our experients confir, that such an iization goal does not necessarily lead to the best uery perforance. Our ain contributions in this paper are the following: We derive an analytical cost odel for evaluating the split of a trajectory into segents (in ters of expected I/Os), given a spatio-teporal range uery. Based on this odel, we introduce a dynaic prograing solution for splitting a given set of trajectories ially. In order to deal with dynaic cases where trajectories are updated increentally, we derive another cost odel that estiates an ial length for segents when increentally splitting a trajectory. Cobining our cost odels, we develop a linear tie trajectory splitting algorith, which in practice perfors close to the ial algorith, and which can be used in dynaic cases. Finally, we deonstrate through an extensive experiental evaluation that our algoriths are both efficient in practical situations and significantly outperfor other trajectory splitting approaches. The rest of the paper is organized as follows. Section 2 provides background and otivation for the paper. In Section 3, we derive a foral cost odel for evaluating the uality of trajectory splits and propose an algorith for finding the ial split with respect to this cost odel. In Section 4, a linear tie algorith for splitting trajectories heuristically is forally derived. A thorough experiental perforance evaluation and coparison is presented in Section 5. Section 6 discusses related work and Section 7 concludes the paper. 2. Background and Motivation The R-tree [6] is typically used to organize ultidiensional spatial objects using iniu bounding hyper-rectangles (MBRs) as approxiations. The leaf nodes store the MBRs of data objects and pointers to the object and their exact geoetry. Internal nodes store a seuence of pairs consisting of an MBR and a pointer to a child node. These MBRs enclose all entries stored in the sub-tree having the referenced child node as its root. To answer a range uery, starting fro the root, the set of MBRs intersecting the uery range is deterined, and then the corresponding child nodes are searched recursively until the data pages are reached. Trajectories are seuences of positions recorded at discrete points in tie. A linear interpolation between two successive locations is typically assued. A trajectory T is a seuence x, y, t ),( x, y, t ), K,( x, y, t ), where ( k k k (x i, y i ) is a spatial location, and t i is a tie instant. A consecutive seuence of points of a trajectory T is called a segent of T. A segent of length, i.e., consisting of 2 consecutive points, is called an eleentary segent. In order to index trajectories using an R-tree, each trajectory (or each of its segents if the trajectory is split) is approxiated by a 3-diensional MBR. It is easy to see that splitting trajectories, offers a great potential for iproving the perforance of spatio-teporal range ueries. When splitting a trajectory, the total volue of the approxiating MBRs decreases, and conseuently the approxiations ay less likely intersect range ueries, leading to an overall reduction of the nuber of data pages that have to be retrieved when processing these ueries. The actual aount of volue reduction when splitting a trajectory depends not only on the nuber of splits but also on the split points, as illustrated in Figure. t x Figure. MBRs for different trajectory splits. Based on this observation, Hadjieleftheriou et al. [7] proposed several algoriths for iniizing the total volue of trajectory approxiations given a user-specified nuber of splits s. First, a dynaic prograing algorith called DPSplit is proposed for splitting a single trajectory T using l splits so that the total volue of T s approxiations is iniized. The coplexity of this algorith is O(t 2 l), where t is the nuber of trajectory points in T. For the sae proble, they also described an O(t log t) greedy heuristic called MergeSplit. To split a set of n trajectories {T,,T n }, the authors proposed three algoriths that try to allocate to each trajectory T i a nuber of splits l i (out of the total nuber s of allowed splits), so that the overall volue of the trajectory approxiations is reduced as uch as possible. The first algorith uses a dynaic prograing approach, with a tie coplexity of O(ns 2 ). This algorith produces the ial solution with respect to volue reduction when cobined with DPSplit. They also introduce two heuristics with tie coplexity O(slogn+nlogn) that show satisfactory perforance in ters of volue reduction. All algoriths assue that the best splits of each trajectory into all possible nuber of splits are pre-coputed and stored, adding the overhead of DPSplit or MergeSplit (with l=t 2) for each trajectory. These approaches have several drawbacks:

3 Miniizing the total volue of trajectory approxiations does not necessarily iniize the nuber of expected I/Os when processing range ueries. Introducing ore splits necessarily reduces the total volue of the trajectory approxiations. However, it also increases the nuber of segents that ay siultaneously intersect a uery range, resulting in unnecessary I/Os. Figure 2(a) shows a scenario where a trajectory has an unnecessarily large nuber of splits for the given uery size; Figure 2(b) shows that the sae splits are appropriate for a saller uery size. The ethods reuire as input paraeter the total nuber of allowed splits for the whole set of trajectories. This paraeter is difficult to deterine even for a static set of trajectories. For the iportant dynaic case, where trajectories can continuously grow and new trajectories are added over tie, a fixed overall nuber of splits is not eaningful. Even knowing a good nuber of possible splits, the proposed algoriths are very tie consuing and have a large storage overhead. (a) (b) t y x Spatio-Teporal Query Range Figure 2. Trajectory splits and different uery sizes. We conclude that iniizing the volue of trajectory approxiations is not enough to iniize the expected nuber of I/Os for spatio-teporal range ueries. We clai that we also need to take into account uery sizes. Assuing a uery size ay see unusual and liiting at first glance, but we will see that it has several advantages: () our cost odel will also show that the two straightforward solutions for approxiating trajectories in R-trees not splitting, and splitting at each trajectory point are in fact just special cases, corresponding to trees that are iized for an extreely sall or extreely large uery size, respectively; these correspond to worst case assuptions about the uery size for ost real applications. (2) Having a uery size offers a potential for tuning the index which is an ion that the other splitting alternatives cannot provide. In practical applications, the assued uery size can be deterined as the average uery size coputed fro a workload of ueries. (3) An average uery size is also a ore natural and robust paraeter than a user-specified total nuber of splits as in [7], and is not restricted to static datasets. 