The V*-Diagram: A Query-Dependent Approach to Moving KNN Queries

Transcription

1 The V*-Diagram: A Query-Dependent Approach to Moving KNN Queries Sarana Nutanong Rui Zhang Egemen Tanin Lars Kuli {sarana,rui,egemen,lars}@csse.unimelb.edu.au Department of Computer Science and Software Engineering University of Melbourne Victoria, Australia NICTA Victoria Laboratory Australia ABSTRACT The moving nearest neighbor (MNN) uery finds the nearest neighbors of a moving uery point continuously. The high potential of reducing the uery processing cost as well as the large spectrum of associated applications have attracted considerable attention to this uery type from the database community. This paper presents an incremental safe-region-based techniue for answering MNN ueries, called the V*-Diagram. In general, a safe region is a set of points where the uery point can move without changing the uery answer. Traditional safe-region approaches compute a safe region based on the data objects but independent of the uery location. Our approach eploits the current nowledge of the uery point and the search space in addition to the data objects. As a result, the V*-Diagram has much smaller IO and computation costs than eisting methods. The eperimental results show that the V*- Diagram outperforms the best eisting techniue by two orders of magnitude.. INTRODUCTION Current location-based services provide accurate position information with a high degree of temporal precision. Consider the following two scenarios. A driver in a GPS-euipped car issues a continuous uery to find the nearest gas station while driving in a city. A tourist uses a location-aware mobile device to issue a continuous uery for the nearest restaurant while waling to a museum. The ueries are sent to a server that processes the ueries and returns the answers. In these scenarios, the server has to continuously maintain the answer set which may change depending on the location of the uery point. These ueries are location-based continuous spatial ueries [23] and the scenarios above are typical eamples of moving nearest neighbor ueries (MNN). A straightforward way to process a MNN uery is using a sampling-based method, which processes the MNN uery as a NN uery at sampled locations. This method does not provide answers between sampled locations. In order to provide an (almost) continuous answer to the uery, a high sampling rate is reuired, which maes the method inefficient due to the freuent processing of NN ueries. The concept of the safe region provides a more effective way to achieve continuous Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, to post on servers or to redistribute to lists, reuires a fee and/or special permission from the publisher, ACM. VLDB 08, August 24-30, 2008, Aucland, New Zealand Copyright 2008 VLDB Endowment, ACM /00/00. answers to location-based spatial ueries. In a safe-region-based method, an answer is returned with a safe region. As long as the uery point stays in the safe region, the answer remains the same. When the uery point moves out of the safe region, another answer with its associated region is returned. Therefore, a safe-regionbased method always (that is, continuously) provides accurate answers without the need for sampling. This approach also reuires much less freuent communication between the mobile client and the server. A classic eample of safe-region-based techniues is the Voronoi Diagram [4]. The Voronoi Diagram is a well nown space decomposition determined by distances to a given discrete set of objects, typically, a set of points. Specifically, the Voronoi Diagram of a set of points P = {p, p 2,..., p n} is defined as a set of cells where each cell V (p i) is a region of space that consists of all points of the data space that are closer to p i than any other point in P. An eample is given in Figure. Figure (a) is a set of points in a 2-dimensional (2D) space and Figure (b) is the Voronoi Diagram. (a) Point set S, {a, b, c, d, e, f } (b) Voronoi Diagram of S Figure : The Voronoi Diagram Processing a NN uery using the Voronoi Diagram involves: (i) locating which Voronoi cell the uery point falls into; and (ii) identifying the associated object. In the above eample, falls in V (a) (the grey region) and therefore a is the nearest neighbor of. The answer remains valid as long as the uery point stays in V (a). When the uery point moves across the boundary of V (a) to V (c), c becomes the NN. The Voronoi Diagram can be generalized to the th -order Voronoi Diagram (VD). In a VD, each region is associated with the set of nearest neighbors, termed NN set or NNs, rather than only the nearest one. The VD can handle MNN ueries in the same manner as the basic first order VD handles NN ueries. Another useful generalization of the Voronoi Diagram is order sensitivity. The ordered VD partitions the space into cells where each cell is associated to a particular ordering of the NN set. The ordered VD can be used to answer MNN ueries that reuire the raning of the NNs by their distances to the uery point.

