A cost model for spatio-temporal queries using the TPR-tree

Transcription

1 The Journal of Systems and Software 73 (2004) A cost model for spatio-temporal queries using the TPR-tree Yong-Jin Choi *, Jun-Ki Min, Chin-Wan Chung Division of Computer Science, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon , South Korea Received 23 May 2002; received in revised form 29 April 2003; accepted 5 May 2003 Available online 5 December 2003 Abstract A query optimizer requires cost models to calculate the costs of various access plans for a query. An effective method to estimate the number of disk (or page) accesses for spatio-temporal queries has not yet been proposed. The TPR-tree is an efficient index that supports spatio-temporal queries for moving objects. Existing cost models for the spatial index such as the R-tree do not accurately estimate the number of disk accesses for spatio-temporal queries using the TPR-tree, because they do not consider the future locations of moving objects, which change continuously as time passes. In this paper, we propose an efficient cost model for spatio-temporal queries to solve this problem. We present analytical formulas which accurately calculate the number of disk accesses for spatio-temporal queries. Extensive experimental results show that our proposed method accurately estimates the number of disk accesses over various queries to spatio-temporal data combining reallife spatial data and synthetic temporal data. To evaluate the effectiveness of our method, we compared our spatio-temporal cost model (STCM) with an existing spatial cost model (SCM). The application of the existing SCM has the average error ratio from 52% to 93%, whereas our STCM has the average error ratio from 11% to 32%. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Spatio-temporal databases; Moving object; TPR-tree; Cost model 1. Introduction With new developments in positioning systems and electronics, many applications for moving objects have appeared. Most research for moving objects has progressed in access methods (Agarwal et al., 2000; Kollios et al., 1999; Pfoser et al., 2000; Saltenis and Jensen, 2002; Saltenis et al., 2000; Tao and Papadias, 2001, 2002). Our paper is related to selection queries considering future positions of moving objects, which are referred to as future queries (Sistla et al., 1997). An example of the future query is as follows: which cars will be inside a query window 20 min from now? Cars correspond to moving objects that move as time passes. Recently, a data model that can deal with the future locations of an object has been proposed (Sistla et al., 1997). Like various studies (Agarwal et al., 2000; Choi * Corresponding author. Tel.: /3577; fax: address: omni@islab.kaist.ac.kr (Y.-J. Choi). and Chung, 2002; Kollios et al., 1999; Saltenis and Jensen, 2002; Saltenis et al., 2000; Tao and Papadias, 2002; Wolfson et al., 1998), our work is based on this model. Saltenis et al. (2000) proposed the TPR-tree that is based on the R-tree and supports spatio-temporal queries for the future locations of moving objects. As the R- tree is one of the most popular access methods in spatial databases, the TPR-tree is a popular access method for moving objects in spatio-temporal databases. In recent years, various methods (Porkaew et al., 2001; Saltenis and Jensen, 2002; Tao and Papadias, 2002) using the TPR-tree have been proposed. Also, a selectivity estimation method for spatio-temporal queries using a concept of the TPR-tree has been proposed (Choi and Chung, 2002). Because the R-tree is one of the most popular methods in spatial databases (Beckmann et al., 1990), the cost models for the R-tree have been proposed (Kamel and Faloutsos, 1993; Leutenegger and Lopez, 2000; Theodoridis et al., 2000). However, although the TPR-tree is one of the most popular methods /$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi: /s (03)

2 102 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) in spatio-temporal databases, any cost model for the TPR-tree has not yet been proposed. As a result, a query optimizer may choose an inefficient plan for a spatiotemporal query. Existing spatial cost models (SCMs) do not accurately estimate the number of disk (or page) accesses for spatio-temporal queries, because they only handle the spatial positions of objects at the current time. So, they are not applicable for the TPR-tree that considers the future positions of moving objects. Therefore, we need a cost model for the TPR-tree considering the future positions of moving objects. In this paper, with the motivation based on the above problem, we propose a spatio-temporal cost model (STCM) that can accurately estimate the number of disk accesses for spatio-temporal queries using the TPR-tree. For the cost model of the TPR-tree, we introduce a spatio-temporal histogram describing the TPR-tree. Also, we present how to maintain the spatio-temporal histogram with a low overhead. Our cost model provides