3. Optial Trajectory Splitting In this section, we derive an analytical cost odel that estiates the expected nuber of I/Os yielded by a given split of a trajectory and a given uery size. Based on this odel, we introduce an algorith for splitting all trajectories in a set of trajectories so that the total nuber of expected disk I/Os for data pages is iniized. 3. A cost odel for splitting trajectories Given a trajectory T= p, p2,, p K t with i ( xi, yi, ti ) p =, we denote a segent of T that starts at point p u and ends at point p v by T[u,v] (using this notation T = T[,t]). A trajectory can be split along its discrete teporal diension into segents ( t ) in 2 possible ways (by choosing split points fro T, excluding t the endpoints p and p t ). A decoposition of T into segents, T=(T[,i ],, T[i -, t]) for a seuence of split positions i,, i -, will be approxiated by a seuence of MBRs B T =(MBR(T[,i ]),,MBR(T[i -, t])), where MBR(T[u,v]) denotes the MBR for segent T[u,v]. We denote the set of the MBR approxiations of all possible decopositions of T into segents by Decop( T, ) = {( B,..., B ) i,..., i : B = MBR( T[, i ]), () B2 = MBR( T[ i, i2]),..., B = MBR( T[ i, t])} For our cost odel, we assue that segents and their MBRs are stored independently, e.g., under an R-tree. That eans that the MBRs of a trajectory are generally stored on different disk pages. Under this assuption and ignoring possible effects of an index directory and caching, each segent s MBR that is intersected by a uery will reuire an independent disk I/O. Denoting the MBR approxiation of a specific decoposition of T into segents by B T = (B,, B ), the nuber of expected disk I/Os reuired to answer a uery is related not only to the total volue of the MBRs in B T but also to the size of. The size of a uery deterines the probability that intersects soe B i B T, which in turn deterines the expected nuber of I/Os that B T contributes to the total I/O cost of processing. Given a range uery the expected nuber of I/Os due to B T can be derived as follows. If intersects B T, then it intersects exactly k segents siultaneously, where k, yielding exactly k I/Os. The utually exclusive events that intersects exactly k segents of B T (and thus resulting in k I/Os) occur with probability P( BT ; k). Conseuently, the overall expected nuber of I/Os, E T B (), is the su of the I/Os due to each event, weighted by the probability of the event, i.e.: E = B T ) k = ( k P( BT ; k) (2) The following lea siplifies this expectation. Lea. Let P( Bi ) be the probability that a uery intersects the i th segent in B T. Then E = B T ) P( B i i= ( ) (3)

4 That eans that the expected nuber of I/Os can be coputed by siply suing up the probabilities of the uery intersecting the MBRs for the trajectory segents independently of each other. (A proof of k P( BT ; k) = P( B ) for the general case of k k i MBRs for spatial data can be found in []). In order to deterine P( Bi ), we consider in a finite data space S the area where a uery can fall and at the sae tie intersect Bi. This area, denoted by Ext (B i ), is given by extending B i by half of the uery extension in each diension (see Figure 3). Clearly, the uery intersects an MBR B i if and only if the uery center is inside the uery extended MBR Ext (B i ). Ext (B i ) B i y /2 Figure 3. A Query Extended MBR. Assuing a unifor distribution of ueries, and ignoring boundary effects, the probability of a uery intersecting a segent MBR B i is proportional to the noralized volue of the uery extended MBR Ext (B i ): P( B ) Vol( Ext ( B )) / Vol( (4) i = By substituting Euation (4) into (3), we obtain: i ( ) Vol( Ext ( B )) / Vol( (5) E = B T i= Miniizing this perforance easure for a single trajectory T, eans finding aong all possible decopositions of T into all possible nubers of segents, the split with the iniu expected nuber of I/Os, i.e., finding in { ( )}. EB T t, BT Decop( T, ) While splitting a trajectory always reduces the total volue of the MBRs approxiating the segents, this is not true for the uery extended MBRs. Figure 4 illustrates a 2-diensional case where the su of the volues of the uery extended MBRs is iniized when splitting the trajectory only once. Introducing ore splits will increase the su of the volues of the uery extended MBRs. Figure 4. Volues of approxiations using,, or 2 splits. Grey areas represent segents' MBRs, and dashed lines show the uery extended MBRs. So far, we have only considered how to split a single trajectory ially. Optially splitting a set of trajectories Θ, theoretically depends also on the directory structure of the particular spatial index used to store the MBRs. This directory structure depends on the page size, the distribution of the trajectories in space and the split algorith used for splitting overfilled node when constructing x /2 i Query x y the index. Modeling these aspects even for purely spatial data sees infeasible, and only epirically justified heuristics for splitting directory nodes in R-trees have been proposed so far in the literature. Therefore (and to be independent of the used index structure), we restrict our odel to the I/O cost due to the data level of an index structure (which doinates the total uery cost for ost R-trees), and ignore the possible effect of an index directory and the distribution of the trajectories in space. In this case, each trajectory T in Θ contributes independently towards the total nuber of expected I/Os for data pages, given a uery, i.e., the expected nuber of I/Os can be coputed as the su of the individual expectations: E ( ) = E T ( ) (6) total B T Θ Given Euation 6, we can find the ial splits for a set of trajectories (with respect to ), by iniizing the splits for each trajectory individually. In general, a trajectory T can be split into segents in different ways, possibly resulting in a different nuber of I/Os when processing. Let E T, ( ) be the iniu expected nuber of I/Os that can be obtained for T by splitting T into segents: E ( ) = in { E ( )} (7) T, BT BT Decop( T, ) A trajectory T can be split into different nubers