2 The VD has the following shortcomings:. Epensive precomputation. The VD reuires precomputing all the VD cells and access to all the data points. Both computation and storage costs are high. 2. No support for dynamically changing values. The VD can only accommodate NN ueries with a specific value; the ordered VD can only accommodate NN ueries with values no larger than the order of the diagram. As a result, the techniue is not suitable for situations where the value of is unnown in advance or can change dynamically. 3. Inefficient update operations. Many cells have to be recomputed for each insertion or deletion on the dataset. The epensive precomputation is especially not justified if the uery point is confined to a small region of the whole data space. For eample, if a car is moving in a small part of a city, then it is unnecessary to compute the VD for the entire city. Furthermore, one may reuire different VDs for different needs. For eample, a driver may need to find a gas station with a restroom facility while another driver needs one with a special type of fuel. Precomputing VDs for all possible scenarios may be prohibitive. In [23], Zhang et al. proposed an algorithm to locally compute a VD cell, which mitigates the precomputation and update problems (shortcomings and 3). However, the algorithm is still relatively epensive and does not address the problem of dynamically changing values (shortcoming 2). In this paper, a techniue called the V*-Diagram for MNN ueries is proposed. The V*-Diagram has the following ey advantages:. It reuires no precomputation. 2. It incrementally computes answers and therefore efficiently adapts to changes such as insertions and deletions of objects, as well as, dynamically changing values of. The V*-Diagram is based on the safe-region concept, but differs from any previous techniue for MNN ueries in the following aspect: previous safe-region-based techniues compute safe regions purely based on the data (for eample, you can compute the VD without referring to the uery point); the V*-Diagram computes safe regions based on not only the data objects, but also the uery point and the current nowledge of the search space. This is one of the main novelties of the techniue. By doing so, both computation and data retrieval of the V*-Diagram are more economical than those of the other techniues. The contributions of this paper are summarized as follows: We propose the V*-Diagram techniue and the associated algorithm, called V*-NN, to support efficient processing of MNN ueries. We show how the V*-Diagram techniue can be applied to the domain of spatial networs. We perform an etensive eperimental study with the results showing that the V*-Diagram outperforms the best eisting techniue [23] by two orders of magnitude. The rest of the paper is organized as follows: Section 2 describes the problem setup. Related wor is discussed in Section 3. We formulate the V*-Diagram in Section 4 and present the algorithm for MNN ueries based on the V*-Diagram in Section 5. In Section 6, we show how the V*-Diagram techniue can be applied to the domain of spatial networs. A performance analysis of the V*-Diagram is given in Section 7. Section 8 presents eperimental results and Section 9 concludes the paper. 2. PROBLEM SETUP Let D be a set of data objects (represented by points) in a d- dimensional space and dist(. ) be the distance function. The nearest neighbor (NN) uery is defined as follows: given D and a static uery point, find a set H that consists of objects from D such that for any p H and any p 2 D H, dist(p, ) dist(p 2, ). The moving nearest neighbor uery (MNN) is defined as follows: given D and a moving uery point, find the NNs of for every position of. Due to the nature of location-based applications, MNN ueries are discussed in the contet of two settings: (i) Centralized processing paradigm. Both the uery issuer and the processor are on the same machine. Then the main performance measure is the uery processing cost. (ii) Client-server paradigm. The uery is issued by a client to a server through a wireless networ (such as a mobile phone networ). The performance measure involves both communication and uery processing costs on the server side, and the former one is more important in delay-sensitive applications. We assume an unnown trajectory which means that the location of gets updated in a periodic manner. We also assume that no VD is maintained but there is a generic spatial inde such as the R-tree [4] built on the data objects, since it can be used for various uery types and is efficient to maintain. This is also argued as a valid assumption in previous wor [23]. 