Geoinformatica (2015) 19:61–84 DOI 10.1007/s10707-014-0206-6

Partition-based range query for uncertain trajectories in road networks Ling Chen & Yanlin Tang & Mingqi Lv & Gencai Chen

Received: 1 July 2013 / Revised: 1 December 2013 Accepted: 3 February 2014 / Published online: 21 February 2014 # Springer Science+Business Media New York 2014

Abstract Query processing for trajectory data is a hot topic in the field of moving objects databases (MODs). Most of the previous research work focused on the Euclidean space, and the uncertain trajectories are represented as sheared cylinders. However, in many applications (e.g. traffic management), the movements of moving objects (MOs) are constrained by the road network environments, which makes the previous methods ineffective. In this paper, we firstly construct an uncertain trajectory model, which is composed of a sequence of segment units with earliest arrival time and latest departure time, based on the assuming availability of a maximum speed on each road segment. Secondly, we present a partition-based uncertain trajectory index (PUTI) to facilitate the search of possible MOs within the space and time range in the road networks based on the uncertain trajectory model. It provides appropriate groups to gather segment units of trajectories according to their network distances. Finally, an efficient algorithm for range query is proposed by leveraging the index. The experiments on two datasets demonstrate that the uncertain trajectory model is effective, and PUTI also significantly outperforms the network distance based MON-tree on range query. Keywords Uncertain trajectories . Range query . Graph partition . Road networks

1 Introduction With the rapid advances in communication and sensing technologies, there is an increasing demand in many applications for supporting location based services (LBSs), such as mobile communication, vehicle guidance, and traffic management, etc. As a result, an enormous amount of research effort goes into the field of moving objects databases (MODs). One of the most representative data in MODs is historical trajectory, available as a sequence of (location, time) values. A typical application for trajectories is range query. For example, in the investigation of a traffic accident, it is very important for officers to find out the potential perpetrators or witnesses, by analyzing the historical trajectories, which are produced by cars passing through the roads near the accident scene. L. Chen (*) : Y. Tang : M. Lv : G. Chen College of Computer Science, Zhejiang University, Hangzhou 310027, People’s Republic of China e-mail: [email protected]

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Traditionally, when querying the moving objects (MOs), a normal assumption is that the entire trajectories are available [3, 4, 6, 14, 16, 20, 21, 22]. However, location acquisition devices (e.g. GPS, roadside sensor, etc.) can only provide location samples at discrete time instants in practice. For example, in taxi management applications, a taxi usually sends its current location to servers every minute or so for the sake of energy efficiency and communication cost reduction. Even the movements of a taxi are restricted to the road networks, the acquisition of its exact location at a specific time instant still cannot be guaranteed. Motivated by these observations, we incorporate uncertainty into trajectories in the road networks, and estimate time based on the maximum speed of each road segment. On the other hand, the existing indexing techniques for MOs in the road networks decompose the networks into roads or road segments, and organize the spatio-temporal location of the MOs on each road or road segment with a specific index, e.g. 2D R-tree [3–6, 29]. One of the shortcomings of these approaches is the way that the space is decomposed. It might lead to indexing too many duplicate trajectories. Besides, these approaches do not support network distance based query. In the light of the limitation, we develop a partition based index structure for uncertain trajectory range query. By taking the above two aspects into consideration, we propose an approach to solve the problem of spatio-temporal range query for uncertain trajectories in the road networks. The basic idea is to partition the network into different regions according to their sums of network distances. Then, the trajectories are split into segment units with path and time uncertainty. Finally, the segment units are co-located according to the network partitions, and the total time interval of segment units in each partition of every MO is indexed. The purposes of partition are as follows. The first is to cut down the disk I/O, because we have to perform disk I/O operations on every road segment without the partitions, like FNR-tree [6]. The second is to query spatial region efficiently by using pre-computed distances between adjacent partitions. The third is to reduce the overlap of time intervals caused by uncertainty. Given a range query including a query point q, a distance threshold dr and a time interval (ts,te), we can quickly find partitions completely covered by the query spatial region and partitions which partially intersect with the query spatial region. For the former kind of partitions, we can easily get the satisfied trajectories through the time interval tree on each partition. For the latter kind of partitions, we firstly query the time interval trees to get candidates, and then refine the candidates for the result. In summary, followings are our main contributions: 1. Present an uncertain trajectory model: By considering the maximum speed on each road segment, we present an uncertain trajectory model, including path uncertainty and time uncertainty. To the best of our knowledge, it is the first model that represents the uncertainty by segment units with earliest arrival time and latest departure time, instead of time-dependent probability functions or space-time prisms. 2. Propose an index method for uncertain trajectories: We propose a partition based index framework, which combines R-tree and B+-tree, for the range query of uncertain trajectories. 3. Propose a range query algorithm for uncertain trajectories: We define the problem of range query for uncertain trajectories in road networks, and provide an efficient range query algorithm that consists of three steps: spatial pruning, time filter and result refinement. 4. Conduct experimental verification: We conduct extensive experimental evaluation which demonstrates the benefits of the proposed methodology. The remaining part of this paper is organized as follows. Section 2 presents the related work about the uncertainty of MOs and trajectory indexes in road networks. Section 3 introduces the

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basic definitions and the uncertain trajectory model. Sections 4 and 5 mainly describe the index structure and range query algorithm in detail. Section 6 presents the performance evaluation, and at last the conclusions are given in Section 7.

