Querying about the Past, the Present, and the Future in SpatioTemporal Databases


 Randolph Morton
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1 Querying aout the Past, the Present, and the Future in SpatioTemporal Dataases Jimeng Sun Dimitris Papadias Yufei Tao Bin Liu Department of Computer Science Carnegie Mellon University Pittsurgh, PA, USA Department of Computer Science Hong Kong University of Science and Technology Clear Water Bay, Hong Kong {dimitris, Department of Computer Science City University of Hong Kong Tat Chee Avenue, Hong Kong Astract Moving ojects (e.g., vehicles in road networks) continuously generate large amounts of spatiotemporal information in the form of data streams. Efficient management of such streams is a challenging goal due to the highly dynamic nature of the data and the need for fast, online computations. In this paper we present a novel approach for approximate query processing aout the present, past, or the future in spatiotemporal dataases. In particular, we first propose an incrementally updateale, multidimensional histogram for presenttime queries. Second, we develop a general architecture for maintaining and querying historical data. Third, we implement a stochastic approach for predicting the results of queries that refer to the future. Finally, we experimentally prove the effectiveness and efficiency of our techniques using a realistic simulation.. Introduction Research on spatiotemporal access methods has mainly focused on two aspects: (i) storage and retrieval of historical information, (ii) future prediction. Several indexes [PJT, KGT+, TP], usually ased on multiversion or 3dimensional variations of Rtrees, have een proposed towards the first goal, aiming at minimization of the storage requirements and query cost. Methods for future prediction assume that, in addition to the current positions, the velocities of moving ojects are known. The goal is to retrieve the ojects that satisfy a spatial condition at a future timestamp (or interval) given their present motion vectors (e.g., "ased on the current information, find the cars that will e in the city center minutes from now"). The only practical index in this category is the TPRtree [SJLL] and its variations [SJ2, TPS3] (also ased on Rtrees). Other, mostly theoretical, predictive indexes have een proposed in [KGT99] (applicale only to D space) and [AAE] (with good asymptotical performance, ut inapplicale in practice due to the large hidden constants).. Motivation Despite the large numer of methods that focus explicitly on historical information retrieval or future prediction, currently there does not exist a single index that can achieve oth goals. Even if such a "universal" structure existed (e.g., a multiversion TPRtree keeping all previous history of each oject), it would e inapplicale for several updateintensive applications, where it is simply infeasile to continuously update the index and at the same time process queries. For instance, an update (i.e., deletion and reinsertion) in a TPRtree may need to access more than nodes, which means that y the time it terminates its result may already e outdated (due to another update of the same oject). Even for a small numer of moving ojects and a low update rate, the TPRtree (or any other index) cannot "follow" the fast changes of the underlying data. Another prolem concerns the space requirements. The incoming data are usually in the form of data streams (e.g., through sensors emedded on the road network), which are potentially unounded in size. Therefore, materializing all data is unrealistic. Furthermore, even if all the data were stored, the size of the tree would render query processing very expensive since any algorithm would have to access at least a complete path from the root to the leaf level. Finally, in several applications, such as traffic control systems, the main focus of query processing is retrieval of approximate summarized information aout ojects that satisfy some spatiotemporal predicate (e.g., " the numer of cars in the city center minutes from now"), as opposed to exact information aout the qualifying ojects (i.e., the car ids), which may e unavailale, or irrelevant..2 Contriutions Motivated y the aove oservations, in this paper we propose a comprehensive method that (i) can approximately answer aggregate spatiotemporal queries aout the past, the present and the future, (ii) can continuously follow the (typically, very frequent) changes in data distriution, (iii) it is efficient and adjustale to the
2 urden of the system, i.e., it does not strain the system, especially during periods of intensive workloads, and (iv) reduces the space consumption y summarizing the data. Our first contriution is an adaptive multidimensional histogram (AMH), which is used for approximate processing of presenttime queries. AMH utilizes idle CPU cycles to perform incremental maintenance. Its precision depends only on the availale system resources, and does not deteriorate with time (meaning the AMH does not need to e periodically reuilt). The second contriution is a general architecture, called the historical synopsis, for maintaining and querying historical data. In this architecture, when a histogram ucket is updated, the previous version is kept in a mainmemory index. If the availale memory is exceeded, part of the index containing the least recent information migrates to the disk. As a result, queries referring to the present and recent past (which are more frequent) can e answered without any I/O. Third we present a stochastic approach, ased on exponential smoothing, that answers queries aout the future using oservations from the present and the recent past. Unlike previous approaches for spatiotemporal prediction, this technique does not assume knowledge of velocity vectors, which in most cases is unavailale. Finally, we experimentally evaluate the proposed techniques under a reallife simulation, where updates and queries are interleaved with histogram maintenance operations. We study the effect of the system workload on the estimation accuracy, and verify the need for approximation methods that adapt to the availale recourses. The rest of the paper is organized as follows. Section 2 reviews previous work related to ours. Section 3 provides the prolem definition and an overview of the proposed methods. Section 4 presents the concrete algorithms of AMH, while Section descries the historical synopsis. Section 6 presents the stochastic approach for prediction, and Section 7 contains the experimental evaluation. Finally, Section 8 concludes the paper with directions for future work. 2. Related Work Because the proposed techniques comine multiple goals, they relate to a numer of different research topics. In particular, since the core of our method is AMH, in Section 2. we survey related work on multidimensional histograms. Section 2.2 presents methods for updating such histograms using query feedack. Section 2.3 discusses other multidimensional approximation techniques. Section 2.4 overviews spatiotemporal prediction techniques and their limitations in several applications. Finally, Section 2. presents methods for (exact) retrieval of spatiotemporal aggregation information. 2. Static multidimensional histograms Histogram construction techniques split the data space into uckets, usually ased on the assumption that data within a small region are (almost) uniform. The various methods differ on the partition policy. The equidepth histogram of [MD88] first partitions along one dimension so that each ucket has the same numer (i.e., frequency) of ojects. Then, these uckets are partitioned again along another dimension. Mhist [PI97] splits on the most critical dimension according to a partitioning metric (e.g., variance). Minskew [APR99] partitions the space into a regular grid. Neighoring cells are then grouped into rectangular, nonoverlapping uckets y a greedy algorithm that tries to minimize the spatialskew (i.e., the variance of the cell density in each ucket). Genhist [GKTD] allows overlapping uckets. As with Minskew, it starts with a regular grid and creates uckets for the cells with high density. Then it removes a percentage of data from these cells to make the area around them smoother. Finally it decreases the resolution and repeats the process recursively. The SQ [AN] and Euler histograms [SAA2], in addition to locations, take into account ojects extents, and are more accurate for nonpoint data. All the aove histograms apply to spatial or multidimensional dataases, where the data are (almost) static. When the data distriution changes due to updates, the ucket structure may ecome outdated, and the histogram must e reuilt. In several spatiotemporal applications, however, there is simply not enough time to scan the entire data (possily several times) in order to uild the histogram from scratch. Furthermore, the new histogram may already e outdated, due to the changes that occurred during its reuilding. 2.2 Queryadaptive multidimensional histograms Query adaptive histograms avoid reuilding y using query feedack to refine the uckets. STGrid [AC99], ased on the heuristic that uckets with higher frequencies contriute more to the estimation error, splits uckets with high frequencies and merges uckets with low frequencies. Restructuring is performed at certain intervals (a parameter of the histogram) for entire rows or columns, selected y a split or a merge threshold (also a parameter). Its structure, however, cannot accurately capture aritrary distriutions. STHoles [BGC] alleviates this prolem y allowing nesting of uckets. If a query identifies large frequency variance within a ucket, a "hole", i.e., a new ucket, is created inside the original one. This "drilling" operation replaces the splitting mechanism of STGrid. SASH [LWV3] decomposes the multidimensional space into suspaces of lower dimensionality. For each suspace a separate histogram is uilt and maintained using query feedack mechanisms.