an advantage that this model can be used by various cost models for moving objects. Experimental results show that our STCM provides an accurate estimation over various queries for synthetic moving objects. However, for supporting a more realistic experimental environment, we used real-life spatial data, Tiger/lines data (US Bereau of Census, 1994) and Sequoia data (Stonebraker et al., 1993) popularly used in spatial database research, to generate moving objects. To our knowledge, the proposed method is the first work that specifically addresses the cost model of the TPR-tree for moving objects. So, to evaluate the effectiveness of our method, we compared our STCM with an existing SCM. While the application of the existing method has the average error ratio from 52% to 93%, our method has the average error ratio from 11% to 32%. Our contributions are summarized as follows: We propose an accurate cost model for spatio-temporal queries using the TPR-tree. To our knowledge, the proposed method is the first work that specifically addresses the cost model of the TPR-tree for moving objects. We present a practical method to maintain the spatiotemporal histogram describing the TPR-tree. The spatio-temporal histogram can represent the movements of moving objects and can be maintained with a low overhead. We provide extensive experimental results using various queries. For a more realistic experimental environment, we use real-life spatial data to generate moving objects. The experimental results show that our proposed method achieves a considerable accuracy. The paper is organized as follows. In Section 2, we describe the related work, which consists of spatiotemporal databases, the TPR-tree, and the cost models of the R-tree in spatial databases. In Section 3, we explain how to estimate the number of disk accesses for spatio-temporal queries using the TPR-tree. In Section 4, we show our experimental results and discuss them in detail. Finally, conclusions are made in Section Related work First, we briefly describe spatio-temporal databases and the TPR-tree, explain the cost models for the R-tree in spatial databases, and then explain the histogram update. Spatio-temporal databases manage objects that move as time passes. With new developments in technology, many applications have been proposed to deal with the problem of movement. These applications require spatio-temporal database management systems with modeling (Forlizzi et al., 2000; Sistla et al., 1997) and processing of queries to moving objects (Agarwal et al., 2000; Choi and Chung, 2002; Kollios et al., 1999; Pfoser et al., 2000; Saltenis et al., 2000; Tao and Papadias, 2001, 2002; Wolfson et al., 1998). Sistla et al. (1997) proposed a data model to represent moving objects. This model introduces a dynamic object that continuously changes its position as time passes. Many studies for query processing are based on this model. Saltenis et al. (2000) proposed the TPR-tree that is based on the R-tree and supports spatio-temporal queries for the future locations of moving objects. Fig. 1 shows a time-parameterized bounding interval (called an MBR in the general tree structure) of the TPR-tree and three one-dimensional moving objects ðo 1 o 3 Þ bounded by the bounding interval. The time-parameterized bounding interval is the most important concept of the TPR-tree and consists of a spatial interval ½x l ; x h Š and a velocity interval ½v l ; v h Š as shown in Fig. 1(a). The spatial interval of the time-parameterized bounding interval is represented at the index load (or creation) time t l. The incline of an arrow means a velocity. As shown in Fig. 1(a), the objects move within the range of thick dotted lines as time passes. Saltenis et al. (2000) proposed how to minimize the time-parameterized bounding interval when a moving object is updated. As shown in Fig. 1(b), consider a moving object o 3 updated at t c. As the newly updated velocity of o 3 is the minimum value, v l is changed. Similarly, the spatial interval is minimized to bound the future locations of moving objects after t c and consists of values at t l. Fig. 2 illustrates the time-parameterized bounding intervals organized to form a TPR-tree. Fig. 2(a) (c) show 13 moving objects, 5 nodes of level bounding moving objects, and 2 nodes of level 2 bounding nodes of level 1, respectively. And, Fig. 2(d) shows the corresponding TPR-tree. s means a time-parameterized