2 Related work Since the problem investigated in this paper relates to both uncertain trajectory models and queries for trajectories in the road networks, we review the related work in these two categories in Sections 2.1 and 2.2 respectively. 2.1 Managing uncertainty of moving objects In recent years, the uncertainty issues in MODs have been proposed in two main environments: the Euclidean space [2, 9, 15, 16, 19, 22–26], and the road network space [3–5, 10, 11, 12, 28, 29, 30]. Researchers have addressed the problem of generic representation of uncertainty, along with a framework for syntactic categorization of spatio-temporal queries. Lange et al. [13] identify two broad categories of uncertainty trajectory models of MOs: one is PDF-based model, and the other is shape-based model. There are some literatures focusing on uncertain trajectories in the Euclidean space environment. Pfoser and Jensen [19] consider the implications of the fact that the object’s motion was constrained by the maximum speed between two updates, and demonstrate that the spatial zone of the MO during two consecutive updates is an ellipse. However, when the difference between two updates or the maximum distance is large, the error may become very great since it limits the uncertainty to the past of MOs. Another uncertain trajectory model is represented as a three-dimensional sheared cylinder by Trajcevski et al. [26]. This is obtained by associating a fixed uncertain threshold r at each time instant with each line segment of the trajectory. For a given uncertain trajectory and two end-points, the volume of the trajectory between two updates is the union of all the disks with a radius r centered at the updates along the line segment which connects two end-points. However, the author did not give an appropriate way to estimate the value of r. In practice, it is difficult to determine the value of r that can assure high accuracy. Based on the above two uncertain trajectory models, a set of novel but natural spatiotemporal operators are proposed to express spatio-temporal queries [16, 26]. Trajcevski et al. [23–25] propose probabilistic range query and continuous nearest neighbor query algorithms. Chung et al. [2] map the uncertain movements of all MOs to a dual space for indexing, and employ an approximate approach to examine the candidates. Researchers also make contributions to meeting the demands of road network environments, and existing uncertain trajectory models can be divided into three categories. The first kind of uncertain trajectory models is associated with a location update policy. De Almeida and Güting ([3, 4] present the geometry of the MOs in networks with uncertainty, and extends the ADT framework [7] in terms of uncertainty in data types and operations. They also propose a location update policy, including ITLU, DTTLU and STTLU, to reduce the uncertainty of MOs. Ding [5] represent the uncertain region [3, 4] as Uncertain Trajectory Unit, and propose an index framework, the UTR-tree, which is based on MON-tree [3, 4], to index the full uncertain trajectories of network constrained MOs. However, since both GPS devices and roadside sensors usually cannot support the update policy, it is difficult for applications to use these methods directly. In addition, network distance based range query is not supported in these literatures.

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The second kind of uncertain trajectory models are represented as time-dependent probability functions, and the impact of uncertain model on query processing is considered. Zheng et al. [29] represent the uncertainty of MOs in the road networks as time-dependent probability distribution function, assuming the availability of a maximal speed on each road segment. The uncertain trajectory model has two types of uncertainty, i.e. path uncertainty and location uncertainty, and the UTH index is introduced to process probabilistic spatio-temporal range query, which is based on FNR-tree [6]. Since the probability that the MOs are in the spatial and time range is computed in query processing, the efficiency is relatively low. The third kind of uncertain trajectory models are represented as the space-time prisms. Kuijpers and Othman [11] propose the space-time prism in the road networks with general maximal speeds of edges based on the model in Euclidean space [19]. For each location along a given edge, a vertical line-segment is bounded by the earliest-possible and the latest-possible time that the MO can be at that location. However, there is only an alibi query provided in the paper, which might be because the complexity of other query types is high. Kuijpers et al. [10] also propose the algorithms to calculate network-based space-time prisms. 2.2 Query for trajectories in road networks As mentioned above, considerable attention has been paid to the index methods for MOs. Most of these works deal with indexing past, present, or near-feature positions of MOs that move freely in a two-dimensional space. There are also methods for indexing trajectories of MOs in the networks [3, 4, 6, 14, 16, 20, 21, 22, 27]. The main idea in these previous works is to decompose a three-dimensional problem into two sub-problems in lower dimensions and then use a combination of two level R-trees to index the trajectories. The trajectories are represented with reference to a network, i.e. with the relative positions of the MOs on network edges [7]. These indexes can be classified into two categories. The first kind of indexes considers the roads or road segments as index units. The FNR-tree [6] utilizes a 2D R-tree to index road segments. For every leaf node in the 2D R-tree, there is a 1D R-tree to index the MOs whose trajectories cross the road segments included in the leaf nodes within a certain period of time. A major disadvantage of the FNR-tree is its limitation in trajectory modeling. Since only the time intervals are stored in the 1D R-tree, it is assumed that the MOs could not stop, change speed or direction in the middle of a road segment. This limitation is addressed by the MON-tree [3, 4]. The MON-tree is composed of a 2D R-tree (top R-tree) that indexes the network edges and a set of 2D R-trees (bottom R-trees) that index the movements of MOs along the edges. An additional hash structure is used to map each polyline [3, 4] to its corresponding tree, which can speed up insertions. Given a 3D spatio-temporal query, the top R-tree is used to find the precise intersections between the spatial part of the query and the network. Based on these intersections, a set of sub-queries is generated for each intersected part of each involved edge, and the corresponding bottom R-trees are accessed to respond to the sub-queries. The second kind of indexes considers the partitions as index units. Xuan et al. [27] and Liu et al. [16] propose range query algorithms for in-network trajectories based on Voronoi, which is a structure widely used in nearest neighbor query process. They consider the Voronoi partition as a processing unit rather than an edge or a route. Xuan et al. [27] also propose continuous range query. Sandu Popa et al. [21] propose PARINET and T-PARINET index to efficiently retrieve the trajectories of MOs moving in the networks. The structure of PARINET [21] is based on a combination of graph partitions and a set of composite B+tree local indexes. The trajectories are decomposed using graph partition theory, and the process can be tuned for

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a given query load and data distribution in the network. T-PARINET [22] is designed for the continuous index of trajectory data flows. Since roads or road segments are basic units in the networks and partitions consist of adjacent road segments, the second type is more efficient on range query. However, all the above indexes in the networks do not take the uncertainty of trajectories into consideration. It is usually assumed that an update is issued each time the MOs change their speed or pass on a different road, and a linear interpolation is employed over each interval. Hence, these existing indexes are inappropriate to express real trajectories. Additionally, network distance based query is not supported.