3 Queryadaptive histograms are ased on the assumption that the frequency of queries is much higher than the frequency of updates. Thus, after the data distriution changes, there is a sufficient numer of queries to "train" (i.e., refine) the histogram. Although the accuracy for these queries will e low, susequent queries will e estimated using the refined histogram, which gradually ecomes very accurate. Clearly, this assumption does not hold for updateintensive applications, where the numer of queries for each instance of the data is, typically, very small. 2.3 Other multidimensional approximation methods Several multidimensional methods emed the data into another domain and summarize the emedded data in order to provide a compact data representation. The DCTased histogram [LKC99] maintains a large numer of small uckets using discrete cosine transform, which cannot e incrementally maintained. The histogram of [MVW98] extracts the most descriptive wavelet coefficients. Although it can e incrementally maintained [MVW], its performance degrades with the numer of updates, so that eventually it has to e reuilt. Another technique [TGIK2] compresses the data distriution into a multidimensional sketch; when the query arrives, the histogram is extracted from the sketch. The main drawack of this approach is that the extracting process is expensive and, therefore, not suitale for online queries. [GMP2] incrementally maintains a small sample set (acking sample) of underlying data and histograms are then constructed/maintained y the acking sample. However, it requires scanning the entire data to regenerate the acking sample when the distriution changes, which is not possile in our case. Furthermore, some other applications of samplingased techniques [GM98, WAA] are also questionale, since it is not clear how to select and effectively maintain a representative multidimensional sample in the presence of very intensive updates. Finally, all previous methods capture a snapshot of the data and cannot e used for historical information retrieval. 2.4 Spatiotemporal prediction methods Methods for spatiotemporal prediction estimate the numer of ojects that will satisfy some spatial condition during a future interval, ased on the current location and velocity information. Choi and Chung [CC2] and Tao et al. [TSP3] present proailistic models for uniform data, which are then applied to nonuniform distriutions using some conventional multidimensional histogram. An experimental comparison of several techniques can e found in [HKT3]. All existing methods assume linear movement and that the velocities of all ojects are known (the same assumptions that hold for the TPRtreeased indexes [SJLL, TPS3]). This restricts their applicaility in several applications ecause: (i) movement is not always linear, (ii) even if it is linear, it is not usually known (e.g., moile devices transmit only their location to the ase station), and (iii) even if it is linear and known, it changes so fast that prediction using velocity information is meaningless (e.g., cars moving on road networks). 2. Spatiotemporal aggregation methods Several methods (e.g., [PTKZ2, ZTG2]) focus on spatiotemporal aggregation. Although the target queries are similar to our historical queries, the context is different. In particular, they assume a "conventional" processing framework where every query invokes disk I/Os and returns an exact answer. Here we sacrifice accuracy for efficiency and attempt to answer the most frequent queries using only the main memory. 3. Prolem Definition and System Overview We assume a set of ojects moving in the twodimensional space and a central server collecting information aout their locations over time. Update data received y the server contain the previous and the new location of individual ojects, ut not their velocity or id. Moving ojects may issue updates in periodic intervals or ased on the deviation from the previous transmitted position. Static ojects do not send information to the server; until an oject transmits a new location it is assumed to e in the last recorded position. We adopt a 2D grid that partitions the data space into w w (where w is a constant called the resolution) regular cells with width /w on each axis. Each cell c ( c w 2 ) is associated with a frequency F c, which is the numer of ojects (at the present time) in its extent. The frequencies of all the cells are maintained uptodate in memory, and serve as the data of the highest granularity. The resolution is determined y the numer of moving ojects in the system, as well as the desirale tradeoff etween accuracy and efficiency. A spatiotemporal aggregation query q(q R,q T ) retrieves the numer of ojects that fall in a rectangle region q R at a timestamp q T. If (i) q T = (i.e., the present time), q constitutes a present timestamp (PT) query; (ii) q T <, q is a historical timestamp (HT) query; and (iii) q T >, q is a future timestamp (FT) query. When the resolution of the grid is high, exhaustive examination of all the cells that are relevant to a query is expensive. AMH (adaptive multidimensional histogram) remedies this prolem y grouping adjacent cells with similar frequencies into a small numer of (rectangular) uckets (similar to other histograms such as minskew [APR99] and genhist [GKTD]). Bucket information is updated as new tuples arrive, and ucket extents (i.e., grouping of cells) continuously adapt to the data distriution changes through incremental maintenance performed during the
4 idle CPU time (i.e., when there are no incoming data or queries). A inary partition tree accelerates ucket maintenance and processing of PT queries. To support historical retrieval, uckets that are outdated (due to data or extent changes) are inserted in a mainmemory index (T) optimized for HT queries. Each ucket is associated with a lifespan denoting the temporal interval during which the ucket information is valid. When the size of the index exceeds the availale memory, part of the tree containing the least recent uckets is migrated to the disk, such that queries aout the recent past can e answered using the memoryresident portion of the index. This migration occurs in locks (rather than individual uckets) at a time, and incurs minimal I/O overhead. AMH together with index T constitute the historical synopsis, whose general structure is shown in Figure 3.. Figure 3.: Historical synopsis The historical synopsis is directly used for answering PT and HT queries. On the other hand, FT queries are processed y an exponential smoothing technique, which retrieves information aout the present and the recent past, in