of segents, ranging fro to t. Conseuently, the inial nuber of I/Os over all possible splits, E T (), is given by the best possible split of T for ranging fro to t : E ) = in { E ( )} (8) T ( T, t As a conseuence of this cost odel, it is easy to see that an R-tree obtained by not splitting any trajectory is euivalent to an R-tree that is iized for a uery size where even splitting the largest trajectory will result in a larger expected nuber of I/Os than not splitting, which is only possible if the uery is very large. Siilarly, an R- tree obtained by splitting every trajectory into all its segents is euivalent to an R-tree that is iized for a uery size where every split of a trajectory introduces a gain in the expected nuber of I/Os, which is only possible if the uery is very sall. Because of space liitations, we have to oit the analytical details here, but clearly, those uery sizes are not representative for ost real applications. However, they are iplicitly and unchangeably integrated into these respective R-trees. Dynaic Prograing Algorith To solve Euation 8, we propose a dynaic prograing solution, which finds the best possible split of T for each value of. Using our notation T[u,v] to denote a segent of T fro point p u to point p v, we can re-write ( ) as E T[, t], proble, we have to show the following property: E T, ( ). In order to apply dynaic prograing to our

5 Theore. Given a trajectory T= p, p2, K, pt and a uery, it holds that E ) in{ E ( ) E ( )} (9) T [, t], ( = T[, u], + T[ u, t], < u< t Proof. (Sketch) Using Euation 5, Euation 7 can be rewritten as Vol( Ext( Bi )) ET [, t], ( ) = in. BT Decop( T[, t], ) i= Vol( Expanding the su in this euation gives us = Vol( Ext ( Bi )) Vol( Ext ( B )) ET [, t], ( ) in +. T B Decop( T[, t], ) i= Vol( Vol( Let the start position of the last segent of an ial decoposition of T be u, and let B = MBR( T[ u, t]) denote the MBR of this last segent. Then we obtain Vol( Ext ( B )) Vol( Ext ( B )) i ET [, t], ( ) = in + T B DecopT ( [, u], ) i= Vol( Vol( = ET [, u], ( ) + ET [ u, t], ( ) This euation holds since the last segent (starting at u) is fixed by assuption, and the reaining prefix of T, T[,u] ust conseuently be split into segents so that the su of volues of the extended MBRs for the first segents is inial, in order for the whole su to be inial. To find the ial decoposition of T without knowing u, we have to consider all possible values of start positions u in the range < u < t for the last segent of T, as stated in the theore: E ) in{ E ( ) E ( )} T [, t], ( = T[, u], + T[ u, t], < u< t Theore states that in order to find the ial solution for a trajectory T using segents, it is sufficient to consider all ial sub-solutions using segents for the prefixes T[,u], < u < t, (which can be found by recursively applying Euation 9), and cobine the with the solution for the reaining segent T[u,t]. The runtie of a dynaic prograing algorith that accordingly deterines the split of one trajectory T into segents (i.e., splits) is O(t 2 ( )) where t is the nuber of points in T. Conseuently, to find the best possible split for T aong all possible values of, the algorith has to be applied for the axiu possible value of, i.e., for =t. To split n trajectories ially, the algorith has to be applied n ties. This tie coplexity is the sae as the tie coplexity for the DPSplit precoputation step used in Hadjieleftheriou et al. s algoriths [7]. Our algorith, however, does not need to execute an additional, tie consuing and storage intensive search algorith on top of this solution to obtain a globally ial solution with respect to our cost odel. 3.3 Directory Level Node Splitting So far, we have only considered access to data pages. Assuing the MBRs approxiating trajectory segeents are stored independently on disk pages. For this estiation, it is not essential that the MBRs enclose trajectory segents. Trajectories only deterine the possible points that can be considered when splitting the, resulting in different sets of MBRs. For R-tree based indices, directory pages ay be split during index construction and during updates. Different heuristics have been proposed for that purpose, such as the uadratic and the linear split [6], or the R*-tree split []. These algoriths generate a certain subset of all possible splits of an MBR and iniize evaluation functions, which are typically based on volue and overlap of the resulting MBRs. The goal of these heuristics is essentially to iniize the probability that ueries will intersect both resulting MBRs thus reducing the nuber of subtrees that have to be traversed. The rationale behind our cost odel can be applied to directory level splits as a heuristic evaluation function as well: Given the average uery size used to split trajectories, we can choose aong the possible splits of a directory node the split that iniizes the volue of the resulting uery extended MBRs using Euation 5. More precisely, we replace the node split evaluation function using our cost odel, keeping the original algorith for generating candidate node splits. 4 Heuristic Trajectory Splitting The above split strategy reuires coplete trajectories to be available in order to find the ial splits. For any applications, however, trajectories are updated continuously. Another liitation is that for large datasets containing long trajectories, even if they were copletely available, the dynaic prograing solution ay be too inefficient to be practical. For such applications, a ore efficient and increental ethod is needed, which ideally can produce near ial results. 4. A Cost Model for Optial Segent Size In this section we assue trajectories where points are continuously added over tie. We forally derive an approxiation of the ial split of a trajectory that can be coputed increentally. Consider first the special case of trajectories for objects oving with constant speed in a constant direction that are sapled at constant tie intervals (see Figure 5 for a 2d illustration). For brevity, we call such trajectories constant-slope trajectories. We will show that the ial split of these trajectories, according to our previous cost odel, will result in segents of eual size. Using this property, pieces of arbitrary trajectories can be approxiated by constant slope trajectories and split nearially in linear tie. Sapling at constant tie intervals does not really constitute a restriction here since we assue a linear interpolation between sapling points so that constant tie intervals can always be achieved by a suitable re-sapling.