3 Uncertain trajectory model Definition 1 (Road Network): A road network is represented by a graph G=(V, E), where V and E are the sets of vertices and edges, respectively. Each edge e∈E is associated with a pair (l(e), s(e)), corresponding to the length and maximum allowed speed of e. Definition 2 (Trajectory): A trajectory of a MO O in the road network G is a sequence of positions with timestamps: To ={(p1,t1),(p2,t2),…,(pi,ti),…,(pn,tn)}, where pi is the sampling position on e∈E at time ti (i=1,2,…,n). The uncertain trajectory model in our work is similar to the model in [29], except that we do not need the time-dependent probability distribution function, because we aim at finding all the trajectories that possibly pass through the spatial region during the specified time interval. The uncertainty of trajectories in the road network space is quite different from that in the Euclidean space, since the movements of MOs are constrained by specific roads. Figure 1 shows an example of range query for uncertain trajectories in road network. Each edge (ei,l(ei),s(ei)) in the network has two attributes: the length l(ei) and the maximum allowed speed s(ei). In the figure, Ta and Tb are two uncertain trajectories. The black round dots represent sampling points of Ta. T b1 dist(q,t=90)=920

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The white round dots represent sampling points of Tb. The corresponding thick lines stand for the real routes of the MOs (i.e. Ta and Tb). Here, T a ¼ fT a1 ðp1 ; 10Þ; T a2 ðp2 ; 40Þ; T a3 ðp3 ; 70Þ; T a4 ðp4 ; 100Þ; T a5 ðp5 ; 130Þ; T a6 ðp6 ; 160Þg; T b ¼ fT b1 ðp1 ; 0Þ; T b2 ðp2 ; 20Þ; T b3 ðp3 ; 40Þ; T b4 ðp4 ; 60Þ; T b5 ðp5 ; 80Þ; T b6 ðp6 ; 100Þg: A range query is issued from query point q for the trajectories within network distance dr=1000 during time interval (90, 100). If the MOs follow the shortest paths and move with uniform velocity, both Ta and Tb are not in the result. The reason is that Ta will choose path {(e1,300,40),(e2,650,50)} between T a3 and T a4 , and Tb will leave (e8,400,45) at time 88.9. However, as shown in Fig. 1, the routes of Ta and Tb actually meet the query condition. In the situation described above, the query results will miss some important MOs. Therefore, we incorporate uncertainty into trajectories to ensure that all qualified MOs are outputted. We represent an uncertain trajectory in road networks as a sequence of road segments with time intervals during which the MO may be on the corresponding road segments. Specifically, we define two types of uncertainty. 1. Path Uncertainty: There might be multiple paths between two sampling points of a MO. Along with some of these paths, the MO can arrive the second sampling location before the second sampling time at maximum speed of each road segment. Given two trajectory sampling points (pi,ti) and (pi+1,ti+1) of a MO O in the road networks, the set of possible paths PPi between (pi,ti) and (pi+1,ti+1) consists of paths whose minimum time costs are less than ti+1 −ti. Then: n o

 PPi ðOÞ ¼ P j ∈paths pi ; piþ1 t P j ≤tiþ1 −ti ; where t(Pj) stands for the minimum time cost for path Pj, 1≤j≤l, l is the number of all possible paths. 2. Time Uncertainty: After all the elements of PPi have been obtained, the exact location of the MO at a specific time instant is still not known. Since the trajectory will be decomposed into segment units for simple representation and partition, we alter the location uncertainty [29] to time uncertainty for each edge, i.e. projecting the uncertainty to the time dimension instead of spatial dimension. The first advantage of this change is that the expression of the uncertain model is simpler. The second is that it is easier to apply to the network partition. This is because the network partition is also relevant to spatial uncertainty, so considering both network partition and spatial uncertainty will be more complexity. The third is that the time dimension is more selective than the spatial dimension in most cases of range query. We use te(vki,j) and tl(vki,j) to denote the earliest k (k=0,1,…,n) along Pj. arrival time and latest departure time for a given vertex vi,j 0 n Particularly, te(vi,j)=ti, and tl(vi,j)=ti+1. The earliest arrival time and latest departure time of other vertices can be inferred recursively as follows:       k te vki; j ¼ te vi;k−1 þ t vi;k−1 j j ; vi; j ;       −t vki; j ; vkþ1 ; tl vki; j ¼ tl vkþ1 i; j i; j

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k k−1 k where t(vk−1 i,j ,vi,j) represents the minimum time costs from vertex vi,j to vertex vi,j. For an edge k k+1 k with a start vertex vi,j and an end vertex vi,j , its earliest arrival time is te(vi,j) and its latest departure time is tl(vk+1 i,j ).

Definition 3 (uncertain trajectory) As the uncertain trajectory model contains path uncertainty and time uncertainty, we use a sequence of all possible segments with earliest arrival time and latest departure time to represent an uncertain trajectory: 

 T u ðOÞ ¼ edgeid; ps1 ; pe1 ; ts1 ; t e1 ; d 1 ; …; edgeid; psi ; pei ; tsi ; tei ; d i
o …; edgeid; psn ; pen ; t sn ; ten ; d n ; where edgeid denotes an edge in the graph (represented by a segment identifier); psi denotes the start position of the edge where the MO arrives; pei denotes the end position of the edge where the MO leaves or stops; tsi stands for the earliest arrival time at psi ; tei stands for the latest departure time at pei ; and di stands for the direction of MOs moving on the edge (i=1,2,…,n). Therefore, in our
uncertain trajectory model, the MOs can change their speed in the middle of an  edge. We define edgeid; psi ; pei ; tsi ; t ei as a segment unit of the uncertain trajectory.