3. RELATED WORK 3. KNN algorithms Many NN search algorithms have been proposed based on spatial hierarchical structures. One of their common features is the application of the branch-and-bound strategy on the tree structure. The tree can be traversed in a depth-first (DF-NN) [6] or a best-first (BF-NN) [5] manner. BF-NN can retrieve more nearest neighbors incrementally if increases. Figure 2: R-tree and BF-NN Figure 2 shows an eample R-tree and the BF-NN algorithm. Figure 2(a) shows a set of points and how they can be grouped in an R-tree, where objects are indeed in a hierarchy of minimum bounding rectangles (MBRs). The corresponding R-tree is shown in the upper part of Figure 2(b). Starting from the root, BF-NN traverses all the entries in increasing order of their mindist, where the mindist of an entry is the minimum distance between the entry and the uery point. To do so, a priority ueue is maintained to eep all the retrieved data points and active nodes. A node is active if its parent has been accessed but itself has not. The traversal stops if the first elements retrieved from the priority ueue are all data points. An eample run of BF-NN is shown in the lower part of Figure 2(b). At step, all entries in the root are inserted into the priority ueue p. Then the entries are deueued and the corresponding nodes are retrieved in order. The first deueued item is S and its two child entries X and W are put bac into p. Then R is deueued. Nodes U and V are put in p and so on. At step 4, data

3 point d is the head of p. Data point d must be the first NN because all other entries has distances to (which are bounded by mindist of their nodes) larger than dist(, d). d can be retrieved as the first NN. If another NN is needed, the process continues until another data point is the head of p. At step 6, f is discovered as the second NN. By this means, an arbitrary number of NNs can be incrementally obtained. If the value of is fied, an aggressive pruning can be performed on the nodes in p to reduce the ueue size, though the page access cost cannot be further reduced due to the search-space optimality of the algorithm [6]. Other techniues such as idistance [] has superior performance than these hierarchical structure based algorithms in high dimensional space. However, it cannot retrieve NNs in an incremental manner. 3.2 Techniues for processing MNN ueries We have discussed two approaches for MNN ueries in Section : one is sampling based and the other one is safe-region based. SR-NN. Song and Roussopoulos [8] introduced a method which will be referred to as SR-NN in this paper. SR-NN reduces the access costs in the sampling-based approach by retrieving redundant data entries and caching. It greatly reduces the cost of uery reevaluation, but does not solve the problem of inaccurate answers between sampled locations. Thus, SR-NN does not provide truly continuous answers. The th -order Voronoi Diagram (VD). Safe-region based techniues produce continuous answers and reduce processing and communication costs. The Voronoi Diagram [4] is a classic eample and can be used to process MNN ueries as described in Section. For MNN ueries, the VD is used. However, the VD the shortcomings described in Section. TPNN and CNN. Tao and Papadias [9] proposed the timeparameterized NN (TPNN) uery. Assuming a linear trajectory of the uery point, a TPNN uery finds (i) the current NN set, (ii) a position on the trajectory where the NN set changes and (iii) the objects that cause the change. This is done by finding the earliest point on the trajectory that has a different NN set from the current one. This point is also nown as the influence point (or euivalently influence time when the speed is nown), which can be considered as the boundary of a safe region. Tao and Papadias [20] also considered the Continuous NN (CNN) uery, which finds the NN for every single point on a predefined linear trajectory. This is achieved by identifying all influence points on the trajectory. The main difference between CNN and TPNN is that CNN obtains all the influence points on the trajectory but TPNN finds only the first one. Both TPNN and CNN are limited to nown linear trajectories. RIS-NN. Zhang et al. [23] proposed an algorithm called Retrieve-Influence-Set NN (RIS-NN) to locally compute VD cells using a spatial inde. RIS-NN uses the TPNN uery [9] to find each edge of a VD cell, 360 degrees around the uery point. Figure 3 shows how RIS-NN discovers all edges of V (a) from the eample in Figure (a). Initially, a is found to be the NN of ; the cell is initialized to the whole data space, that is, rectangle ABCD. At step, a TPNN uery is eecuted with the trajectory from to the top left corner of the space ( A) and d is returned as the object that changes the NN result along A. The