6 Assue a trajectory T consisting of t points, or euivalently, consisting of t consecutive eleentary segents s,, s t- as well as a decoposition of T, B T ={B,,B }. The su in Euation 5 can be expressed differently by thinking of the volue of each Ext (B i ) as being generated by the eleentary segents contained in B i, via a function f that expresses an eual contribution of each eleentary segent s to the volue of the uery extended MBR it belongs to: Vol( Ext( Bi containing s)) f ( s) = () # eleentary segents in B containing s Lea 2. Let B T ={B,,B } be a decoposition of a trajectory T, and f be defined as in Euation. Then, = t Vol ( Ext ( Bi )) f ( s) = f ( si ) () i = i = s contained in Bi i = Proof. Trivial. By construction of f, it holds that f ( s) = Vol( Ext ( B i )) for each Bi in B T. s contained in B i The significance of Lea 2 is that a decoposition that iniizes the right hand side also iniizes the left hand side. Miniizing the right hand side is in general not an easier proble, since the f values for eleentary segents depend on where the actual split points for a split of T are. However, for constant-slope trajectories, the f values depend only on the nuber of eleentary segents in an enclosing MBR B i, i.e., we can copute the volue of Ext (B i ) using only the increents in each diension ( x, y, ) that define the slope of the trajectory, and the nuber c of eleentary segents in B Vol ( Ext ( Bi )) = ( c x + x ) ( c y + y ) ( c + t ) (2) t t /2 x /2 i i : x x Figure 5. Illustration of a constant-slope trajectory sapled at regular tie intervals. Furtherore, we can now look at the values of f for MBRs of varying nuber of contained eleentary segents c by looking at its definition as a function g of c, ( c x + x ) ( c y + y ) ( c + t ) g( c) = (3) c The significance of this function is that g(c) has a real iniu, which eans that there is an ial segent length c (i.e., ial nuber of consecutive eleentary segents that for the segents) for constant slope trajectories T, in the sense that if T is decoposed into segents of this length, the value f(s) will be inial for each eleentary segent s. This in turn eans that this decoposition iniizes the right hand side of Euation, giving us an ial decoposition according to our cost odel in Section 3. In this ial decoposition, all the segents have the sae length c, which is independent of the length of T. The value c deterines where the split points have to be; it is only dependent on the increents in each diension in T and the uery size, which also eans that we can ially split constantslope trajectories in an increental anner, i.e., after soe points of T have been added. We will use this fact later to derive a heuristic increental splitting algorith for arbitrary trajectories by conceptually approxiating the with several constant-slope sub-trajectories. Theore 2. Given a uery, and increents ( x, y, ) (that define the slope of a constant-slope trajectory), the function g (Euation 3) has a global, real iniu c with respect to c. Proof. Function g can be re-written as k c3 k c k3c + k4 g( c) = c where k = x y, k = x y + y + x 2 t x y k = x + y +, k = 3 y t x t x y 4 x y t Dividing by c gives g ( c) = ( k 2 c + k2c + k3 + k4 ) c Applying the first derivative to find extreas, we get dg( c) = (2k c + k2 + ( k4 )) = dc c2 which is euivalent to finding the solutions to ( 2k 3 2 c + k2c + ( k4)) = (4) This cubic euation has an analytical solution c in the doain of positive real nubers. The second derivative is always greater than, so g(c) has a iniu at c= c. Intuitively, we can use the value c that iniizes g(c) to construct an ial decoposition of a constantslope trajectory T (according to the cost odel in section 3) by dividing it into segents of eual length, containing c any eleentary segents. In practice, since c does not depend on the nuber of trajectory points and can be coputed using only the increents x, y, of T and the uery size, we can split T into segents of length c continuously as points are added to T over tie. Theore 3. Given a uery, and a constant-slope trajectory T = p, p2,, p defined by increents ( x, y, ), K t we can find c that iniizes g(c) (according to Theore 2). Assuing that t is divisible 2 by c, the decoposition of T into segents of eual length deterined by c is a solution to Euation 8, i.e., a decoposition that iniizes the expected nuber of I/Os. Proof. Without loss of generality, let c be an integer. 3 Let BT = B,..., B ) be the decoposition of T where c ( each Bi contains the sae nuber c of eleentary seg- 2 This is justified by the fact that we assue long trajectories where points are continuously added. 3 If c is not an integer, we can re-saple T so that c can be expressed as an integer w.r.t. the new eleentary segent size.