4 Indexing structure In this section, we propose an index structure, partition-based uncertain trajectory index (PUTI), to support efficient range query for uncertain trajectories in road networks. As shown in Fig. 2, PUTI consists of a top level 2 D R-tree, whose leaf entries contain pointers to 1 D Rtree. In PUTI, the 2 D R-tree is used to index the partitions of the networks, and each partition has a 1 D R-tree to index the time intervals during which the MOs might be in the partition. Typically, R-tree [8] cannot store the remaining detail information of trajectories in the partitions. Therefore, we combine 1 D R-tree and B+tree through a unique key to represent the spatial and temporal information in each partition. Section 4.1 explains how we partition the network into appropriate regions, and the detail index components are presented in Section 4.2. Section 4.3 describes how to insert trajectories into PUTI and how to delete trajectories from PUTI. 4.1 Network partition Graph partition is an important problem that has been extensively studied in the last decades. In our work, we take two kinds of information into consideration: the network distance and the connections among adjacent partitions. In addition, since the spatial region is computed according to network distance, the graph partition must maintain the network topology. METIS [17] is a public implementation which aims to efficiently find partitions with high quality based on specific heuristics, which meets our demands for graph partition with vertex weights. It partitions the network into several regions, such that the sum of vertices weights in each partition is approximately equal to each other and the connections among regions are least. We implement the network partition method based on the multilevel k-way algorithm [17] in METIS. Figure 3a illustrates the network partition method. Given an undirected graph

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Fig. 2 An overview of PUTI structure example

G=(V, E) representing the road network, firstly, we weight an edge ei by its minimum time cost l(ei)/s(ei)), and construct G’s line graph L(G) (i.e. each vertex of L(G) represents an edge of G, and two vertices of L(G) are adjacent if their corresponding edges share a common endpoint in G). Let L(G), the weights of vertices and the number of partitions p be the input of the multilevel k-way algorithm in METIS, we can get p contiguous and balanced partitions as described in [17]. After converted back, we finally obtain p balanced partitions of G. This is because that the partitions in the line graph are vertex balanced [17], and the edge lengths in the original graph are equal to the weights of vertices in the line graph. Figure 3b shows a real example of road network partition of San Joaquin city with 1,000 regions. The different partitions are represented by regions with different colors. 4.2 PUTI structure To index the uncertain trajectories, we borrow the ideas from [16], which are proposed to solve the problem of the range query for trajectories in road networks based on NVD. The concepts of valid partition and partially valid partition are the same as valid region and partially valid region in [16]. However, the proposed index structure PUTI is different with that in [16] from the following aspects. 1. In [16], trajectories are composed of vertices with timestamps. In PUTI, to capture the uncertainty, trajectories are composed of sequences of segment units. 2. Since spatial dimension is less selective than time dimension in most cases (i.e. time filter might make the result much smaller than spatial filter for the candidate partitions), we intend to index the time intervals of trajectories in each partition, instead of finding the available edges firstly.

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(b) An example of network partitions with 1000 regions Fig. 3 Network partitions. a The main process of network partition. b An example of network partitions with 1,000 regions

3. METIS partition costs less on both offline computation of precomputed dist component and online computation of valid partitions and partially valid partitions. This is because METIS guarantees that the partitions are balanced. PUTI is based on partitions for uncertain trajectories in road networks, and it consists of four major components, as illustrated in Fig. 4. 1. Edge Hash Table: The edge hash table describes the corresponding relationships between edges and partitions. Since all the positions are given in the form of edge id and offset, we simply adopt hash table to quickly compute which partition the query position belongs to. The edge hash table can fit into main memory to reduce the disk access cost. 2. Partition Component: The partition component records all the border points in each partition. It is similar to the NVPs Component [16]. It can also fit into main memory to reduce the disk access cost. 3. Precomputed Dist Component: The precomputed dist component records the shortest path distances between any two border points in each partition. This component can be stored in main memory or disk or both according to the size of the network. 4. Time Interval Component: To quickly determine if a candidate partition contains objects of interest, we maintain a 1 D R-tree and a B+tree to organize the time periods and corresponding segment units in each partition. 1 D R-tree cannot store the segment units in the tree, because the leaf nodes have the same structure with intermediate nodes in a Rtree. The trajectory ids, earliest arrival time and latest departure time of MOs in the

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Fig. 4 The data structure of PUTI

partitions are stored in the leaf nodes of the R-tree. The string that consists of earliest arrival time, latest departure time, and the trajectory id is regarded as the key of the segment units within earliest arrival time and latest departure time. We exploit Berkeley DB1 to store the key and the data to reduce disk I/O, because the B+tree in Berkeley DB can guarantee that the data within short time interval is stored in the same data page.

4.3 Insert and delete trajectories The trajectories of MOs can be inserted into the time interval component as follows. First, we calculate the possible paths PPi ðOÞ of the MOs, in the form of sets of segment units as described in Section 3. Then, we co-locate segment units according to the network partitions. If there units in the same partition 
is no path uncertainty, the  segment
 are put together as S p ¼ edgeid; ps1 ; pe1 ; ts1 ; te1 ; d 1 ; … edgeid; psm ; pem ; tsm ; tem ; d m , and the values of psi and pei are 0 or 1, except for the MOs starting from or ending at the middle of edges. The time interval ðt s1 ; tem Þ and relevant segment unit set are inserted into 1 D R-tree and B+tree respectively. If path uncertainty exists and all possible paths between two trajectory sampling points are in the same partition, we deal with all the segment units as above. If the path uncertainty exists and some possible paths between two trajectory sampling points are not in the same partition, the segment units are co-located according to the partitions. For segment units in the same partition, the earliest arrival time and latest departure time of all the segment units in the same partition are inserted into R-tree as the time interval, and these segment units are inserted into B+-tree. Algorithm 1 shows the pseudo-code for inserting a trajectory into the PUTI. In the algorithm, trajEdgeSet is a data structure that contains some segment units. The partition currentPartition that the first sample belongs to is obtained by edge hash table in partitionHash(trajNode.edgeid) function. When inserting a trajectory into PUTI, there are three cases. Case 1 (lines 7–9): the adjacent trajectory sampling points (we call them trajectory 1