order to predict the future. Figure 3.2 shows the astract query processing mechanism and Tale 3. lists the frequentlyused symols. In the rest of the paper we descrie the various components of the architecture, starting with AMH in Section Adaptive MultiDimensional Histogram Given a regular w w cell partition, AMH generates n rectangular uckets whose edges are aligned with the cell oundaries. For each ucket k ( k n), we denote (i) as n k the numer of cells it covers, (ii) as f k the average frequency of these cells (i.e., f k =(/n k )Σ ( cell in k) F c ), and (iii) as v k their variance (i.e., v k =(/n k )Σ ( cell in k) (F c f k ) 2 ). AMH aims at minimizing the weighted variance sum (WVS) of all the uckets, or formally: WVS= i=~n (n i v i ) (4) 4. AMH structure Each ucket stores: (i) its rectangular extent R (ausing the terminology slightly, we also use R to denote its area), (ii) the average frequency f of all the cells in R, and (iii) the average of squared frequency g of these cells: g=(/n k ) ( cell c in R) (F c ) 2. Thus, the numer n of cells covered y a ucket can e represented as R w 2, where w is the cell resolution. A ucket s variance v equals v=g f 2, and as a result, WVS can e computed using R, f, g of all uckets. AMH maintains a inary partition tree (BPT), where each leaf node corresponds to a ucket. An intermediate node is associated with a rectangular extent R that encloses the extents of its (two) children. Initially, BPT contains a single leaf node, and the histogram has one ucket covering the entire data space. New uckets (leaf nodes) are created through ucket splits (elaorated shortly), ut the total numer n of uckets never exceeds a system parameter B. As an example, Figure 4.a shows the frequencies of 2 cells (i.e., w=) at timestamp, Figure 4. demonstrates the extents of 6 uckets, and Figure 4.c illustrates the corresponding BPT. y y 4 y 3 y 2 y y y 4 y 3 y 2 y Symol w F c (B) n R k n k f k g k v k Figure 3.2: Query processing Description resolution frequency of cell c (maximum) numer of uckets ucket extent or area numer of cells in ucket k frequency (i.e., numer of ojects) in ucket k average squared frequency of ucket k frequency variance of ucket k Tale 3.: Frequent symols x x 2 x 3 x 4 x x x 2 x 3 x 4 x (a) Cell frequency () Bucket extents node R f g v 7 [ x x ][ y y ] /36 [ 8/ /6 9/6 x x 2 ][ y y 2 ] 2/4 2/ [ x 3 x 4 ][ y 4 y ] 3/4 43/4 3/6 4 [ x 3 x 4 ][ y 2 y 3 ] 39/4 383/4 / [ x x ][ y 2 y ] 2/4 /4 3/6 WVS=7.8 [ x 3 x ][ y y ] 3/3 3/3 3 4 (c) The BPT (d) Bucket information Figure 4.: AMH at time
5 Intermediate node 7, for instance, covers uckets, 2, implying that they were created y splitting 7. Figure 4.d lists the detailed information of each ucket (v is not explicitly stored, ut computed from f and g). 4.2 AMH information update A location update contains the old and the new position of a moving oject. When such an update is received, AMH executes an infoupdate process to modify the frequencies of the cells and uckets that cover the corresponding locations. Particularly, a cell can e identified using a hash function, while a ucket is located y following a single path of BPT, guided y the extents of the intermediate nodes. Assume, for example, that the frequency of cell (x,y 2 ) in Figure 4.a changes from 6 to at time 2, due to concurrent movement of several ojects. After modifying the cell frequency, infoupdate otains the frequency (squared frequency) difference 4 (64) etween the new and old values. To complete the update, it identifies ucket 6 covering the cell (following the path,, 9 ), and updates its f and g as f 6 = (f 6 n 6 +4)/n 6, g 6 = (g 6 n 6 +64)/n 6, where n 6 is the numer of cells covered y 6, and f 6 (g 6 ) is their average (square) frequency. Assuming that BPT is fairly alanced, the algorithm (shown in Figure 4.2) has average cost logb, where B is the maximum numer of uckets. Algorithm Infoupdate (x, y, d) /* (x,y) are the cell coordinates, d is the frequency difference */. find the cell c that contains (x, y) using a hash function 2. f=d ; g= (F c +d) 2  F c 2 ; F c =F c +d 3. descend BPT to find the ucket that contains (x,y) 4. f =(R w 2 f + f)/(r w 2 ); g =(R w 2 g + g)/(r w 2 ) // R is the extent area of, w is the cell resolution End Infoupdate Figure 4.2: infoupdate algorithm Updates may alter the frequency distriution, increasing the frequency variances and the approximation error. Figure 4.3a shows the updated cells at time 2, and Figure 4.3 illustrates the information of the corresponding uckets. Notice that v, v are significantly larger than those in Figure 4.d, causing considerale increase of WVS (from 7.8 in Figure 4.d to 73.8). To remedy this prolem, AMH performs ucket reorganization when the CPU is idle (i.e., no data/query is pending), which involves ucket merging and splitting. 4.3 AMH ucket merge A merge is performed when the numer of uckets has reached the maximum value B (assumed to e 6 in these examples), and aims at grouping cells of multiple uckets, whose frequencies have converged since the last reorganization, into a single ucket. Towards this, AMH selects the leaf node of BPT with the lowest merge penalty (MPen); MPen is defined as the increase of WVS (see equation 4) if the node is merged with its siling. Assume, for instance, the cell frequencies in Figure 4.3a, the ucket information in Figure 4.3, and the tree in Figure 4.c. Merging ucket 3 with its siling 4 removes oth nodes from BPT and makes 8 a leaf (i.e., a new ucket in the histogram). The change incurs n 8 v 8 (n 3 v 3 +n 4 v 4 ) increase in WVS (i.e., the merge penalty for 3, 4 ), where n i is the numer of cells covered y node i, and v i is its frequency variance. A ucket s siling can sometimes e an intermediate node, e.g., the siling of ucket 6 is 8, in which case MPen should e computed as n 9 v 9 (n 3 v 3 +n 4 v 4 +n 6 v 6 ). Notice that, merging 6 with 8 would reduce the ucket numer y 2 (i.e., removing 3 leaf nodes 3, 4, 6, spawning 9 ). The leaf node with the lowest merge penalty can e identified using a single postorder traversal of BPT s intermediate nodes (i.e., each intermediate node is processed after its children). Consider, for example, Figure 4.c, where node 7 is the first node processed. We compute the following information (from its children, 2 ): (i) the average frequency of all cells in its extent f 7 = (f R + f 2 R 2 )/ R 7, (ii) the average squared frequency g 7 = (g R + g 2 R 2 )/R 7, (iii) the merge penalty of its children MPen 7 = n 7 v 7 (n v +n 2 v 2 ), where n i =R i w 2, v i =g i f 2 i. Next, similar computations are performed for 8, 9,, (in this order). Note that i) the leaf node 3 ( 4, 6 ) is updated from the corresponding cells; ii) the computation for 9 ( ) is ased on the alreadycomputed information of intermediate node 8 ( 9 ). Similarly, the information of is computed y 7 and. Finally, the algorithm merges the children of the node that incurs the smallest penalty (in this case 9 ). Figures 4.4a and 4.4 illustrate the resulting ucket extents and BPT, respectively. Figure 4. shows the pseudocode of the ucketmerge algorithm. Its running time is B, since it visits all nodes of BPT. y y 4 y 3 y 2 y 9 node R f g v 3 9 [ x x 2 ][ y 3 y ] 32/6 284/6 68/ [ x x 2 ][ y y 2 ] 2/4 2/4 8/6 9 3 [ x 3 x 4 ][ y 4 y ] 39/4 38/4 3/ [ x 3 x 4 ][ y 2 y 3 ] 39/4 383/4 /6 2 [ x 3 x ][ y y ] 2/3 2/3 62/9 [ 6 6 x x ][ y 2 y ] 4/4 42/4 3/6 x x 2 x 3 x 4 x WVS=73.8 (a) Cell frequency () Bucket information Figure 4.3: AMH at time 2 y 4 9 y 3 7 y 2 y x x 2 x 3 x 4 x (a) Buckets after merging () BPT after merging Figure 4.4: Example of ucket merge (cont. Figure 4.)