7 ents s. The resulting values f(s) (Euation ) for each eleentary segent s is by Theore 2 inial, i.e., no other MBR size can give saller f(s) values. Conseuently, the su f ( ) is inial for the decoposition BT c s i i=,... t aong all possible decopositions. By Lea 2, this decoposition is also a solution to Euation 8. So far, in this section, we have assued trajectories of constant slope that are sapled at constant tie intervals. This assuption is not true for ost trajectories in practical applications. However, we can still apply our odel to an arbitrary trajectory T by approxiating it with a constant-slope trajectory T in the following way. We can copute the increents x, y, that define the slope of T as the average of the corresponding increents of T, t e.g., in x direction: x = x i, where x i represents t i= the difference in x direction between two consecutive points of T. Obviously, the saller the variance in the increents of T is, the better is the approxiation T. Although the error of the approxiation can be large for long trajectories, this is not true for sub-trajectories in case of ost real world applications since objects usually keep oving in a siilar direction with a siilar speed for certain periods of tie. The fact that we can typically approxiate a long trajectory well, using several constant-slope sub-trajectories, allows us to design an increental splitting algorith that perfors nearly ial in practice (unless objects ove extreely erratically). 4.2 Linear Tie Trajectory Splitting To split a trajectory T increentally, we can buffer a certain nuber of incoing points of T, say fro point p u to point p v, and copute the average increents x, y, for the points in the buffer to obtain a constantslope approxiation T [u,v] for the trajectory segent T[u,v] in the buffer. Using the proof of Theore 2, we can then deterine the ial nuber c of eleentary segents that should be grouped together in an ial decoposition of T [u,v], and then use this nuber to decopose T[u,v] accordingly. To apply this ethod, we have to deterine a suitable nuber of points that should be buffered before applying the split policy. This nuber ay depend on several factors including the average uery size, the speed, the direction changes, and the sapling rate of the oving object. For different trajectories, and even for different segents of the sae trajectory, a different buffer size ay be appropriate. The cost odel for ially splitting a trajectory fro Section 3 can be used as a heuristic to deterine dynaically a suitable buffer size. We can deterine when an MBR around a trajectory segent T[u,v+] is not ial, according to the following condition. E ) > E ( ) E ( ) (5) T [ u, v+ ],( T[ u, v], + T[ v, v+ ], This condition is true, when the expected nuber of I/Os using one MBR around the segent of T[u,v+] is larger (i.e., worse w.r.t. perforance) than the nuber of expected I/Os when introducing a split before the last eleentary segent of T[u,v+]. In this case, it akes sense to consider splitting T[u,v+] since there is at least one possible split (before the last eleentary segent) that will result in a better I/O expectation. This split is, however, in general not the best possible way of splitting the current segent T[u,v]. Iteratively collecting points until Euation 5 becoes true, then introducing a split at exactly that position, and repeating this until the trajectory ends, will, in general, create segents that are consistently larger than the segents obtained by an ial split. Euation 5 is good at detecting significant changes in speed and direction of a trajectory. For nearly constantslope segents of a trajectory, the condition tends to be true only after several ties the ial segents size has been accuulated. We have confired this behavior experientally, but we can also understand it ore forally. Consider the difference between the left hand side and the right hand side of Euation 5. E ) [ E ( ) E ( )] (6) T [ u, v+ ],( T [ u, v], + T [ v, v+ ], For the case of constant-slope trajectories, we can copute the expected I/O values in this expression as the volues of the uery extended MBRs around T[u,v+], T[u,v], and T[v,v+] respectively, using Euation 2. The nuber of eleentary segents in T[u,v] is c = v u. After siple arithetic transforations, we obtain: E 2 T[ u, v+ ], ) ET [ u, v], ( ) ET [ v, v+ ], ( ) = 3k c + 3k c+ 2k2 ( c k where k, k 2, k 4 are defined as in the proof of Theore 2. Conseuently, Euation 5 holds if 3k 2 c + 3k c+ 2k2c > k4. (7) On the other hand, we know fro the proof of Theore 2 that the function g (Euation 3) has a global iniu c for the ial nuber of eleentary segents at k c3 + k c2 k or, euivalently if = 2k + k (8) 3 2 c k2c = 4 Substituting Euation 8 in Euation 7, tells us when the condition in Euation 5 is true in ters of the nuber of eleentary segents for a constant-slope trajectory, i.e., it is true if k c2 + 3k c + 2k c > k c3 + k c2 (9) It is easy to see that this ineuality holds if the value of c is larger than or eual to the value of c for all c 2 (i.e., if the buffer contains at least 2 eleentary segents, which is reuired in practice before considering a split). In suary, this eans that the nuber of eleentary segents c, collected up to the point where condition 8 becoes true (i.e., the trajectory buffer at that point), is always a ultiple of the ial segent size. Using a dynaically deterined buffer size according to these considerations, we propose a linear tie trajectory splitting algorith, called LinearSplit. The algorith 4