http://www.oracle.com/technetwork/products/berkeleydb/overview/index.html

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nodes in the pseudo-code) are on the same edge. If the trajectory node is the last node of the trajectory, the segment units trajEdgeSet are inserted into the time interval component. Case 2 (lines 10–15): the adjacent trajectory nodes are on the adjacent edges. The earliest arrival time and latest departure time of the first edge are updated. If the second edge is in a new partition, the segment units in the old partition are inserted into the time interval component. Case 3 (lines 17–18): there are more than two edges between the adjacent trajectory nodes. The pathUncertain(startNode, endNode, limitedTime) function is to find all the possible paths that the MO can arrive the vertex endNode from the vertex startNode within the time limitedTime. To implement this, we use a priority queue H to maintain the explored paths along with the minimum time costs for traversing them. In each iteration, it retrieves a path P from the top of H and expands P by using its adjacent edges that have not been visited yet. In detail, when popping a path from H, if the MO can pass through the explored edge within limited time at the maximum speed, we insert the edge to the path and push the path into H. The process is repeated until H is empty. In the partitionSegUnit(paths) function, firstly, the earliest arrival time and latest departure time of all segment units are computed. Secondly, these segment units are co-located according to the partitions and segment units in each partition are inserted into the time interval component as described above.

Algorithm 1: Insert a trajectory Input: traj: an original trajectory 1. trajEdgeSet 2. trajEdgeSet.trajId traj.trajId traj.node[0] 3. trajNode partitionHash(trajNode.edgeid) 4. currentPartition 5. preEdge trajNode.edgeid 6. for each trajNode in traj from node1 do 7. if the adjacent nodes are on the same edge and trajNode is the last node then 8. insert trajEdgeSet into time interval component 9. currentPartition partitionHash(trajNode.edgeid) 10. else if the adjacent nodes are on the adjacent edges then 11. compute segment unit info and time interval for the preEdge 12. if currentPartition != partitionHash(trajNode.edgeid) or trajNode is the last node then 13. insert trajEdgeSet into Time Interval Component 14. currentPartition partitionHash(trajNode.edgeid) 15. end if 16. else pathUncertain(startNode,endNode,limitedTime) 17. paths 18. partitionSegUnit(paths) trajNode.edgeid 19. preEdge 20. end for

Since deletion operation is applied much less frequently for historical trajectories, we mention it briefly. First, we find the edges intersected with the trajectory for deletion. Then,

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after obtaining the related partitions through edge hash table, we get the leaf nodes with trajectory id through time interval R-trees. Finally, for every deleted leaf node of R-trees, we associate the time intervals with the trajectory ids as keys and delete the corresponding values in B+-trees.

5 Range query processing Definition 4 (range query for uncertain trajectories): Given a query point q (an edge id and offset), a network distance restriction dr, and a time restriction (ts,te), the range query is to find all the MOs that possibly pass by the region r(q,dr) during the time interval (ts,te), where r(q,dr) stands for a set of positions whose network distances from q are less than dr. Our algorithm consists of three steps: spatial pruning, time filter, and result refinement. In the spatial pruning step, our approach locates the partition that contains query point q, and obtains valid and partially valid partitions. In the time filter step, for valid partitions, the partial final result is outputted, and for partially valid partitions, candidates are selected. In the result refinement step, final result set is computed from the candidates of partially valid partitions through retrieving the segment units in B + -tree. Algorithm 2 shows the pseudo-code of the range query on PUTI. Lists of trajectory result, valid and partially valid partitions are initially set to empty. The detailed process of the range query algorithm is depicted as follows. 1. Spatial Pruning (lines 3–5): Since the range dr is explicitly specified in the syntax of the query, we firstly calculate the partitions intersect with the region r(q, dr). We use edge hash table to find the first partition cur_partition that q belongs to in the containPartition(q) function, and then use Euclidean distance to estimate the partitions candidate_partitions intersect with the region r(q, dr) by invoking 2 D R-tree index on partitions in the spatialFilter(q, dr) function. In the partitionExpansion(next_partition, dr) function, with cur_partition and the precomputed dist component, we get the valid partitions and partially valid partitions partition_list from candidate_partitions through the partition extension part of the algorithm [16]. We sort the partitions in ascending order according to the minimum distances from partitions to query point q. As a result, we can get the trajectories approximately in the order of their network distance from q. 2. Time Filter (lines 6–9): Only those MOs that travel inside the selected partitions during the query time interval may belong to the final result. In the timeFilter(partitionId, ts, te) function, we get a part of the final result by time interval range query on 1 D R-tree for the valid partitions, and a set of candidates which is significantly smaller than the data in the partitions for the partially valid partitions. So far, the trajectory id result from valid partitions can be returned, because the partial result is the neatest trajectories. 3. Result Refinement (lines 10–23): For each partially valid partition, we consider it as a local network including the query point. In the localExpansion(q, localNetwork) function, incremental network expansion is employed to get exact edges valid_edge_list within the spatial restriction. In the