perpendicular bisector of a and d, B ad, contributes one edge to the cell. The cell is updated to the polygon BCDEF. At steps 2 and 3, two TPNN ueries are eecuted with the trajectories from to the new corners ( E and F ). Two new edges are found according to B af and B ae. Since VD cells for data points are conve polygons, this process continues until all corners of the cell have been checed and they all have the same NN as. RIS-NN mitigates the precomputation problem (shortcoming (a) Step (b) Steps 2 and 3 (c) Steps 4 and 5 (d) Final steps Figure 3: Computing a Voronoi cell locally ) of the VD because it only accesses local data, but this algorithm is still epensive because it performs multiple (on average 2 [23]) TPNN ueries, where each TPNN uery involves a costly tree traversal. RIS-NN does not solve the problem of dynamically changing values (shortcoming 2); changing to a larger value incurs recalculation of the VD cell. The computation of the previous VD cell cannot be reused and hence this algorithm is not incremental. Due to the fact that only one VD cell is maintained at a time, RIS-NN handles dataset updates (insertions and deletions of objects) more efficiently than the traditional VD techniue. However, in a case where an update effects the NN answers, the current VD cell has to be recalculated. IRU. Kuli and Tanin [3] introduced an algorithm called incremental ran updates (IRU) to compute regions where the raning of all the objects (based on their distances) is the same. This is euivalent to computing the ordered VD cell with = n, where n is the total number of objects. Rather than computing the whole nvd, IRU incrementally computes a neighboring nvd cell from the current cell. In [3], an nvd cell is termed a fied-ran region (FRR) since for any point in the region, the raning of all the objects based on their distances is fied. Based on the observation that only ran-adjacent objects can swap their rans, defining the FRR of n objects reuires at most n bisectors 2 of the n pairs of ran-adjacent objects. Continuous monitoring of the raning of the objects is done by maintaining a ran-sorted list of objects and its corresponding list of bisectors of pairs of ran-adjacent objects (ran-adjacent bisectors). Each time a bisector is crossed by the uery point, the rans of the two corresponding objects are swapped and the list of ran-adjacent bisectors are updated. An eample is given in Figure 4, where the grey region is the current FRR that is in. Let us assume that is the location of a moving uery point which starts at and stops at 2. In Figure 4(a), is at and the raning is initially a, c, b, f, e, d and the corresponding list of bisectors is B ac, B cb, B bf, B fe, B ed. Then In this paper, the ran of an object means the object s position in a list of objects sorted by their distances to some other object. We use raning and ordering interchangeably due to the usage of both in the literature. 2 The bisector of two objects a and b is the set of points where each point is euidistant to a and b.

4 (a) Initial step (b) First update (c) Second update Figure 4: Incremental ran update crosses B ef in Figure 4(b). This causes e and f to swap their rans. Therefore B bf and B de are replaced by B be and B df, respectively. In Figure 4(c), B ac is crossed. This causes a and c to swap their rans and B bc is replaced by B ab. It is shown in [3] that only O(n) instead of O(n 2 ) bisectors are maintained during the iterations in the IRU algorithm. However, IRU still accesses all data objects to obtain the sorted list and checs n bisectors every time moves. Related spatial-networ ueries. Several NN techniues for static uery points were proposed in [8, 9, 5, 7]. There are also MNN techniues specific to the domain of spatial networs. Kolahdouzan et al. [2] proposed an algorithm that utilizes the Voronoi Networ Nearest Neighbor (VN 3 ) []. Cho et al. [3] proposed a techniue that issues static NN ueries at the intersection points on the uery path. Related moving-object ueries. Hu et al. [7] proposed a saferegion-based techniue for static window and NN ueries on moving objects. Each moving object maintains its own safe region and only report if its new location changes the results of any of the ueries. Yu et al. [22] presented a uery-indeing techniue for monitoring NN ueries for moving objects and a given set of ueries, where each uery object maintain its own critical region to eep trac of the NN set. The similarity between [7,22] and the MNN techniues [3,4,8 20,23] including the V*-Diagram is the use of the safe-region concept to reduce the access cost. Benetis et al. [2] presented algorithms to process NN and reverse NN ueries for moving objects with nown trajectories. Result