3 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) Fig. 1. Time-parameterized bounding interval of the TPR-tree for one-dimensional moving objects: (a) t l, (b) t c. Fig. 2. Moving objects organized in a TPR-tree with fanout ¼ 3, and the corresponding TPR-tree: (a) moving objects, (b) nodes of level 1 (or leaves), (c) nodes of level 2, (d) TPR-tree. bounding interval. Let us consider an update of the object o 13 at t c on the TPR-tree. Fig. 3(a) and (b) show the change of node 7 by o 13 and the change of node 2 by node 7, respectively. As seen in Fig. 2(d), the dark rectangles mean the modified entries. For better understanding on the cost model for the TPR-tree, let us observe the cost models for the R-tree that have been studied intensively in spatial databases. Kamel and Faloutsos (1993) proposed analytical formulas to calculate the average number of disk accesses for range queries using the R-tree. The paper assumes that queries are rectangles uniformly distributed over the unit square space (Kamel and Faloutsos, 1993). It presents the effects of the sizes of the nodes and the query for the R-tree performance estimation. Theodoridis et al. (2000) proposed a more specific analysis to accurately estimate the average number of disk accesses for range queries using the R-tree. The analysis, which is based on (Kamel and Faloutsos, 1993), basically presents the intersecting probability between nodes and a query for each level of the R-tree. Let f be the average fanout of the R-tree, N the number of objects, h ¼ 1 þdlog f ðn=f Þe the height of the R-tree, and s l;i the average extent of node rectangles of the R- tree at level l on dimension i. The following formula considers the average number DA of disk accesses for a query q 1 q 2 using the R-tree. DA ¼ 1 þ Xh 1 l¼1 N Y 2 f l i¼1 ðs l;i þ q i Þ! ð1þ The formula assumes that spatial data is randomly distributed in the unit spatial space. So, the paper presents a spatial histogram to handle skewed data. The spatial histogram is a simple grid structure that describes the spatial data distribution. In the experiment with a grid of cells, the paper showed that the average relative error is between 0% and 20% for real-life spatial data (Theodoridis et al., 2000). We use two histograms to accurately estimate the number of disk accesses for spatio-temporal queries using the TPR-tree: a spatial histogram for handling the skewed spatial distribution and a spatio-temporal histogram for describing the TPR-tree. Although we additionally use a spatio-temporal histogram, it is maintained with a low overhead. Also, the size of the spatio-temporal histogram is remarkably small, compared to the size of the spatial histogram. The histogram is one of the most popular estimation methods, because

4 104 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) Fig. 3. Change of nodes in the TPR-tree: (a) change of node 7, (b) change of node 2. it approximates any data distribution and requires reasonably small storage with low error rates (Acharya et al., 1999). We will discuss the spatio-temporal histogram describing the TPR-tree and the method to maintain the spatio-temporal histogram in Section 3. We use the spatial histogram to manage the skewed spatial distribution of moving objects, as used in (Theodoridis et al., 2000). In addition, we should consider the fact that the positions of moving objects are updated very frequently. If new update information is not reflected to the spatial histogram, the query optimizer may choose an inefficient plan because an inaccurate estimation is used. However, it is clearly impractical to construct a new spatial histogram whenever moving objects are updated. The histogram is not required to be updated from the raw objects, but it can be updated from sample data (Gibbons et al., 1997). In spatio-temporal databases, the histogram can be maintained practically by using sample data. The sample data only includes information related to the histogram construction. Considering the moving object, the information consists of the identification, the last update time, the spatial position of the time, and the velocity. The construction of a new histogram from the sample data reduces the I/O overhead remarkably, compared to the overhead of the construction of a new histogram from all of the data. Therefore, we construct the spatial histogram only from the sample data of moving objects. We construct the whole spatial histogram periodically. When an object is modified, previous techniques for the histogram update can update the part of the histogram related to the modified object without constructing the whole histogram. However, because the spatial positions of the non-modified objects implicitly change as time passes, to update only the part of the histogram related to the modified object is not proper. We construct the whole spatial histogram per time unit. We consider moving objects spatial positions, located at the corresponding time, to construct the spatial histogram. In Section 4, we demonstrate how this update strategy for the spatial histogram can be possible. Tayeb et al. (1998) used Quadtrees for indexing the future locations of one-dimensional moving objects, and Agarwal and Procopiuc (2002) proposed an indexing method using the duality transformation. Duality maps a point in the two-dimensional space to a line in the twodimensional space and vice versa. However, these indexing approaches have a space overhead because of the duplication of each object. 