6 Algorithm BucketMerge. minmpen= ; newleaf=null 2. let =the first intermediate node in the postorder traversal 3. do 4. if ( has at least one leaf node). compute f, g, MPen from its children 6. if (MPen <minmpen) 7. minmpen=mpen ; newleaf=; 8. =next intermediate node in the postorder traversal 9. until (=root). merge the children of newleaf End BucketMerge Figure 4.: ucketmerge algorithm 4.4 AMH ucket split A ucket split creates refined partitions in regions where uckets have large variance. It improves AMH s approximation accuracy y (i) reducing the ucket frequency variance, and (ii) decreasing the ucket extent. Towards this, the algorithm splits the ucket with the highest split enefit (SBen), i.e., the maximum decrease of WVS achieved y splitting this ucket at the est position. Consider, for example, ucket in Figure 4.4a. Since a split guarantees that the resulting uckets cover an integer numer of cells, there are three possile ways to split (one on the x, and two on the yaxes, respectively). The largest reduction of WVS is achieved y splitting on the xdimension, resulting in uckets 2, 3 (Figure 4.6a) and split enefit n v (n 2 v 2 +n 3 v 3 ). The split enefits of the other uckets are also computed and the one with the largest SBen (in this case ) is split (at its est split position). The ucketsplit algorithm repeats the split process until (i) new data/queries arrive, (ii) the numer of uckets reaches the maximum threshold B or (iii) there does not exist any ucket with positive split enefit, implying that the current ucket numer is sufficient to model the cell frequency distriution. Continuing the example, ucketsplit computes SBen for the new uckets ( 2, 3 ) and splits (whose enefit is currently the largest) into 4,. Since the ucket numer has reached B (=6), the algorithm terminates. Figures 4.6a, 4.6 illustrate the final ucket extents and BPT, respectively. y y 4 y 3 y 2 y PT query q x x 2 x 3 x 4 x (a) Final Buckets () BPT after splitting Figure 4.6: After splitting, (cont. Figure 4.4) Figure 4.7 summarizes the ucketsplit algorithm. Each ucket reorganization involves at most one ucket merging (i.e., if the numer of uckets equals B), and a small numer of ucket splits. After its termination, the process starts again if the CPU is still idle. Compared to traditional histograms requiring expensive total reuilding, AMH deploys repetitive, simple, interruptile, partial reorganizations, and, therefore, it is ideal for dynamic spatiotemporal environments, where updates/queries may occur in (unpredictale) ursts. Algorithm BucketSplit. for every ucket 2. compute its split enefit and est split position 3. sort all split enefits in descending order into a list L s 4. while (ucket numer n <B and no new data/query and the first ucket in L s has positive split enefit). remove the first ucket from L s 6. split it into, 2 7. otain split enefits and est split positions for, 2 8. insert, 2 into the appropriate positions in L s 9. return End BucketSplit Figure 4.7: ucketsplit algorithm 4. AMH PT query processing A presenttime query is answered y examining all uckets whose extents intersect the query region q R. Such uckets are located y traversing BPT and following the nodes that overlap q R. For example, the query q in Figure 4.6a intersects only ucket 9, which can e reached y following the intermediate nodes,. For each intersecting ucket k, we compute the overlap area R intr etween its extent and q R, and otain a partial result f k R intr w 2 (note that R intr w 2 equals the numer of cells in k that are covered y q R ). Then, the final result is computed as the sum of the partial results of all intersecting uckets, without accessing the underlying cells.. Historical Synopsis In this section, we discuss HT query processing using the historical synopsis that consists of (i) a dynamic histogram maintaining the currently valid uckets (e.g., AMH), and (ii) a past index T storing the osolete uckets.. General architecture A ucket in AMH dies if (i) the frequency in any of its cells is updated, or (ii) its extent changes due to a split or merge. In case (i), a new ucket with the same extent, ut different frequency statistics (i.e., f and g) is created. In case (ii) new ucket(s) are created due to merge or split. Each ucket contains a lifespan [l s, l e ), where l s (l e ) is the time of creation (death). All the uckets in AMH are alive, and their l e equals now. Dead uckets are inserted into index T and their cell content is discarded. In Figure 4., for example, all the uckets o,...,o 6 are alive and have lifespans [,now). Then, in Figures 4.3 uckets, 3,, 6 die at time 2 ecause some cells in their extents change frequencies at this timestamp. This