validity is thus epressed as a function of time. Similar to our wor, Benetis et al. included methods to handle insertions and deletions of data points. These techniues [2, 7, 22] focus on monitoring changes caused by location updates of data objects. The emphasis of the MNN techniues, on the other hand, is on the changes caused by location updates of the uery point. Summary. The MNN techniues are summarized in Table on five features: providing continuous answer, incremental evaluation, accessing only local data (instead of all data), woring on unnown uery path, and providing order-sensitive NNs. Only our proposed algorithm, V*-NN, has all these features. Techniue Table : Comparison of MNN techniues Continuous Incremental Local access Unnown path Order-sensitive SB-NN [8] VD [4] Ordered VD [4] TPNN [9] CNN [20] RIS-NN [23] IRU [3] V*-NN 4. THE V*-DIAGRAM We formulate the V*-Diagram in this section. The V*-Diagram is a safe-region-based techniue. Previous techniues compute safe regions purely based on the data. The V*-Diagram computes safe regions based on not only the data, but also the uery point and the current nowledge of the search space. The V*-Diagram assumes a metric space and a spatial hierarchical inde on the dataset. Hence the BF-NN algorithm can be used to retrieve NNs incrementally as discussed in Section 3.. In the V*-Diagram, the (+) NNs of the moving uery point are maintained, where is the number of auiliary objects to help the V*-Diagram wor effectively (more analysis on is in Section 5.). The V*-Diagram comprises two types of regions, which are discussed in Section 4. and Section 4.2. These regions are then put together to form a combined safe region, discussed in Section 4.3. Commonly used symbols are summarized in Table 2. Symbol Meaning Table 2: Symbols n The number of objects in the database. The number of reuested nearest neighbors. The number of auiliary objects. The moving NN uery point. b The position where the latest BF-NN call is made. The current position of the uery point. p A data object. p The current th NN of. z The ( + ) th NN of b when is at b. S The safe region with regard to p. B pi p j The bisector of two objects p i and p j. 4. Safe region with regard to a data object To help eplain the concept of a safe region with regard to a data object, the notions of search sphere, nown region and reliable region are first introduced. Recall the BF-NN algorithm in Section 3.. Each object/node retrieved from the priority ueue corresponds to an implicit search sphere (centered at the uery point), which delimits the current search coverage, and the sphere epands gradually as more nodes/objects are accessed. Numbers are assigned to those spheres in Figure 2(a) according to the steps in Figure 2(b) that access the corresponding nodes. For eample, sphere 2 corresponds to step 2 where node S is retrieved; sphere 5 corresponds to step 5 where object d is retrieved. Intuitively, the search sphere denotes the region we have full nowledge of, because all the objects in the sphere are already retrieved. In the V*-Diagram, BF-NN is called repeatedly to help maintain ( + ) NNs. Let b be the position of where the latest BF-NN call is made. Let z be the ( +) th NN to b. The search sphere corresponding to z centered at b is the latest one (since BF-NN stops when z is obtained), and we call this search sphere the nown region, denoted by W( b, z). We highlight z because it determines the boundary of the nown region. Figure 5 gives an eample. W( b, z) is actually a sphere centered at b with the radius dist( b, z). Point p is one of the ( + ) NNs of b and other objects in W( b, z) are not shown. Net, we formulate a region within which can move while p remains one of the ( +) NNs of. Let denote a later position of after b. Suppose is at the position as shown in Figure 5. We etend the line segment b and it eits W( b, z) at χ. Let sph(v, l) denote a sphere with center v and radius l. As long as p is in sph(, dist(, χ)), it is one of the ( + ) NNs of. This is because any object outside sph(, dist(, χ)) must be farther to than p to and there are at most ( + ) objects inside sph(, dist(, χ)). Since any object in sph(, dist(, χ)) remains one of the ( + ) NNs of, we call sph(, dist(, χ))