3. Cost model for spatio-temporal queries This section explains the cost model for spatio-temporal queries using the TPR-tree. First, we describe the structure of our spatio-temporal histogram and the method for the histogram maintenance. Then, we explain how to analyze the number of disk accesses for spatio-temporal queries using the TPR-tree Spatio-temporal histogram for the TPR-tree This section describes the structure of our spatiotemporal histogram for the TPR-tree and a simple strategy for the maintenance of the histogram. Table 1 presents the symbols used throughout the paper. We describe how to estimate the number of disk accesses for spatio-temporal queries to one-dimensional moving objects, because the estimation method for one-dimensional moving objects can be easily extended to that for two-dimensional moving objects. We illustrate the estimation method for one-dimension corresponding to the

5 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) Table 1 Symbol description Symbol IðI l ; I h Þ sðx 1 ; x 2 ; v 1 ; v 2 s 1 ;...; s n Þ t c t l Qða 1 ; a 2 ; tþ N l S l ðs 1 ; s 2 Þ V l ðv 1 ; v 2 Þ h f Description Interval: I l I h ; I l, low value of I; I h, high value of I Time-parameterized bounding rectangle of the TPR-tree for 2D moving objects: x 1 and x 2, spatial intervals; v 1 and v 2, velocity intervals Node of the TPR-tree: n, number of timeparameterized bounding rectangles; s i, the ith s Current time Index load (or creation) time Query: a 1 and a 2, spatial intervals; t, time interval Number of TPR-tree nodes at level l Average extent of spatial intervals of the TPRtree nodes of level l at t c Average velocity interval of the TPR-tree nodes at level l Height of the TPR-tree Average TPR-tree node fanout Fig. 4. Time-parameterized bounding interval and query to onedimensional moving objects. first dimension of two-dimensional moving objects. Also, we use x, v, a, and s rather than x 1, v 1, a 1 and s 1, respectively, for notational convenience. Fig. 4 shows the relationship between a time-parameterized bounding interval s and a spatio-temporal query Q. The low value t l of Q t should be greater than or equal to the current time t c, where Q t denotes the t component of Q. Our goal is to estimate the number of intersecting cases between s and Q, as shown in Fig. 4. However, existing SCMs do not consider the future locations of moving objects. So, they do not consider the intersecting case between s and Q, as shown in Fig. 4. They only consider the intersecting cases between s and Q at t c. Our spatio-temporal histogram consists of N l, S l, and V l, defined in Table 1, for each level l of the TPR-tree. An average extent s of S l is P N l i¼1 SLðt c; s i Þ=N l, where SLðt c ; sþ ¼ðsx h þs v h ðt c t l ÞÞ ðs x l þ s v l ðt c t l ÞÞ and s w denotes the w component of s. Fig. 5 shows SL for two time-parameterized bounding intervals at t c.an average velocity interval vðv l ; v h Þ of V l is defined as the following: v l ¼ P N l i¼1 s i v l =N l and v h ¼ P N l i¼1 s i v h =N l. A simple method to calculate N l, S l, and V l is to visit all the nodes of the TPR-tree. In an environment where Fig. 5. s and SL as time passes: (a) s 1 and SLðt c ; s 1 Þ at t c, (b) s 2 and SLðt c ; s 2 Þ at t c, (c) s 0 2 and SLðt0 c ; s0 2 Þ at t0 c < t c. data is frequently updated, the spatio-temporal histogram must be periodically recomputed to accurately preserve it. So, this method consumes much time because all the nodes of the TPR-tree must be visited. We propose a simple and practical method that maintains the spatio-temporal histogram describing the TPR-tree with a low overhead. When an object is modified, we should certainly access nodes of the TPR-tree. Our basic strategy is to dynamically update N l, S l,andv l whenever nodes are accessed. This method exactly maintains N l and V l from only the accessed nodes. Although a node overflows or underflows, N l can be easily maintained. When the velocity interval of a node is changed by an updated object o 7 at t c as shown in Fig. 5(b), V l can be maintained by a relationship between the previous velocity interval and the new velocity interval. However, it is difficult to exactly maintain S l. We should consider all the nodes of the TPR-tree at level l. For example, let us consider two time-parameterized