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SegUnitInPartition(trajId, startTime, endTime) function, for each trajectory candidate, we get the segment units segment_unit_list from the B + −tree through the key, which consists of the earliest arrival time, the latest departure time, and the trajectory id. If the trajectory is not in the result, we test whether segment_unit_list intersects with valid_edge_list. If it is true, the trajectory that at least one of whose intersected segment units satisfying the condition tsi ≤te and tei ≥ts is added to final result. Otherwise, we directly step to the next candidate. If the application needs to find the exact MOs in r(q, dr) during the time interval, we further test if the time period of MOs on the part edge with the max allowed speed, which is in r(q, dr), is intersected with the query time interval when the edge is partially in r(q, dr). Specially, when partition_list is empty (i.e. the query spatial region is too small to cover any partition), we treat the only partition which contains the query point as a partially valid partition. A simple example of result refinement is given in Fig. 5. A range query is issued to find the trajectories within network distance 20 from query point Q and time period (40, 50). As shown in Fig. 5, P2 is one of the partially valid partitions, whose border points are A, B, C and D. The red dotted lines stand for the distances from query point to the border points. Firstly, an incremental network expansion is employed to calculate the segment units within network distance 20. The results include AD, DC, AE, EF and FC (the yellow lines) in Fig. 5. Secondly, the segment units which satisfy the time period condition are obtained from the B + tree, according to the key got from the time filter step. The results include AD and BC in Fig. 5. Thirdly, the segment unit AD is the intersection of the above two segment unit sets. For an approximate range query, T1 is in the final results. However, if the target is to find the exact MOs in r(q, dr) during the time interval, we may miss some possible MOs. In Fig. 5, we find that BC is partially in r(Q, 20). Supposed that the segment unit of T4 in P2 is (BC, 45, 60, 0), T4 might be in the final query results. The max allowed speed of BC is 1.5, so T4 would cost 2 time units to move 3 at max allowed speed. As a result, the earliest arrival time and the latest departure time of HC is 47 and 60.

B

Time Time Interval Interval Component Component

P2

H, CH=5 5

A

3

root

E

3

F

4

C 50

8

ID R-tree

8

G

4

T2

T1

T3

10

B+ tree

P2

15

D

Time incresing

10 30

30_50_T2 : (BC, 0, 1, 30, 50, 0) 40_60_T1 : (AD, 0, 1, 40, 60, 1) 100_124_T3 (AE, 0, 1, 100, 110, 0)(EF, 0, 1, 108, 118, 0)(FC, 0, 1, 115, 124, 0)

Q

(a) partition map Fig. 5 An example of the result refinement step in range query

(b) Time Interval Component Index

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Because (47, 60) is intersected with (40, 50), T4 is definitely in the final query results.

6 Experimental evaluation In this section, we evaluate the effectiveness of the uncertain trajectory model and the efficiency of PUTI. All experiments are implemented in C++ language and run on an Intel Core i3 CPU 2.93 GHz PC with 2 GB RAM. The network partitions are generated based on METIS [17]. The storage files are organized by using Oracle Berkeley DB. The page size of Rtree, which is used for indexing time interval and Euclidean space of partitions, is set to 4 KB. There are two choices, i.e. UTR-tree [5] and MON-tree [3, 4], that can be considered as potential competitors with our method. UTR-tree focuses on the same goal with our method, but it has a strict requirement on the trajectories update, and thus it is not appropriate to

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compare it with our method on the same dataset. In order to compare MON-tree with our method, we modify the MON-tree into network distance based MON-tree. In detail, we use INE algorithm [18] and 2 D R-trees to implement network distance based MON-tree. Assuming that the MOs always follow the shortest paths and travel at a constant speed between two consecutive trajectory samples, we can obtain the entire trajectories through Dijkstra algorithm and linear interpolation. First, we use INE algorithm to compute the valid edge set within the network distance dr from query point q. Then, for each edge in the edge set, we search the 2 D R-tree for satisfied trajectories. The MON-tree [3, 4] uses a modified version of R-tree that is capable of handling multiple query windows in one index scan. However, the open source R-tree implementation used in the experiments does not support the parallel subquery. Besides, due to the file open limitation of the OS, we cache the roots of 2 D R-trees. At last, we eliminate the duplicate trajectories. Note that network distance based MON-tree is still called as MON-tree for short in the following paragraphs. The evaluation is divided into three parts. First, we evaluate the performance of the insert operation of PUTI. Second, we conduct elaborated experiments on PUTI to assess the index robustness with variation of the partition number and query size. Third, we compare our method against MON-tree, in terms of accuracy and efficiency. 6.1 Data sets Available real trajectory dataset is not sufficiently representative in terms of trajectory variety and data size. Moreover, the underlying road network required by the experiment is rarely available free of charge. Therefore, we use the generator proposed by Brinkhoff [1] to create synthetic datasets of MOs in the road networks. We use Hancock and San Joaquin as the underlying road networks in our experiments, because there are areas of different road densities in the maps. Hancock has 11,760 vertices and 13,175 edges, while San Joaquin has 33,340 vertices and 39,647 edges. The networks are represented in a segment-oriented model [3, 4]. The road network segments can be categorized into 8 classes, each of which corresponds to a maximum speed. We generate 8 classes of MOs, and each class stands for a kind of vehicle that has a maximum speed. The generator also simulates weather conditions or other similar events that impact the motion and speed of the MOs. The configuration of the generator is default. The Hancock dataset, which has a smaller size, is used to analyze the factors that influence the efficiency of insert operation, and to compare PUTI with MON-tree in terms of recall rate, precise rate, F-measure metric, and efficiency. In order to compute the recall rate and the precise rate of PUTI, we generate the trajectories with high sample rate, i.e. one sample per time unit, as the ground truth. Based on the ground truth data, trajectories with low sample rate, i.e. one sample per m time units, are generated as the inputs for the experiments. San Joaquin dataset, which has a larger amount of data, is used for deeper analysis of the effects of partition number on range query. Table 1 presents the statistics of the datasets. 6.2 Results and discussion 6.2.1 Performance of inserting trajectories Firstly, we evaluate the performance of the insert operation in PUTI with different sampling intervals. The result of this experiment is shown in Fig. 6. We set the number of partitions as 300, since range query can get almost the best performance with 300 partitions. As mentioned

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Table 1 Statistics of two datasets Dataset names

# of segment units

# of MOs at beginning

# of time units

# of MOs created per time unit

Hancock dataset

5,114,865

San Joaquin dataset

27,656,847

1,000

2,000

10

1,000

10,000

30

in Section 4.2, the cost of inserting data is mainly composed of two parts: finding all the possible paths and inserting the segment units into index. It can be seen from Fig. 6a that as the number of trajectories grows up, the cost for inserting trajectories rises too. Specifically, the speed of growth is relatively slow when m is smaller, because the former part costs much less 1600 Total time of insert operation(s)