5 Figure 5: The nown, reliable, and safe regions the reliable region with regard to 3 and any object in the reliable region a reliable object. If p is a reliable object, p is said to be reliable; otherwise, it is said to be unreliable. Mathematically, p being in the reliable region with regard to is epressed as dist(, p) dist( b, z) dist( b, ), () where dist( b, z) dist( b, ) is the length of χ. If we view as a variable, then Euation () actually describes all the possible positions of that guarantee p remaining among the ( + ) NNs. Conseuently, we can formulate the safe region with regard to p as follows. DEFINITION (SAFE REGION WITH REGARD TO A POINT). Given a nown region W( b, z) and a data point p within W( b, z), the safe region with regard to p is: S( b, z, p) = { : dist(, p) + dist( b, ) dist( b, z)}. For Euclidean distance in a 2D space, the boundary of S( b, z, p) is an ellipse as illustrated in Figure 5. The two foci of the ellipse are b and p; the sum of the distances from b and p to any point on the ellipse is dist( b, z). We further have the following results. COROLLARY. In Euclidean space, the safe region with regard to z, S( b, z, z), is the line segment bz. PROOF. For any in S( b, z, z), dist(, z)+dist( b, ) dist( b, z). By the triangular ineuality, we have dist(, z) + dist( b, ) dist( b, z). The set of points that satisfies both ineualities is { : dist(, z) + dist( b, ) = dist( b, z)}. In Euclidean space, this is the line segment bz. COROLLARY 2. p is unreliable iff is outside of S( b, z, p). The proof is straightforward and we omit it. 4.2 Fied-ran region As discussed in Section 3.2, the fied-ran region (FRR) was introduced in [3] to denote a set of possible uery-point locations that share a specific raning of all objects. For the V*-Diagram, the FRR is applied to a subset of all objects, specifically, the (+)NN. The IRU algorithm is used to incrementally maintain the FRR. Given a list L of objects, p, p 2,..., p m, the FRR of L is the set of all points such that for any point v in the set, p, p 2,..., p m are already in sorted (ascending) order by their distances to v. Let H pi p j be defined as {v DS : dist(v, p i) dist(v, p j)}, where DS is the data space. An FRR is a function of a list and is formulated as follows. DEFINITION 2 (FIXED-RANK REGION). F p, p 2,..., p m = m \ i= H pi p i+. 3 We omit with regard to when the contet is clear. F p, p 2,..., p m may be written in the compact format of F(L). In Figure 4(a), the raning of the objects according to their distances to is L = a, c, b, f, e, d. The FRR F(L) is defined as H ac H cb H bf H fe H ed. The boundary of F(L) is defined by five bisectors, B ac, B cb, B bf, B fe and B ed. We use the IRU algorithm described in Section 3.2 to incrementally compute F(L) in which currently resides for the ( + ) maintained objects. An FRR is represented as (i) a list B of the (+ ) ran-adjacent bisectors, B pi p i+, for i =, 2,..., +, and (ii) a reference point (which could be any point in F(L), b in our case). The FRR is incrementally maintained by (i) checing whether crosses a bisector in B, and (ii) if yes, performing updates accordingly. The purpose of maintaining the FRR using the IRU algorithm is to eep the ( + ) objects sorted according to their distances to. One alternative solution to IRU is to computing distances between the ( + ) objects and for every position of, which is sampling-based. Although both methods have the same compleity, the FRR is important to the formulation of a region where the (order-sensitive) NN does not change. 4.3 Integrated safe region We are now ready to formulate the safe region for the MNN uery, called the integrated safe region (ISR). The ISR is the intersection of the current FRR of the ( + ) maintained objects and the safe regions with regard to the nearest objects. We will first define the ISR formally and then prove that ISR satisfies the MNN safe-region reuirements. Let O denote the ( + )NN set of b, L be the list of these ( + ) objects sorted by their distances to, and z still be the farthest retrieved object to b, which is p +. The ISR is then formulated as: \ F(L) ( S( b, z, p i)) (2) i= The computation of the ISR can be greatly reduced based on Lemma and Theorem 3 below. LEMMA. F p i, p j S( b, z, p j) S( b, z, p i) = F p i, p j S( b, z, p j). PROOF. For any point v F p i, p j, v satisfies dist(v, p i) dist(v, p j). (3) For any point v S( b, z, p j), by definition v satisfies dist(v, p j) + dist( b, v) dist( b, z). (4) For any v F p i, p j S( b, z, p i), it satisfies ineualities (3) and (4). By adding the two ineualities, dist(v, p i) + dist( b, v) dist( b, z). (5) Ineuality (5) shows that v S( b, z, p i), given that v F p i, p j S( b, z, p j). It can be concluded that: F p i, p j S( b, z, p j) S( b, z, p i) and hence S( b, z, p i) can be discarded in F p i, p j S( b, z, p j) S( b, z, p i). An eample is given in Figure 6. The grey region F a, c, b, f S(, f, c) S(, f, a) is eactly the same as F a, c, b, f S(, f, c). THEOREM 3. For 2, \ F p, p 2,..., p ( S( b, z, p i)) = i= F p, p 2,..., p S( b, z, p )