8 collects points of a trajectory consecutively. For each new point p v+, it deterines whether the new point should be erged into the current buffer T[u,v] or whether a split at this point would iprove I/O expectation according to Euation 5. If the condition is true, the ial segent size c (Theore 2) is coputed (using a constant-slope approxiation of the current trajectory segent T[u,v]), and we round c to the nearest positive integer *. We split as any segents of size c* c as possible fro T[u,v] and insert the corresponding MBRs into the index. This procedure is repeated as long as new points are added. If a trajectory is copleted, the last segent which ay still be in the buffer has to be inserted as well. Obviously, this algorith splits a trajectory in O(t) tie where t is the nuber of points of a trajectory. The pseudo code for the algorith LinearSplit is given below. Algorith LinearSplit u :=, v := 2; //after the first 2 points of T while (next point pv+ in trajectory T exists) if E ) > E ( ) E ( ) T [ u, v+ ], ( T[ u, v], + T[ v, v+ ], find c for T[u,v] using Theore 2; c* = round(c ); extract the first k= (v u)/c* segents fro T[u,v]; insert their MBRs into the index; u:=u + k * c*; v++; //end of T is reached insert last MBR(T[u,v]) into the index; 5. Experiental Results For the experiental evaluation, we used two datasets, produced by the Network Data Generator [2] and by GSTD [7], respectively. The network data generator siulates different classes of objects, e.g., vehicles and people, oving through streets of a real city (Oldenburg). Different objects have different speeds and lifeties, giving a rich and realistic dataset. GSTD allows generating ore rando patterns suitable to investigate the perforance of the algoriths under ore extree situations. For each generator, we produced datasets containing,, 2,, and 5, trajectories, respectively. For each trajectory in the network datasets, a varying nuber of observations ranging between 5 and 345 were recorded, resulting in 97 observations on average per trajectory. We set GSTD s paraeters so that trajectories were fored by objects, uniforly distributed in the data space, changing speed and direction randoly at any point in tie (the axiu speed was liited though, so that an object could not cross ore than 2% of the total space fro one tie stap to the next). This scenario, when objects are oving extreely erratically, was expected to be particularly challenging for the LinearSplit algorith. Exactly observations were recorded for each trajectory. Therefore, all our datasets had between,, and 5,, observations. All experients were perfored on a 9+ AMD Athlon PC with 52 Mb RAM. We used the R-tree ipleentation provided by the XXL library [3], using a 4Kb page size for all algoriths. For our algoriths, we replaced the split evaluation function using our cost odel as described in Section 3.3. We evaluate the uality of all algoriths by easuring the nuber of disk I/Os on the index s directory and data level per uery, averaged over, uniforly distributed ueries, without considering buffering. We also easure the actual tie reuired to pre-process a dataset, i.e., the tie reuired to split the trajectories, create the MBRs and create the index tree. Our dynaic prograing-based algorith is referred to as OptialSplit ; the linear tie algorith is referred to as LinearSplit. HKTG-k% denotes the volue oriented split policy proposed in [7] (using the DPSplit algorith for splitting trajectories individually), where k% eans that N k total nuber of splits are used for splitting a dataset with N trajectories [7]. Siilar to [7], we set k eual to 5, and 5. 4 We also copare with two baseline algoriths. First, an R-tree, referred to as NoSplit, where each trajectory is approxiated by a single MBR, i.e., trajectories were not split (as discussed above, this is euivalent to a tree, iized for a very large uery size). Second, an R-tree, referred to as FullSplit, where each eleentary segent of trajectory is approxiated by an MBR, i.e., trajectories were split at each observation point (as discussed above, this is euivalent to a tree iized for an extreely sall uery size). 5. Optiality and Robustness w.r.t. Query Size In the first experient, we used datasets of 5, trajectories to study the suitability and robustness of the cost odels and the associated split algoriths. We easured the average nuber of disk I/Os at the data level here, since the objective of the odels is iniizing data level I/Os. We built trees that are iized for different uery sizes of S% of each spatial diension and T tie points. In particular, S = %, 2%,, 6% and T=, 2,, 6. 5 We use I i,j to denote the index that is iized for the uery size with spatial extensions given by S=i% and teporal extension T=j. Siilarly, Q i,j denotes the size of the ueries that were executed against the different indices. The nubers in Tables through 4 represent the average perforance for different indices and different uery sizes. 4 We also set k for the HKTG-k% algorith so that it corresponded to the ial nuber of splits found by our ial split algorith. However, when using our datasets we could not finish building the trees even after a few days. We did test the HKTG-k% with the ial nuber of splits for very sall data sets, up to 5, trajectories. On those dataset, the nuber of data level I/Os of the HKTG-k% algorith is very close to ours when provided with the ial nuber of splits, however, the overall I/O perforance for ueries was uch worse than ours due to a larger overhead for directory level I/Os. 5 The spatial extension of a uery is given as percentage of the total 2d space since the space is finite; the teporal diension is given in absolute tie points since tie is unbounded.