1400 1200

m=10

m=20

m=30

m=40

m=50

m=60

1000 800 600 400 200 0 1

5 10 15 Number of trajectories(x1000)

20

(a) The Effect of #of trajectories and sampling interval on total time of insert operation Time of finding possible pathes(s)

800 700 600

m=10

m=20

m=30

m=40

m=50

m=60

500 400 300 200 100 0 1

5 10 15 Number of trajectories(x1000)

20

(b) The Effect of #of trajectories and sampling interval on time of finding possible paths 1400 # of disk accesses

1200 1000 800 600 400 200 0

10

20

30

40

50

60

Sampling interval m(s)

(c) The Effect of sampling interval ondisk accesses Fig. 6 Performance of inserting trajectories with different sampling intervals. a The Effect of #of trajectories and sampling interval on total time of insert operation. b The Effect of #of trajectories and sampling interval on time of finding possible paths. c The Effect of sampling interval on disk accesses

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time than the latter part. However, when m becomes larger, the speed of growth is obviously accelerated. The reason is that the cost of finding all the possible paths becomes more expensive when m grows up, as shown in Fig. 6b. Namely, there are much more paths to be found out and inserted into the index when the sampling interval is higher. Due to the same reason, when the sampling interval increases, the number of disk accesses of the insert operation goes up too, as shown in Fig. 6c. The effects of the number of partitions on the performance of the insert operation are shown in Fig. 7. We set the sampling interval m=30, since the insert operation obtains average performance when m=30. It can be seen that the time cost of insert operation first increases and then decreases. The reason is as follows. The process of inserting trajectories composes of two subprocesses, i.e. finding possible paths and inserting data into index. The cost of finding possible paths is almost steady since the sampling interval does not change as the number of partitions rises. As for inserting data into index, there are two influencing factors: the number of R-trees or B+-trees where data is inserted and the size of each index. As we can see from Fig. 7a, the former factor costs more than the later factor when the number of partitions changes from 100 to 700, but the case becomes opposite when the number of partitions is larger than 900. Since the more the partitions are, the larger the number of time interval components is, we find that as the number of partitions rises, there are more disk accesses to be executed as shown in Fig. 7b. 6.2.2 Effect of the number of partitions In this section, we analyze the efficiency of the proposed range query algorithm under different query sizes and numbers of partitions. The number of partitions is an important influencing

Total time of inserting operator(s)

2000

p=100 p=500 p=900

1500

p=300 p=700 p=1100

1000 500

0 1

5

10

15

20

Number of trajectories(x1000)

(a) The Effect of # of trajectories and # of partitions on total time of insert operation # of disk accesses

2500 2000 1500 1000 500 0

100

300

500 700 # of partitions

900

1100

(b) The Effect of # of partitions on disk accesses Fig. 7 Performance of inserting trajectories with various numbers of partitions. a The Effect of # of trajectories and # of partitions on total time of insert operation. b The Effect of # of partitions on disk accesses

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factor of PUTI. We set the sampling interval m=30, since range query obtains average performance when m=30. Figure 8 presents the results of San Joaquin dataset with different range query sizes. The query size varies from 1ε to 6ε, where ε=(0.0005×D)×(0.0005×T) (D is the sum of distances of all the road segments in the road network, and T is the total time interval of all the trajectories). As shown in Fig. 8a, with 100–300 partitions, our method can achieve the best performance for the dataset. Because when the number of partitions is too large, the number of hits of candidates in the time filter step becomes low; when the number of partitions is too small, the execution of the time filter step becomes frequent and the computation of the satisfied partitions costs more. Obviously, when the number of partitions grows large to a certain extent, PUTI would retrograde to MON-tree in terms of space expansion. When the query size increases from 1ε to 6ε, the cost of computing satisfied partitions and the number of partitions to deal with are increased. We also find that the smaller the number of partitions is, the slighter the effects of query size on the cost of range query is. Additionally, as shown in Fig. 8b, with the increase of the number of partitions, the number of disk accesses of range query continues to decrease and then becomes stable. This might be because during the time filter step, large amount of partially valid partitions causes large number of fault hits, leading to more disk I/O reads in the refinement step. At the same time, the number of partitions is small when the query size increases from 1ε to 6ε, the number of the disk accesses to refine partially valid partitions also increases.

CPU time per query(s)

3.5 3

1ε

2ε

3ε

4ε

5ε

6ε

2.5 2 1.5 1 0.5 0 50

100

300 500 # of partitions

1000

1500

(a) The Effect of the number of partitions and query size on query time # of disk accesses per query

1200 1000

1ε

2ε

3ε

4ε

5ε

6ε

800 600 400 200 0 50

100

300 500 # of partitions

1000

1500

(b) The Effect of the number of partitions and query size on query disk accesses

Fig. 8 Performance of range query with different query sizes. a The Effect of the number of partitions and query size on query time. b The Effect of the number of partitions and query size on query disk accesses

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6.2.3 PUTI versus MON-tree on recall rate, precision rate and F-measure We compare the recall rate and the precision rate of PUTI and MON-tree with different sampling intervals. We set the number of partitions as 300, since range query obtains almost the best performance when the number of partitions is 300. Figure 9a shows that when the sampling interval becomes higher, the recall rate of MON-tree drops much more quickly than that of PUTI. This is because PUTI finds all possible MOs within the query size, and MONtree misses a lot of MOs which violate the rules of movement as described in Section 6.1. 1.2

PUTI

MON-tree

Recall rate

1 0.8 0.6 0.4 0.2 0 10

30

40 60 Sampling interval m(s)

80

90

80

90

(a) Sampling interval vs. recall rate 1.2

PUTI

MON-tree

Precision rate

1 0.8 0.6 0.4 0.2 0 10

30

40

60

(b) Sampling interval m(s) 1.2

MON-tree

PUTI

F-measure

1 0.8 0.6 0.4 0.2 0

10

30

40 60 Sampling interval m(s)