6 106 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) bounding intervals s 1 and s 2 at t c as shown in Fig. 5. s 1 is the time-parameterized bounding interval without any modification at t c as shown in Fig. 5(a). s 2 is the modified time-parameterized bounding interval by the updated object o 7 at t c as shown in Fig. 5(b). s 1 means the non-accessed node at t c and s 2 means the accessed node at t c. We cannot exactly calculate SL for S l because s 1 as shown in Fig. 5(a) is not accessed since it is not updated. On the other hand, we can easily calculate SL for S l at t c from s 2 in Fig. 5(b) because it is accessed. As shown in Fig. 5(c), let s 0 2 be the previous state of s 2. We do not know SL from s 0 2 that is accessed at the previous time t0 c. Therefore, unlike V l, it is difficult to exactly process SL for S l from the accessed node at t c. So, we propose how to maintain an approximated S l, called Sl a, instead of S l. For example, let us consider node accesses of an index by an object insertion. As shown in Fig. 6, dark rectangles for each level depict the accessed nodes. We can obtain information of the time-parameterized bounding intervals at level l 1 from an accessed node at level l. As a result, by an object insertion as shown in Fig. 6, we can obtain information of nodes including light dark rectangles. This means that we can use SL of non-accessed nodes. When a node except a leaf node is accessed at level l, there exists four possible cases for the node: a simple node access without any modification, a node access with a modified entry by a modified node at level l 1, a node access with a deleted entry by the underflow of a node at level l 1, and a node access with an inserted entry by the overflow of a node at level l 1. The following formulas describe how to maintain N l 1, V l 1, and Sl 1 a as a node of level l except a leaf node is accessed at t c. Simple node access without any modification. S a l 1 s ðn l a l 1 sþ i¼1 SLðtc;@siÞ N l 1 is the accessed node. Node access with a modified entry. Fig. 6. Node accesses by an object insertion. V l 1 v l V l 1 v h S a l 1 s N l 1 V l 1 v l av l þbv l N l 1 N l 1 V l 1 v h av h þbv h N l 1 ðn l 1 1ÞS a l 1 sþslðtc;bþ N l 1 where a is the entry before modification and b is the entry after modification. Node access with a deleted entry. V l 1 v l V l 1 :v h N l 1 V l 1 v l av l N l 1 1 N l 1 V l 1 v h av h N l 1 1 N l 1 N l 1 1 where a is the entry before deletion. Node access with an inserted entry. V l 1 v l V l 1 v h Sl 1 a s N l 1 N l 1 þ 1 N l 1 V l 1 v l av l þbv l þcv l N l 1 þ1 N l 1 V l 1 v h av h þbv l þcv l N l 1 þ1 ðn l 1 1ÞS a l 1 sþslðtc;bþþslðtc;cþ N l 1 þ1 where a is the entry before overflow, b is the modified entry after overflow, and c is the entry inserted newly after overflow. Let us consider a simple example for an update of our spatio-temporal histogram. Table 2 presents information of 5 leaf nodes described in Fig. 2(b) at t l ¼ 0. The spatial position of the object o 13 is 33 and the velocity of the object is )0.7. SL of the node 7 at t l is x h x l ¼ ¼ 5. Now, we explain how to maintain N 1, S1 a s, V 1 v l, and V 1 v h of level 1 when a node of level 2 is visited. As shown in Fig. 2(d), to apply the update of o 13 on the TPR-tree, nodes are accessed as the following order: root node, node 2, node 7, node 2, and root node. The first two accesses (root node and node 2) are simple node accesses without any modification. The next three accesses (node 7, node 2, and root node) are node accesses with a modified entry. As shown in Fig. 7, let us assume that node 7 is modified by o 13 with a velocity )0.8 changed at t c ¼ 1. S1 a s is ð2 þ 3 þ 4 þ 3 þ 5Þ=5 ¼ 3:4att l before the change. Table 3 presents the changes of N 1, S1 a s, V 1 v l, and V 1 v h by two accesses for node 2 of level 2 at t c. For the first simple node access at t c, S1 a s is ðð5 2Þ 3:4 þ 4:2 þ 6:3Þ=5 ¼ 4:14 where SLðt c ; s 6 Þ¼ ð25 þ 0:7 ð1 0ÞÞ ð22 0:5 ð1 0ÞÞ ¼ 4:2 and SLðt c ; s 7 Þ¼ ð33 þ 0:6 ð1 0ÞÞ ð28 0:7 ð1 0ÞÞ ¼ 6:3. As node 2 is accessed again, s 7 is modified. Fig. 7 illustrates the change of s 7. v l of s 7 becomes )0.8. We can easily calculate a spatial position 28:2ð¼ 28:5 0:3 ð1 0ÞÞ from o 12 at t c and then, x l ¼ 28:2þ 0:8 ð1 0Þ ¼29. Similarly, we can calculate x h. For the modified s 7, SLðt c ; s 7 Þ is ð31:7 þ 0:6 ð1 0ÞÞ ð29 0:8 ð1 0ÞÞ ¼ 4:1. Finally, S1 a s is ðð5 1Þ 4:14 þ 4:1Þ=5 ¼ 4:132. And, we can justify that V 1 v l is correctly maintained

7 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) Table 2 Nodes of level 1 (or leaves) s 3 s 4 s 5 s 6 s 7 x l : 5 x l : 9 x l : 15 x l : 22 x l : 28 x h : 7 x h : 12 x h : 19 x h : 25 x h : 33 v l : 0:3 v l : 0:5 v l : 0:6 v l : 0:5 v l : 0:7 v h : 0:5 v h : 0:9 v h : 0:6 v h : 0:7 v h : 0:6 SL : 2 SL : 3 SL : 4 SL : 3 SL : 5 o 1 o 2 o 3 o 4 o 5 o 6 o 7 o 8 o 9 o 10 o 11 o 12 o ) ) )0.2 ) ) )0.3 )0.7 from the fact that the average v l of s 3, s 4, s 5, s 6, and the modified s 7 is ð 0:2 0:5 0:6 0:5 0:8Þ=5 ¼ 0:52. Also, we can use this method as nodes of the TPRtree are accessed by spatio-temporal queries. Spatiotemporal queries only require simple node accesses without any modification. Our spatio-temporal histogram that describes the TPR-tree has the following features. First, the histogram can be accurately maintained by directly obtained information from the TPRtree. Second, the histogram can be maintained with a low overhead due to the very small size of the histogram and the processing of its additional updates only when nodes are accessed Cost model of the TPR-tree Fig. 7. Modification of node 7 by the update of o 13 : (a) before the