7 produces four dead uckets with lifespans [,2) (which are inserted into T), and creates four live ones with lifespans [2,now). At the next timestamp 3, uckets, 3, 4,, 6 die (yielding four uckets with lifespans [2,3], and one 4 with lifespan [,3)) since 3, 4, 6 are merged (Figure 4.4) and, are split (Figure 4.6). Buckets 9, 2, 3, 4, that result from these operations are alive with lifespans [3, now). As shown in Figure., after these operations, T contains 9 dead uckets and AMH 6 live ones. dead stored in T alive stored in AMS node R f g v lifespan [ x x 2 ][ y 3 y ] 7/6 9/6 /36 [,2) 3 [ x 3 x 4 ][ y 4 y ] 3/4 43/4 3/6 [,2) [ x 3 x ][ y y ] 3/3 3/3 [,2) 6 [ x x ][ y 2 y ] 2/4 /4 3/6 [,2) 3 [ x 3 x 4 ][ y 4 y ] 39/4 38/4 3/6 [2,3) 4 [ x 3 x 4 ][ y 2 y 3 ] 39/4 383/4 /6 [,3) 6 [ x x ][ y 2 y ] 4/4 42/4 3/6 [2,3) [ x x 2 ][ y 3 y ] 32/6 284/6 68/36 [2,3) [ x 3 x ][ y y ] 2/3 2/3 62/9 [2,3) 2 [ x x 2 ][ y y 2 ] 2/4 2/4 8/6 [,now) * o 9 [ x 3 x ][ y 3 y ] /2 [3,now) 2 [ x x ][ y 3 y ] 3/3 3/3 [3,now) x 3 [ 2 x 2 ][ y 3 y ] 29/3 28/3 2/3 [3,now) [ x [3,now) 4 3 x 4 ][ y y ] 2/2 2/2 x [ x ][ y y ] / / [3,now) Figure.: Buckets at time 3 (cf. Figures 4., 4.3, 4.6) The size of T grows continuously with time and eventually exceeds the availale memory, in which case part of it migrates to the disk. The migration is performed in locks, meaning that we simply move pages of equal sizes from the memory to the disk. Further, the disk accesses are sequential except for, possily, the first page transferred. Accordingly, pointers to these pages are modified to the corresponding disk addresses (from their original mainmemory addresses). Each page of T that is currently in memory is assigned a migration rank, such that pages are migrated in ascending order of their ranks. A separate mainmemory structure (e.g., priority queue) can e maintained to select the page with the lowest rank efficiently. In our implementation, the migration rank of a page is set to the latest ending timestamp l e of all its entries lifespans. In general, pages with low ranks should contain old uckets, and e less likely accessed y queries. Finally, pages that are already on the disk may e eventually (physically) erased or moved to tertiary storage, when they ecome completely osolete..2 Implementation using a packed Btree The past index T should support the following operations efficiently: (i) insertion of a dead ucket, and (ii) selection of uckets intersecting a HT query. Our first implementation adopts a packed Btree, which indexes the ending time l e of the uckets lifespans. Specifically, each leaf entry stores all information of a ucket (i.e., R, f, g, and its life span), while an intermediate entry stores a lifespan [l s, l e ), which encloses those of the leaf entries in * * its child node. All the nodes in the Btree are full, except for the rightmost one at each level, which does not have a minimum utilization requirement. Particularly, we refer to the rightmost leaf node as the active leaf, for which a special pointer is maintained. Since uckets are inserted into T in ascending order of their l e, each insertion simply appends the new ucket into the active leaf, and terminates if the leaf incurs no overflow. Otherwise, a new active leaf is created with the most recently inserted ucket, and this change propagates to the parent levels. The amortized insertion cost is constant. Figure.2 illustrates the corresponding packed Btree for the dead uckets in Figure., assuming a capacity of 3 entries per node. D [,2) [,3) [2,3) [,2) 3 [,2) [,2) 6 [,2) 3 [2,3) 4 [,3) 6[2,3) [2,3) [2,3) A B C lifespan active leaf Figure.2: The packed Btree In some cases, HT may have to access AMH, if there are some uckets that were created at the query timestamp, ut still remain alive. Consider, for example, the query with region q R =[x 2,x 3 ][y 2,y 3 ] and timestamp q T = on the uckets of Figure. (the corresponding AMH is shown in Figure 4.6). The query must visit all the uckets whose extents (lifespans) intersect q R at time, namely, those marked with * in Figure.. The final result is the sum of the partial results of all such uckets. The partial result from ucket 2 (which is still alive in AMH), is otained y a PT query as discussed in Section 4.. To locate the intersecting uckets (i.e., and 4 ) in the Btree, the algorithm accesses the nodes whose lifespans contain q T (=), i.e., nodes D, A, B in Figure.2. At the leaf level the partial results of qualifying uckets are otained in the same way as PT queries (i.e., y computing the size of the intersection area with q R ). The packed Btree implementation contains a simple and very efficient insertion algorithm, ut processes HT queries using only temporal pruning. In the next section, we descrie an alternative structure that takes advantage of spatial conditions to accelerate query processing..3 Implementation using a 3D Rtree Our second implementation is ased on a mainmemory adaptation of the 3D R*tree [BKSS9]. In this structure, each intermediate entry contains a 3D ox which encloses the extents and lifespans of all the uckets in its sutree. Given a HT q (which can also e regarded as a 2D rectangle defined y q R at time q T ), we only need to visit those nodes whose 3D oxes intersect that of q. Compared to the packed Btree, the 3D Rtree implementation achieves lower query cost since it utilizes oth spatial and temporal conditions to prune the search space. On the other hand, it incurs higher update overhead
8 due to the complex insertion operations of the Rtree (which are required to achieve spatiotemporal locality). Both indexes use the same migration mechanism discussed in Section., i.e., when the main memory capacity is reached, the least recent locks of the index are transferred to the disk. Thus, their update cost difference refers to CPU overhead. In summary, the B tree method is preferale for very high streaming rates, while the Rtree for applications with heavy query workload. The tradeoff etween update and query efficiency is explored in the experimental evaluation. 6. Prediction Model As discussed in Section 2.4, spatiotemporal prediction ased on current locations and velocities has limited applicaility in the majority of real applications. Consider, for instance, a traffic supervision scenario, where cars moving on a road network periodically transmit their information. The assumption that the velocity will remain constant etween the current and the query time q T is not realistic, since the cars that reach the end of the road segment on which they travel, must update their velocity (i.e., they must turn or stop). As shown in the experimental evaluation, for typical traffic simulation settings, up to 9% of the cars may update their velocities per timestamp, implying that velocityased prediction is meaningless even for the next timestamp. The motivation of our model is that, although the individual oject velocities may change aruptly, the overall data distriution varies gradually (and slowly) with time, due to the continuity of movement. Furthermore, since we keep past and present data, we can use this information to identify the trend of movement and estimate the result of a future query. Consider, for instance, that the current time is, and we want to process the query q(q R,), that predicts the numer of ojects in q R at the next timestamp. The output of q can e represented as a function of the result of a PT and a set of recent HT queries with the same region q R. Exponential smoothing, a wellknown forecasting method in time series analysis [G8], is ased on the same reasoning. According