9 Each row represents the perforance of a uery Q i,j using all the constructed indices (listed in the coluns). If the odels are appropriate, given a uery size, the best perforance should occur when using the index built for that uery size, i.e., in the diagonal of the tables. Note that we copare in the rows the perforance of different trees for a given uery. Coparing different ueries for the sae tree here (i.e., looking at coluns) only shows the obvious fact that saller ueries result in saller nubers of I/Os than larger ueries. Note that, if we would include the NoSplit and FullSplit trees, their perforance would be shown in coluns to the left, respectively right side of the tables, since they correspond to trees iized for an even saller, respectively larger uery size than the given ones. Their perforance is worse for every uery than the values for the given trees, which are interediate trees with respect to the degree of trajectory splitting. The perforance of both algoriths on both datasets is ualitatively very siilar. The best perforance (shaded cells) occurs exactly where expected for the OptialSplit (Tables and 3). Even for LinearSplit (Tables 2 and 4) this is true, except for surprisingly only one case when using GSTD data (Q 2,2 in Table 4), since this data sets has by construction a very erratic behavior. Overall, the LinearSplit heuristic approxiates the OptialSplit very well not only in ters of where the iu is but also in ter of absolute nubers of I/Os, bu at a uch lower coputational cost. Furtherore, we can also observe that the uery perforance degrades on average only by 3% when ueries were run against indices iized for ueries two ties saller or larger than the used uery size. This indicates that the algoriths are uite robust with respect to the assued average uery size, but looking at the perforance of a uery against even ore different trees (which in practical worst cases could be the FullSplit or NoSplit tree), it is clear that the uery size paraeter offers a great opportunity for tuning an index to a particular workload of uery sizes. We will explore this in ore detail in the following experients. Table. Robustness of OptialSplit for Network Data # I/Os Tree iized for S(%) and T (duration) I, I 2,2 I 4,4 I 8,8 I 6,6 Q, Q 2, Q 4, Q 8, Q 6, Queries Table 2. Robustness of LinearSplit for Network Data # I/Os Tree iized for S(%) and T (duration) I, I 2,2 I 4,4 I 8,8 I 6,6 Q, Q 2, Q 4, Q 8, Q 6, Queries Table 3. Robustness of OptialSplit forgstd Data # I/Os Tree iized for S(%) and T (duration) I, I 2,2 I 4,4 I 8,8 I 6,6 Q, Q 2, Q 4, Q 8, Q 6, Queries Table 4. Robustness of LinearSplit for GSTD Data # I/Os Tree iized for S(%) and T (duration) I, I 2,2 I 4,4 I 8,8 I 6,6 Q, Q 2, Q 4, Q 8, Q 6, Queries 5.2 Nuber of Disk I/Os 5.2. Varying Query Size To study the perforance of different uery sizes we used again databases with 5, trajectories. The definition of the used ueries, siilarly to [7], is as following: Snapshot Query Sizes Spatial extent in each di. (S%) Sall (S 3 Mediu (SM) 3 9 Large (SL) 9 27 Duration (T) Range Query Spatial extent in Duration Sizes each di. (S%) (T) Sall (R Mediu (RM) Large (RL) To provide a thorough and realistic perforance analysis we now easure the nuber of I/Os at both the directory and the data level, and we assue a uery load of ueries with varying sizes, where each spatial extent and duration within the liits of a uery type is eually freuent within that uery type. Two different approaches are used with respect to our algoriths. In the ultiple tree approach, we built an index for each uery type separately (i.e., 6 per algorith), using the average size of a uery type as input paraeter (e.g., for the RS uery type, the average uery size is S=6% and T=2). 6 A uery was then run against the tree that was iized for the uery s uery type. In the single tree approach, we built only one index for all uery types per algorith, i.e., we deterine the average uery size over all given uery types (i.e., S=7.3% and T=4.83), and use this uery size as paraeter for the cost odels. The resulting index is then used to answer all ueries. Figure 6(a) (d) show the average nuber of I/Os per uery for different data sets and approaches. Note that 6 Note that the trajectory data does not have to be replicated; only different index directory structures were created.

10 using ultiple trees or just a single tree affects only OptialSplit and LinearSplit, and the values for other algoriths are conseuently the sae for the sae dataset (a) Multiple Trees Network Data I/O Perforance - M ultiple Trees (Network Data) SS SM SL RS RM RL Query Types (b) Multiple Trees GSTD Data I/O Perforance - M ultiple Trees (GSTD Data) 727 SS SM SL RS RM RL Query Types (c) Single Tree Network Data I/O Perforance - Single Tree (Network Data) In all scenarios, our approaches consistently outperfor all others, and LinearSplit shows perforance close to OptialSplit, confiring again the suitability of the linear split heuristic. For SS and RL ueries on the Network data, the FullSplit algorith perfors copetitively to our approaches, however for other uery types the perforance can be uch worse. Our approaches have a significantly lower nuber of I/Os on the data level than the other algoriths (except FullSplit which has no separate data level). 7 Note also that our algoriths in general result in less directory I/Os than the NoSplit and the HKTG-k% algoriths even though our trees are typically larger since we introduce ore splits (except FullSplit which always has the largest tree). The perforance of our algoriths using ultiple trees is very close to the perforance of using only a single tree for a workload of all uery types, confiring again the robustness of our approach Varying Database Size To easure sclability, we created indices for different datasets with,, 2,, and 5, trajectories. We ran ediu sized range ueries of type RM against all indices, where our indices where built for the average RM uery size. The results are shown in Figure 7(a) and (b) (a) Network Data I/O Perfor ance (Network Data) 3 SS SM SL RS RM RL Query Types (d) Single Tree GSTD Data I/O Perforance - Single Tree (GSTD Data) K 2K 5K DB Size (b) GSTD Data I/O Perfor ance (GSTD Data) SS SM SL RS RM RL Query Types Fro left to right: ) NoSplit, 2) HKTG-5%, 3) HKTG-%, 4) HKTG-5%, 5) OptialSplit, 6) LinearSplit, 7) FullSplit; Shaded botto parts of bars: leaf level I/Os; Blank upper parts of bars: directory level I/Os Figure 6. Average #I/Os for different uery types. Each bar in the figures represents the average nuber of I/Os per uery and consists of two parts: the botto (shaded) part corresponds to the average nuber of hits on the data level while the top (blank) part corresponds to the average nuber of hits on the directory levels of the indices except for the FullSplit algorith where the trajectory inforation is copletely stored in the directory and conseuently all hits are directory level hits. 4 K 2K 5K DB Size Fro left to right: ) NoSplit, 2) HKTG-5%, 3) HKTG-%, 4) HKTG-5%, 5) OptialSplit, 6) LinearSplit, 7) FullSplit; Shaded botto parts of bars: leaf level I/Os; Blank upper parts of bars: directory level I/Os Figure 7. Average #I/Os per uery, varying DB size The I/O perforance or our algoriths is always significantly better than the NoSplit and the HKTG-k% algoriths. For the saller datasets FullSplit perfors co- 7 Reducing data level I/Os also reduces false hits, which also saves CPU tie for coputationally intensive algoriths that are invoked to deterine whether a trajectory segent approxiated by an intersected MBRs actually intersects a given uery.