modification, (b) after the modification. This section estimates the average number of disk accesses for spatio-temporal queries using the TPR-tree. We use the spatio-temporal histogram proposed in Section 3.1. We use Fig. 8 to explain three possible relationships between a spatio-temporal query Q and an average time-parameterized bounding interval s based on V l and S l of the spatio-temporal histogram. A spatiotemporal query Q consists of a spatial interval [a l ; a h ] and a time interval [t l ; t h ] as shown in Fig. 4. As in the definition of S l, the spatial interval of s is considered at t c. If the spatial interval of s intersects with the spatial interval I 3, s always intersects with the query. If the spatial interval of s intersects with the spatial interval I 2 (or I 4 ), s can intersect with the query or cannot intersect with the query. If the spatial interval of s intersects with Fig. 8. Average time-parameterized bounding intervals and query. Table 3 Changes of N 1, S a 1 s, V 1 v l, and V 1 v h t l N 1 ¼ 5 N 1 ¼ 5 S1 a s ¼ 2þ3þ4þ3þ5 ¼ 3:4 S a 5 1 s ¼ ð5 1Þ4:14þ4:1 ¼ 4:132 5 V 1 v l ¼ 0:2 0:5 0:6 0:5 0:7 ¼ 0:5 V 5 1 v l ¼ 5ð 0:5Þ ð 0:7Þþð 0:8Þ ¼ 0:52 5 V 1 v h ¼ 0:5þ0:9þ0:6þ0:7þ0:6 ¼ 0:66 V 5 1 v h ¼ 5ð0:66Þ ð0:6þþð0:6þ ¼ 0:66 5 t c

8 108 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) the spatial interval I 1 (or I 5 ), s does not intersect with the query. We make a definition to specifically explain the spatial interval I 3. Definition 1. Let t c be the current time, v the velocity interval, Q the query. For the time-parameterized bounding intervals limited by v, CQ is the maximum spatial interval I at t c satisfying the following condition. All the possible time-parameterized bounding intervals with the spatial interval containing a sub-interval of I at t c should intersect with Q. For example, let us consider a query and timeparameterized bounding intervals limited by a velocity interval vðv l < 0; v h > 0Þ. As shown in Fig. 8, the spatial interval I 3 at t c depicts CQ. The following is the formula derived for the above definition. CQðt c ; v; Qða; tþþ has an interval I as follows: I l ¼ al ðt l t c Þv h if v h < 0 a l ðt h t c Þv h otherwise I h ¼ ah ðt l t c Þv l if 0 < v l a h ðt h t c Þv l otherwise Now, we should consider a relationship between an average time-parameterized bounding interval s and CQ. The concept of the relationship is basically the same as the concepts proposed in (Kamel and Faloutsos, 1993; Leutenegger and Lopez, 2000). We make a definition to specifically explain the relationship between the center point of the spatial interval of s and CQ. Definition 2. Let t c be the current time, s the spatial extent, v the velocity interval, Q the query. For the timeparameterized bounding intervals limited by v and the spatial interval with s at t c, ECQ is the maximum spatial interval I at t c expanded by the center points of the timeparameterized bounding intervals that intersect with Q. For example, let us consider a query and the timeparameterized bounding intervals limited by a velocity interval vðv l < 0; v h > 0Þ and a spatial interval with extent s at t c. Fig. 9 shows ECQ at t c. ECQ is easily derived by CQ. ECQðs; JÞ has an interval IðI l ¼ J l ðs=2þ; I h ¼ J h þðs=2þþ where J ¼ CQðt c ; v; Qða; tþþ. We use CQ and ECQ to estimate the number of nodes intersecting with Q. The number is determined by not only the objects intersecting with Q but also the objects that do not intersect with Q and are adjacent from Q. As shown in Fig. 10, the average time-parameterized bounding intervals that enclose objects passing CQ at t c always intersect with Q. However, the bounding intervals that enclose objects passing I 2 (or I 4 ) can or cannot intersect with Q. So, we can consider that Q intersects with the bounding intervals enclosing objects that pass Fig. 9. ECQ. Fig. 10. CQ and ECQ. ECQ extended from CQ. And, we estimate the number of nodes from the number of objects passing ECQ. The estimation method for one-dimensional moving objects is easily extended to the estimation method for two-dimensional moving objects because the estimation for one-dimension is independent of that for the other. Fig. 11 shows the disk access estimation algorithm for spatio-temporal queries which use the TPR-tree indexing two-dimensional moving objects. We use CQ 2 and ECQ 2 for two-dimensional moving objects, instead of CQ and ECQ used for one-dimensional moving objects, respectively. The DiskAccess algorithm estimates the average number of disk accesses using the spatio-temporal histogram for the TPR-tree and the spatial histogram for the skewed spatial distribution. In Fig. 11, line 1 is the access to the root node of the TPR-tree and line 2 generates the height h of the TPR-tree where N is the number of objects. In the loop of lines 3 7, the number of disk accesses from level 1 to level h 1is summated. Lines 4 5 generate a spatial rectangle R at t c Fig. 11. Algorithm DiskAccess for the TPR-tree.