to this method, the estimated result S' of the query (q R,) can e modeled as a time series such that: S' = αs o + α( α)s  + α( α) 2 S α( α) n S n where (i) S is the actual result of the PT query (q R,) (at the current time), (i) S i is the actual result of the HT query (q R,i) at the i th previous timestamp, (iii) n determines the length of previous history that will e used for prediction, and (iv) a is the smoothing parameter in the range (,). The formula is ased on the idea that recent timestamps are more important for prediction than older ones. The relative weight is adjusted y the value of a: as it approaches, the significance of the most recent timestamps increases. The model is applicale in our case ecause the changes of the query result are smooth, or more specifically, the series has a locally constant mean which shows some drift over time. If we want to predict the result at a future timestamp i>, (i.e., no actual values for S to S i are availale), we have to use a stepystep technique, i.e., first compute S', then use this value to estimate S' 2 and so on. In general, the value of S' i can e represented y the following recurrence: S' i = αs o + (α)s' i where S' i is the predicted value for the query at future timestamp i. Figure 6. illustrates the pseudocode for the prediction algorithm ased on the aove discussion. In our implementation we fix a to.2 and n to 6. Although, these parameters can e set onthefly for each individual query (i.e., the values that give the est estimation for the current and recent timestamps, since the actual results are known), we chose to fix gloal values for all queries in order to avoid the overhead of tuning. In particular, it has een suggested [H86] that a =.2~.3 gives accurate results for a variety of prolems, while n = 6 represents a good tradeoff etween accuracy and query cost (the longer we go into the past the higher the cost). Algorithm Prediction (qr,qt) /* qr: query region; qt: prediction timestamp; n: the least recent timestamp taken into account for prediction, α : smoothing parameter */. S =PT(qr,) // compute result of corresponding PT query 2. for i= to n 3. S i = HT(qr,i) // result of HT at i th previous timestamp 4. t = n; S current = S n ;. while (t qt) 6. if (t ) // present or past 7. S next = α S t + ( α) S current 8. else // future 9. S next = α S + ( α) S current. S current = S next. t = t+ 2. return S current // predicted value at qt End Prediction Figure 6.: Algorithm for prediction 7. Experimental Evaluation This section evaluates the proposed methods, using a Pentium IV.8GHz CPU and 26 MBytes of main memory. Section 7. descries the experimental settings, Section 7.2 tunes the size of AMH and Section 7.3 evaluates its aility to follow the data distriution effectively. Section 7.4 compares AMH with the conventional, total reuilding approach. Section 7. evaluates the approximation error and compares exponential smoothing with velocityased prediction. Finally, Section 7.6 studies the effect of update intensity on the performance of the historical synopsis and the approximation error.
9 7. Settings The first dataset, denoted as spatial, simulates a scenario involving k moile ojects. The initial positions of the ojects are sampled from the real dataset MG, and their destinations are randomly selected from another real dataset LB (oth availale from www/tiger/). Each oject moves from the initial to the destination point following a straight line and reports a fixed numer of location changes to the server. After it reaches its destination, it selects a new random destination from the initial dataset and this process is repeated for a total of "round trips". Although the endpoints of each trip are different, the overall distriution after the completion of a trip is either MG or LB. The second dataset, denoted as road, is created y the spatiotemporal generator of [B2]. The input of the generator is the road map of Oldenurg (a city in Germany), containing 6 nodes and 73 edges, and the output is a set of point ojects moving on this network. Each point is represented y its locations at successive timestamps. The datasets are created y setting the parameters (see [B2] for their detailed semantics) of the generator as follows. There are 6 classes of ojects (e.g., cars, pedestrians), so that each class moves with a particular speed range. The total numer of ojects (in all 6 classes) at any timestamp is, (the generator decides the numer of ojects per class according to some function eyond users control). At each timestamp 2k ojects disappear (i.e., they reach their destinations), while 2k new ones appear at random nodes. Given these settings, on average, 9% of the ojects issue updates per timestamp (i.e., they reach the end of the road segment that they are on and they turn or stop). The total numer of updates in oth road and spatial datasets is 2.M. The server maintains a regular partition of the space in cells (as discussed in Sections 3 and 4). When a location update is received from an oject, the server (i) decreases (y ) the frequency of the cell corresponding to the old position, and (ii) increases the frequency of the cell at the new location. The first column of Figure 7. shows the density of ojects per cell at the initial, median and ending timestamp in a single trip, for the spatial dataset. The second column illustrates the corresponding ucket structure, which continuously follows the data distriution. Figure 7.2a shows the road network of Oldenurg and 7.2 the density of the cells at the initial timestamp. Although the velocity updates for road are much more frequent than the spatial dataset, the overall distriution does not change significantly over time ecause movement is constrained to the network. Each query has two parameters: the side length q L and the query timestamp q T. In all cases, the query extent is a square with side length q L, uniformly distriuted in space. For HT (FT), queries q T is uniformly distriuted in the past (future) timestamps. An FT query is processed y issuing 6 HT and PT query with the same extent. The error rate is measured as: act app / act, where act (app) equals the actual (approximate) answer. (a) Initial data (MG) (c) Median data () Initial histogram (d) Median histogram (e) Final data (LB) (f) Final histogram Figure 7.: Spatial dataset (a) Road Network () Data on road Figure 7.2: Road dataset 7.2 Setting the numer of uckets Unlike other multidimensional histograms (e.g., genhist) that require tuning for several parameters, AMH involves a single parameter B (maximum numer of uckets). In order to tune B, we issue 2k PT queries, distriuted in the whole history with length q L =6% of the spatial universe. We allow AMH to reorganize every incoming tuples. This reorganization is limited to five