11 petitively to our approaches, but its perforance degrades uch faster with increasing database size. Note that, even though the OptialSplit always results, by design, in the sallest nuber of data level I/Os, this does not guarantee the best overall perforance. In soe cases LinearSplit exhibits the best perforance due to a saller nuber of directory level I/Os, which is due to the heuristic nature of directory node splitting policies. 5.3 Index Building Tie 5.3. Varying Query Size The index building ties for all algoriths are shown in Figure 8(a) and (b), where we use a tree for every uery type. NoSplit is clearly the fastest since it has to insert only one MBR per trajectory. For the HKTG-k% algoriths, ost of the tie is spent splitting the trajectories. FullSplit has to insert one MBR per eleentary segent of each trajectory consuing also a significant aount of tie. Our algoriths exhibit a good balance between trajectory splitting tie and insertion tie, outperfored only by the trivial NoSplit. Obviously, as the uery size increases, our index building ties decrease since trajectories are split less and fewer MBRs are inserted. Preprocessing Running Ti e (sec. )- M utiple Trees (a)(network Data) Data SS SM SL RS RM RL Query Types Preprocessing Running Ti e (sec. ) - M ultiple Trees (b)(gstd Data) Data SS SM SL RS RM RL Query Types Fro left to right: ) NoSplit, 2) HKTG-5%, 3) HKTG-%, 4) HKTG-5%, 5) OptialSplit, 6) LinearSplit, 7) FullSplit Figure 8. Preprocessing tie in seconds Varying Database Size To easure the scalability of the index building tie with respect to database size, we used again our indices for the databases containing,, 2,, and 5, trajectories, where our indices where built for the average RM uery size. The results are shown in Figure 9. As expected the index building tie for all algoriths increases as the database size increases. Our algoriths scale linearly at a uch slower rate than all other ones (again with the exception of the trivial NoSplit algorith) Preprocessing Running Tie (sec. ) (Network Data) K 2K 5K DB Size Figure 9. Preprocessing tie in seconds 6. Related Work Network Data Fro left to right: ) NoSplit, 2) HKTG-5%, 3) HKTG-%, 4) HKTG-5%, 5) OptialSplit, 6) LinearSplit, 7) FullSplit Most spatio-teporal index structures proposed in the literature are based in one way or another on R-trees [9]. They can be classified into three ain approaches. In the first approach, tie is siply treated as an additional spatial diension [6]. For trajectories, this leads to inefficient indices since the MBRs tend to be very large, covering large portions of epty space and leading to a high degree of overlap aong the MBRs. Another structure under this approach is the TB-tree [2]. Its insertion split strategy is oriented towards trajectory preservation so that leaf nodes only contain segents that belong to the sae trajectory. The ain disadvantage, however, is that concessions to the ost iportant R-tree property, node overlap ust be ade. Indeed, experiental results show that it is outperfored by a regular R-tree for spatio-teporal range ueries, in particular for sall ueries. In the second approach, tie and space are treated differently within a cobined indexing schee, e.g., [4] and [4]. In [4] a two level index is proposed. Where the space is first partitioned into non-overlapping cells, and for each cell, an R-tree is used to index the teporal intervals at which objects were in those cells. In [4], the space is first partitioned into zones, and the locations of objects are only represented by zone ids, resulting in a less accurate but ore efficient representation, anaged by the SEB tree. These types of approaches are not copatible with our cost odels since they don t use MBRs. The third approach also treats tie differently fro space. The idea is to have virtual and increentally aintained 2-diensional R-trees for each point in tie []. This approach, however, suffers fro a prohibitively large overhead when indexing very dynaic scenarios, and is not suited for trajectory data. Recent work aiing at iproving the first approach has proposed two orthogonal strategies: replacing MBRs by different approxiations, and splitting trajectories. In [9], the authors propose to tri the corners of trajectories MBRs in order to obtain a bounding octagon pris, instead of a bounding hyper-rectangle, which is a tighter approxiation. The experiental results, however, do not provide clear evidence that a considerable gain is obtained for spatio-teporal range ueries, when copared to an