9 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) according to Definitions 1 and 2. Line 6 estimates the number n of moving objects satisfying to the query at level l. Line 7 estimates the number of nodes of the TPR-tree at level l. 4. Experiments In this section, we present the experimental environment and the experimental results of our proposed method. Moving objects are synthetically generated by using real-life spatial data to make a realistic experimental environment. We evaluate the accuracy of results by using queries with various spatial rectangle sizes and various time interval lengths Experimental environment Our experimental environment is similar to that in (Choi and Chung, 2002) based on (Kollios et al., 1999; Saltenis et al., 2000). Objects with the maximum speed 3.0 km/min (180 km/h) are set to move in 1000 km 1000 km two-dimensional space. We generated realistic movements of objects using skewed speed distributions and skewed direction distributions. We used a real-life spatial data, Tiger/lines data (US Bereau of Census, 1994) popularly used in spatial database research, to generate moving objects. As shown in Fig. 12(a), we represent the initial spatial positions of moving objects using road data of California area Tiger/lines data. The number of moving objects is Fig. 12(b) describes the velocity distribution to update the velocities of moving objects. To update the velocity of a moving object, we randomly choose a point in a circle with a radius 3 corresponding to the maximum speed. The position of the chosen point with respect to the center point of the circle determines the movement of the object. The distance between two points indicates the new updated speed of the object. The direction from the center point to the chosen point indicates the new updated direction of the object. Also, we used vertices of road data of Tiger/lines data to generate realistic movements of objects with skewed speed distributions and skewed direction distributions. In general, most research presents the concept of time units to easily represent the passage of time (Choi and Chung, 2002; Kollios et al., 1999; Pfoser et al., 2000; Saltenis et al., 2000; Tao and Papadias, 2001). We used 1 min as the time unit and updated approximately 2% of the whole moving objects every time unit. The histogram can be updated from sample data (Gibbons et al., 1997). So, we used 1% of the whole moving objects as the sample objects for the spatial histogram. The sample data only includes information related to the histogram construction. The information for a sample object needs only the identification, the last update time, the spatial position of the time, and the velocity. This requirement decreases the size of the sample data remarkably. We updated the spatial histogram from the sample data every time unit. This histogram update strategy is possible because the I/O overhead for the spatial histogram update from the sample data is very low, compared to the update overhead of approximately 2% of the whole moving objects per time unit. A spatial grid for constructing the spatial histogram is set to cells. All experiments were conducted on a Pentium IV 2.4 GHz PC with 1 Gbytes of main memory. It took less than 1 ms to execute the DiskAccess algorithm and less than 20 ms to reconstruct the spatial histogram from updated sample date. Although we additionally use a spatio-temporal histogram, the spatio-temporal histogram is maintained with a low overhead as explained Fig. 12. Synthetic moving objects generated by Tiger/lines data: (a) initial spatial locations, (b) velocities.