10 merge operations, each followed y one or more splits (depending on whether intermediate BPT nodes are merged). As expected (and shown in Figure 7.3a), the average error rate decreases with B. The error is higher for road ecause the local uniformity assumption (within each ucket) does not accurately capture the real oject distriution. The same prolem exists for all multidimensional histograms (see Section 2) that decompose the space ased on local uniformity. To the est of our knowledge, there does not exist any histogram specifically targeting network data. 4% error rate 3% % spatial ucket numer road 2 CPU time µ sec 2 ucket numer (a) Error rate vs. B () Reorganization cost vs. B Figure 7.3: Tuning B (q L =6%, q T =) In order to evaluate the overhead, we use the same settings and measure the total CPUtime spent on reorganization operations. Figure 7.3 shows the average CPU cost per location update as a function of B for the spatial dataset (the CPU cost for road is aout the same and omitted). The overhead increases linearly with the maximum numer of uckets, since oth merge and split operations examine all uckets. Based on these experiments, we set B=, since it provides good accuracy (average error less than % and, for spatial and road, respectively) with reasonale overhead. 7.3 Roustness with time The goal of this experiment is to evaluate the roustness of AMH with time. We fix B=, apply the same settings as in Figure 7.3, and issue 2k PT queries, uniformly distriuted in history (q L =6%). Figure 7.4 shows the error rate as a function of the total numer of updates for the spatial dataset. The periodic ehaviour of the error is ecause the initial and final distriutions are more skewed than the intermediate ones (see Figure 7.). 6% error rate 2% 8% 4% k numer of location updates.m M.M 2M 2.M Figure 7.4: Error vs. numer of updates (spatial) Figure 7. repeats the experiment for the road dataset, which does not show the periodic effect of spatial, ecause the data distriution is constrained y the underlying road network. In oth cases the overall error rate remains stale as time evolves, indicating that AMH captures the dynamic data distriution very well through successive reorganizations. 3% k numer of location updates.m M.M 2M 2.M Figure 7.: Error vs. numer of updates (road) 7.4 Comparison with conventional histograms Now, we compare AMH with minskew [APR99] (a common enchmark in the histogram literature [GKTD, WAA]), constructed using the same regular partition as AMH. Since minskew is a static histogram, we reuild it every k location updates, measure the cost (aout. seconds) for each reuilding and assign a percentage tp of this cost to reorganization operations in AMH. For instance, if tp=%, we use the same amount of time for oth AMH reorganizations and minskew reuilding. The difference is that the reorganization operations of AMH are uniformly distriuted among the k location updates. Figure 7.6a () illustrates the average error rate of spatial and road datasets over 2k uniformly distriuted (in history) PT queries as a function of tp. AMH is much more accurate than minskew, even when consuming / of the CPU resources, ecause it continuously adapts to the data distriution using cheap operations. On the other hand, minskew is accurate only during the short period after the reuilding and its ucket structure soon ecomes outdated. 3% error rate AMH minskew time proportion... 3% error rate 2% % time proportion... (a) spatial () road Figure 7.6: Error rate vs. tp (q L = 6%, q T =, B=) 7. Evaluation of query and prediction error In order to evaluate the error of different query types, we issue 3 query sets, each containing 2k queries of the same type (HT, PT, or FT). Figure 7.7 shows the error rate as the function of the query length q L (from 2% to of the spatial universe), respectively. The error of FT queries (average of predictions for the next 
11 timestamps) is higher ecause it also includes the inaccuracy of prediction. The results for the road dataset are less accurate, ecause (as discussed in Figure 7.3) the local uniformity assumption within each ucket does not capture well the actual data distriution. error rate % % % HT PT FT 3% error rate 3% 2% % % % 2% 4% 6% 8% 2% 4% 6% 8% ql ql (a) spatial () road Figure 7.7: Query error comparison (B=) Next, we compare our prediction method, ES for exponential smoothing, with the stateoftheart model [TSP3] that uses only information aout current location and velocity. In this model, nonuniform data are handled y a 4D minskew histogram (2D for spatial, 2D for velocity). The velocity of an oject at timestamp i is determined y its location at timestamps i and i+. The histogram is constructed using and 6 6 regular partitions on the spatial and velocity dimensions, respectively. The numer of uckets is set to 2 (more uckets are needed due to the 4D space) and reuilding occurs every timestamp (so that the error is due to the model and not the histogram deterioration). In other words, oth the space consumption and the CPU overhead are much higher than AMH. Figure 7.8 shows the error rate for 2k uniformly distriuted FT queries as a function of q T (i.e., future prediction time), fixing q L to 6% of the spatial universe. Although in spatial the movement is linear, velocityased prediction is less accurate than ES (Figure 7.8a). Compared to the results reported in [TSP3], the error of the method is higher, ecause it now includes the updates that occur etween the current and the query time (in [TSP3] it is assumed that there are no such updates). For the road dataset (Figure 7.8), where updates are much more intense, the quality of prediction deteriorates fast with q T, confirming our oservations (in Section 6) that velocityased prediction is meaningless in such cases. 