80

90

(c) Sampling interval vs. F Figure 9 PUTI versus MON-tree on Recall Rate, Precision Rate and F-measure. a Sampling interval vs. recall rate. b Sampling interval vs. precision rate. c Sampling interval vs. F

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It can be seen in Fig. 9b that the precision rate of PUTI is a little lower than that of the MON-tree, but the difference is subtle. For example, when the sampling interval is 30 s, the recall rate of PUTI is 16.68 % higher, and the precision is only 0.3 % lower. This might be because that the path uncertainty contains more possible paths and the time uncertainty elongates the time interval for each segment unit, leading to adding more trajectories that are not in the ground truth to the final result, as the sampling interval increases. When the sampling interval increases to 80 s, the precision rate of PUTI exceeds that of MON-tree. Figure 9c shows the comparative results of the standard F-measure metric. It demonstrates that our method outperforms MON-tree when the sampling interval varies. For example, when the sampling interval is 40 s, the F-measure of PUTI is 0.993, 13.76 % higher than that of MON-tree. As the time interval increases, the gap of F-measure between them becomes larger. 6.2.4 PUTI versus MON-tree on query efficiency We also compare the efficiency of PUTI and MON-tree for queries with different sizes. The range query includes two dimensions, i.e. spatial and time dimensions, and the query size varies from 1ε to 6ε, where ε=(0.001×D)×(0.001×T). The number of partitions is 300. As Fig. 10 shows, PUTI with different sampling intervals consistently outperforms MON-tree on CPU time and disk accesses, and the gap increases as query size varies from 1ε to 6ε. It is because that PUTI only gets the result from R-tree in a valid partition, while MON-tree

CPU time per query(s)

0.8 0.7

m=10

m=30

m=80

MON-tree

m=60

0.6 0.5 0.4 0.3 0.2 0.1 0

1ε

2ε

3ε

4ε

5ε

6ε

Query size

(a) Query size vs. CPU time # of disk accesses per query

800 700 600

m=10

m=30

500

m=80

MON-tree

m=60

400 300 200 100 0

1ε

2ε

3ε 4ε Query size

5ε

6ε

(b) Query size vs.# of disk accesses Fig. 10 PUTI versus MON-tree on Query Efficiency. a Query size vs. CPU time. b Query size vs. # of disk accesses

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computes the satisfied MOs on each edge respectively. As the query size increases, the number of valid partitions grows up either. As a result, with the increase of the network distance, the cost of PUTI rises slowly while that of MON-tree rises significantly. This is because that the trajectories of MOs fully cover most of the edges that MOs pass by, and PUTI regards the time dimension as a more selective dimension. We can also find from Fig. 10 that the CPU time and the number of disk accesses grow slowly when the sampling interval increases. That might be because that longer sampling interval causes more segment units with longer time interval to be indexed and queried.

7 Conclusions and future work In this paper, we propose a partition-based approach to solve the range query problem on uncertain trajectories in road networks. First, we define the problem of range query for uncertain trajectories in road networks. Second, we construct the uncertain trajectory model including path uncertainty and time uncertainty, by considering the maximum speed on each road segment. Third, we build a partition based index framework, i.e. PUTI, for range query on uncertain trajectories. Finally, we evaluate our uncertain trajectory model and PUTI with different datasets of trajectories in real road network maps. PUTI has two important advantages over the existing approaches. The first advantage lies on the side of the performance. The experimental evaluation shows that our approach significantly outperforms the MON-tree, and PUTI shows robust performance with different query sizes, sampling intervals, and partition numbers. The second advantage is that PUTI supports a concise and flexible uncertain trajectory model which reduces the cost and guarantees the accuracy. Therefore, our approach can be applied to the investigation of traffic accidents and provide supports to data mining on trajectories. Since our uncertain trajectory model is based on the max speed of the road segments, the MOs that have lower speed would cause a larger error. On the other hand, the long sampling intervals would make the accuracy and efficiency low. The range query is the only query type proposed in this paper. There might be other query type based on PUTI to meet the demands of the applications. Therefore, our work can be extended in three directions in the future. The first direction is to construct the uncertain trajectory model with real-time traffic condition instead of max speed of the road segment, in order to improve the precision of the query result. The second direction is to support other query types of historical trajectory data based on our model, e.g. continuous range query and k nearest neighbor query. The third direction is to handle queries about the current and near-future position of MOs in the networks. Acknowledgments This work was funded by the Ministry of Industry and Information Technology of China (No.2010ZX01042-002-003-001), Natural Science Foundation of China (Nos. 60703040, 61332017), and Science and Technology Department of Zhejiang Province (Nos. 2007C13019, 2011C13042, 2013C01046).

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Ling Chen received his Ph.D degree in computer science from Zhejiang University, China in 2004. He joined Zhejiang University in 2004. From 2006 to 2007, he was a research fellow in The University of Nottingham, UK. Currently, he is an associate professor in the college of computer science. His research interests include databases,

distributed systems, HCI, AI, and pattern recognition.

Yanlin Tang was a master student in the College of Computer Science and Technology, Zhejiang University, China when this research was performed. Her research interests include uncertainty and query in temporal-spatial databases.

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Mingqi Lv obtained his Ph.D. Degree in computer science from Zhejiang University, China in 2012. Currently, he is a research fellow at Nanyang Technological University, Singapore. His research interests include ubiquitous computing, data mining, and HCI.

Gencai Chen is a professor in the College of Computer Science, Zhejiang University, China. From 1987 to 1988, he studied in the Department of Computer Science, State University of New York at Buffalo, USA. He won the special allowance, conferred by the State Council of China in 1997. He is currently vice director of the Software Research Institute, Zhejiang University. His research interests mainly include database, data mining, CSCW, and pattern recognition.