10 110 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) below. We evaluated the average update cost about 1,000,000 updates of moving objects. Without the maintenance of our spatio-temporal histogram, it took about 4.0 ms to simply apply an update to the TPR-tree. However, it took about 4.2 ms to apply an update to the TPR-tree and maintain the spatio-temporal histogram at the same time. The size of the spatio-temporal histogram for the TPR-tree is about 5% of the size of the spatial histogram. Experiments were conducted for 500 time units. The size of the query spatial rectangle (QS) varied as 0.25%, 0.5%, 1%, 2%, 4% of the size of the data space. The spatial position of the query was randomly chosen in the data space. The length of the query time interval varied as 0, 10, 20, 30, 40. The t l of the time interval was randomly chosen between t c and t c þ 25. To generate various queries, we dealt with 25 combinations: All combinations for five types of QS and five types of the query time interval as mentioned above. For each time unit, 25 queries are generated. The actual results of queries are compared to the results estimated using the proposed method in this paper. % Error is used to assess the accuracy of estimation results: actual result estimated result errorð%þ ¼ j j 100 actual result To build an efficient TPR-tree, Saltenis et al. (2000) introduced two parameters: the index usage time U and the querying window W. When U is a half of the average update interval length of moving objects and W is the maximum length of the query time interval, the TPRtree shows good performance over various experiments because the leaf nodes are bounded periodically and properly (Saltenis et al., 2000). Thus, we set values of W and U to build the TPR-tree as mentioned above. W is set to 40 since the maximum length of the query time interval is 40 and U is set to 25 since we update approximately 2% of the whole moving objects every time unit. The quality of the TPR-tree is affected by the selection of the parameter values, whereas our cost model is not affected. Regardless of the quality of the TPR-tree, our cost model accurately estimates the number of disk accesses since the model uses the spatiotemporal histogram directly acquired from the TPRtree. The average capacity of the TPR-tree is set to the typical 67 percent value as in (Theodoridis et al., 2000) Experimental results Fig. 13 shows the experimental results evaluated from queries (25 queries/time unit 500 time units). The results represent the average relative error with respect to the time interval length of the query and the spatial rectangle of the query. Each query type is evaluated by the average relative error of 500 queries (1 query/time unit 500 time units). Fig. 13 shows the Fig. 13. Average relative error for the different QS sizes as the time interval length of the query is varied (Tiger/lines data). experimental results of moving objects using Tiger/lines data. The average relative error is from 14% to 32%. As the spatial rectangle size of the query increases, the average relative error decreases, as in the general experimental results (Acharya et al., 1999). However, as the time interval length of the query increases, the average relative error gradually increases. This is because our method considers the movements of moving objects that pass the left/right sides of the query as time passes, as shown in Fig. 4. That is, the more the time interval length increases, the more the error in calculation using the left/right sides of the query increases. Since there has been no previous work that specifically addresses a cost model for spatio-temporal queries using the TPR-tree, we compared our STCM with an existing SCM using the formula (1) in Section 2. We also used a grid of cells for the spatial histogram of SCM and updated the spatial histogram from the sample data every time unit. The comparative experiments are based on moving objects using Tiger/lines data. Fig. 14(a) shows the estimation results of SCM and STCM with respect to the time interval length of the query applied to QS for 0.25%. As expected, SCM that does not consider the property of moving objects produces high error rates. The average error rate of a query with a time interval length of 40 is about 93%. However, STCM considering the property of moving objects has reasonably accurate estimation results with an average error rate about 32% for a time interval length of 40. Fig. 14(b) shows the estimation results with SCM and STCM with respect to the time interval length of the query applied to QS for 4%. Like Fig. 14(a), STCM has better estimation results, compared to SCM. In the same way, we made another realistic experimental environment by using Sequoia data (Stonebraker et al., 1993) for the initial spatial positions and the movements of moving objects. The number of moving objects using Sequoia data is Fig. 15 shows the similar experimental results to moving objects using Sequoia data. Fig. 15 shows the average relative error

11 Y.-J. Choi et al. / The Journal of Systems and Software 73 (2004) Fig. 14. Average relative error for SCM and STCM as the time interval length of the query is varied (Tiger/lines data); (a) QS size: 0.25%, (b) QS size: 4%. of disk accesses for spatio-temporal queries using the TPR-tree, compared to SCM. 5. Conclusions Fig. 15. Average relative error for the different QS sizes as the time interval length of the query is varied (Sequoia data). with respect to the size of the spatial rectangle for different time interval lengths. In addition, we assessed the average relative error of SCM and STCM with respect to the time interval length for two QSs. The comparative experiments are based on moving objects using Sequoia data. For Fig. 16(a), QS is set to 0.25%. As expected, STCM has accurate estimation results, compared to SCM. Fig. 16(b) shows the impact of the time interval length applied to QS for 4%. Like Fig. 16(a), STCM accurately estimates the number Spatio-temporal databases have been studied intensively in recent years. In this paper, we proposed a cost model to estimate the number of disk accesses for spatio-temporal queries using the TPR-tree. We used analytical formulas that accurately estimate the movements of moving objects. Our analytical formulas provide an advantage that these formulas can be basically used by various cost models to moving objects. To conduct experiments in a realistic environment, we generated synthetic moving objects by using real-life spatial data with a reasonable skew distribution. In the experiments, the proposed method provided accurate estimation results over various queries with different spatial area sizes and time interval lengths. To our knowledge, the proposed method is the first work specifically addressing the cost model of the TPR-tree for moving objects. Experimental results show that our cost model accurately estimates the number of disk accesses for spatio-temporal queries using the TPR-tree. Fig. 16. Average relative error for SCM and STCM as the time interval length of the query is varied (Sequoia data); (a) QS size: 0.25%, (b) QS size: 4%.

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