3% error rate 2% % % ES Velocityased prediction prediction timestamp % error rate 8% 6% 4% prediction timestamp (a) spatial () road Figure 7.8: Error rate vs. q T (q L = 6%) 7.6 The effect of update intensity Having demonstrated the accuracy of the proposed methods, we now test the historical synopsis architecture in realtime environments involving intensive updates. In this scenario we assume that updates have higher priority and, therefore, reorganization is delayed during periods of high workload. In order to evaluate the ehaviour of the two implementations (packed Btree and 3D Rtree), we issue 2k HT, PT and FT queries uniformly distriuted in history. Figure 7.9 shows the error as a function of the update rate, which varies from k to k updates per second. For k updates the accuracy of oth implementations is the same, implying that they can oth follow the data distriution effectively. In the other cases, however, the 3D Rtree incurs higher error, ecause it consumes a significant amount of time for insertion operations (of the dead uckets in the tree). On the other hand, the efficient insertion algorithms of the packed B tree, allow the system to devote more CPU time to histogram reorganization, leading to lower error. Note that the error rate of spatial increases faster than that of the road dataset, ecause the data distriution varies significantly with time, implying that if the uckets do not get reorganized, they will soon ecome outdated. However, if a large percentage of the workload consists of queries, then the 3D Rtree synopsis is preferale, since it utilizes the spatial condition to effectively prune the search space. Figure 7. shows the average cost of all query types for the two implementations. In oth cases, PT queries are processed y AMH and have the same performance. HT (and, consequently, FT) queries incur almost half the cost in Rtrees. For typical settings, we found that if the query/update ratio in the workload exceeds 3/7, the 3D Rtree is etter (i.e., it allows for more reorganization operations). 2% error rate % % % Packed Btree 3D Rtree 2% error rate % % % k k k k k k update rate update rate (a) spatial () road Figure 7.9: Error rate vs. update rate (q L = 6%, B=) CPU time msec PT HT FT Figure 7.: Query cost comparison (q L = 6%, B =)
12 8. Conclusions Existing spatiotemporal access methods have several disadvantages that severely limit their applicaility: (i) they focus explicitly on either historical information retrieval, or future prediction; (ii) they are very expensive to update and query online, since oth types of operations incur a large numer of disk accesses; (iii) their prediction assumptions are unrealistic for most practical scenarios. Motivated y these prolems, we present a comprehensive approach for processing queries that refer to any time in history. Instead of keeping detailed information aout individual ojects, the proposed architecture maintains an incremental multidimensional histogram, which answers presenttime queries. Outdated uckets are stored in a mainmemory index, in order to answer queries aout the recent past without any I/O operations. The "oldest" parts of the index migrate to the disk in locks, so that the total I/O cost of updates is minimized. Finally, future queries are answered y a stochastic method that uses the recent history to predict the future, without any assumptions aout velocity. Extensive experiments confirm the effectiveness of our techniques under realistic settings. Acknowledgements This work was supported y grants HKUST 68/E and HKUST 697/2E from Hong Kong RGC. References [AAE] Agarwal, P., Arge, L., Erickson, J. Indexing Moving Points. PODS, 2. [AC99] Aoulnaga, A., Chaudhuri, S. Selftuning Histograms: Building Histograms Without Looking [AN] at Data. SIGMOD, 999. Aoulnaga, A., Naughton, J. Accurate Estimation of the Cost of Spatial Selections. ICDE, 2. [APR99] Acharya, S., Poosala, V., Ramaswamy, S. Selectivity Estimation in Spatial Dataases. SIGMOD, 999. [B2] Brinkhoff, T. A Framework for Generating NetworkBased Moving Ojects. GeoInformatica 6(2), 38, 22. [BKSS9] Beckmann, N., Kriegel, H., Schneider, R., Seeger, B. The R*tree: An Efficient and Roust Access Method for Points and Rectangles. SIGMOD, 99. [BGC] Bruno, N., Chaudhuri, S., Gravano, L. STHoles: A Multidimensional WorkloadAware Histogram. SIGMOD, 2. [CC2] [G8] Choi, Y., Chung, C. Selectivity Estimation for SpatioTemporal Queries to Moving Ojects. SIGMOD, 22. Gardner, E. Exponential Smoothing: The State of the Art. Journal of Forecasting (4): 28, 98. [GKTD] Gunopulos, D., Kollios, G., Tsotras, V., Domeniconi, C. Approximating MultiDimensional Aggregate Range Queries over Real Attriutes. SIGMOD, 2. [GM98] Gions, P., Matias, Y. New SamplingBased Summary Statistics for Improving Approximate Query Answers. SIGMOD, 998. [GMP2] Gions, P., Matias, Y., Poosala, V. Fast Incremental Maintenance of Approximate [H86] Histograms. TODS 27(3):26298, 22. Hunter, J. The Exponentially Weighted Moving Average, Journal of Quality Technology, 8(4), 2327, 986. [HKT3] Hadjieleftheriou, M., Kollios, G., Tsotras, V. Performance Evaluation of Spatiotemporal Selectivity Estimation Techniques. SSDBM, 23. [HKTG2] Hadjieleftheriou, M., Kollios, G., Tsotras, V., Gunopulos, D. Efficient Indexing of Spatiotemporal Ojects. EDBT, 22. [KGT99] Kollios, G., Gunopulos, D., Tsotras, V. On Indexing Moile Ojects. PODS, 999. [LKC99] Lee, J., Kim, D., Chung, C.. Multidimensional Selectivity Estimation Using Compressed Histogram Information. SIGMOD, 999. [LWV3] Lim, L., Wang, M., Vitter, J. SASH: A Self Adaptive Histogram Set for Dynamically Changing Workloads. VLDB, 23. [MD88] Muralikrishna, M., DeWitt, D. EquiDepth Histograms For Estimating Selectivity Factors For MultiDimensional Queries. SIGMOD, 988. [MVW] Matias,Y.,Vitter,J,Wang,M., Dynamic Maintenance of WaveletBased Histograms. VLDB, 2. [MVW98] Matias, Y., Vitter, J., Wang, M. WaveletBased Histograms for Selectivity Estimation. SIGMOD, 998. [PI97] Poosala, V., Ioannidis, Y. Selectivity Estimation Without the Attriute Value Independence Assumption. VLDB, 997. [PJT] Pfoser, D., Jensen, C., Theodoridis, Y. Novel Approaches in Query Processing for Moving Oject Trajectories. VLDB, 2. [PTKZ2] Papadias, D., Tao, Y., Kalnis, P., Zhang, J. Indexing [SAA2] [SJ2] SpatioTemporal Data Warehouses. ICDE, 22. Sun, C., Agrawal, D., Aadi, A. Exploring Spatial Datasets with Histograms. ICDE, 22. Saltenis, S., Jensen, C. Indexing of Moving Ojects for LocationBased Services. ICDE, 22. [SJLL] Saltenis, S., Jensen, C., Leutenegger, S., Lopez, M. Indexing the Positions of Continuously Moving Ojects. SIGMOD, 2. [TGIK2] Thaper, N., Guha, S., Indyk, P., Koudas, N. Dynamic Multidimensional Histograms. SIGMOD, 22. [TP] [TPS3] [TSP3] Tao, Y., Papadias, D. The MV3RTree: A Spatio Temporal Access Method for Timestamp and Interval Queries. VLDB, 2. Tao, Y., Papadias, D., Sun, J. The TPR*Tree: An Optimized SpatioTemporal Access Method. VLDB, 23. Tao, Y., Sun, J. Papadias, D. Selectivity Estimation for Predictive SpatioTemporal Queries. ICDE, 23. [WAA] Wu, Y., Agrawal, D., Aadi, A. Using the Golden Rule of Sampling for Query Estimation. SIGMOD, 2. [ZTG2] Zhang, D., Tsotras, V., Gunopulos, D. Efficient Aggregation over Ojects with Extents.PODS, 22.
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