6 PROOF. Lemma shows the case of = 2, that is, F p, p 2 S( b, z, p 2) S( b, z, p ) = F p, p 2 S( b, z, p 2). If the theorem holds for = l, that is, l\ F p, p 2,..., p l ( S( b, z, p i)) = i= F p, p 2,..., p l S( b, z, p l), then the theorem can be verified for = l + as follows. l+ \ F p, p 2,..., p l+ ( S( b, z, p i)) i= = F p, p 2,..., p l F p l, p l+ l\ ( S( b, z, p i)) S( b, z, p l+) i= = F p, p 2,..., p l F p l, p l+ S( b, z, p l) S( b, z, p l+). By applying Lemma, S( b, z, p l) can be removed from the above epression. Hence, we obtain a final result of F p, p 2,..., p l F p l, p l+ S( b, z, p l+) = F p, p 2,..., p l+ S( b, z, p l+). The theorem therefore holds for any integer value of greater than or eual to 2. Since F(L) = F p, p 2,..., p F p, p +..., p +, based on Theorem 3, epression (2) can be reduced to F(L) S( b, z, p ). Therefore, the ISR can be defined as follows. DEFINITION 3 (INTEGRATED SAFE REGION (ISR)). Let O be the ( + )NN set of b, L be the list of these ( + ) objects sorted by their distances to, z be the farthest retrieved object to b, and p be the th object in L. The integrated safe region with respect to b, z, p and L is defined as: I( b, z, p, L) = F(L) S( b, z, p ) (6) Net, we prove that the ISR defined above satisfies the reuirements of being a safe region for the MNN uery, that is, the NNs as well as their order do not change when remains in the ISR. THEOREM 4. If the ISR I( b, z, p, L) is not an empty set, every point in I( b, z, p, L) has the same order-sensitive NNs. PROOF. According to Definition 3, () since I F(L) (parameters of I omitted), the raning of the ( + ) objects is fied for all points in I, which satisfies the order-sensitivity reuirement; (2) every point in I is also in the safe regions with regard to the first objects in L. As a result, there can be no object outside W( b, z) nearer to than any of the first objects in L. Therefore, for any in I, has the same order-sensitive NNs as b. recent point where (+) NNs are retrieved from the database. As long as remains in F a, c, b, f S(, f, c) (the grey region), then: (i) no object outside W(, f) is nearer to the two objects: a and c; and (ii) the raning of a, c, b, f is unchanged. The diagram that contains the information used in computing the ISR is called the V*-Diagram. It consists of: (i) the bisectors of the ran-adjacent pairs in L; and (ii) the boundary of the safe region with regard to p. We may also use the V*-Diagram to refer generally to the whole techniue based on it, including the algorithms. Although the V*-Diagram in the eample of Figure 6 only computes a single ISR, it actually allows incremental computation of new ISRs, which is further discussed in Section ALGORITHMS In this section, we present V*-NN, an algorithm for MNN ueries based on the V*-Diagram, followed by a discussion on the effect of, the number of auiliary objects. We also present the algorithms to handle insertions/deletions and dynamically changing values. The V*-NN algorithm uses the following data structures and variables to compute and maintain the ISR.. L: a list of ( + ) objects always sorted in ascending order by their distances to ; these objects are the ( + ) NNs retrieved at b. 2. z: the farthest retrieved object in the nown region when is at b. 3. p : the th object in L. 4. S : the safe region with regard to p. 5. B: a list of bisectors of pairs of ran-adjacent objects (ranadjacent bisectors) in the order corresponding to L. We do not eplicitly maintain F(L) because it is represented by B, and checing whether moves out of the current F(L) is also done by checing whether crosses any bisector in B. V*-NN produces answers continuously as shown in Algorithm. It has an initialization part (lines to 3) and a continuous processing part (lines 4 to 9). The initialization part calls the algo- Algorithm : V*-NN( 0,, ) b 0 2 (L, z, S, B, ISR) Compute-V*( b,, ) 3 ReportResult(L.Head()) 4 while (Event GetEvent()) do 5 Event.Position 6 switch Event.Type do 7 case RanUpdate 8 Bisector Event.Bisector 9 L.OrderSwap(Bisector.Inde) B.Update(L,Bisector.Inde) if Bisector.Inde then 2 ReportResult(L.Head()) if Bisector.Inde [, ] then p L.Item() S S( b, z, p ) ISR ConstructISR(S,B,) case ReliabilityUpdate b (L, z, S, B, ISR) Compute-V*( b,, ) Figure 6: Integrated safe region eample ( = 2, = 2) As eemplified in Figure 6, four objects retrieved by a 4NN uery ( = 2 and = 2) at are a, c, b, f. Point is the most rithm Compute-V* (Algorithm 2) to compute the initial ISR using the starting point 0 of the trajectory as b. Then the continuous processing part starts.

7 Algorithm Compute-V* (Algorithm 2) runs as follows. It first calls the BF-NN algorithm to retrieve ( + ) objects with b as the uery point; with the retrieved objects, it sets z and p accordingly; then, it computes S( b, z, p ) and ran-adjacent bisectors based on L and assigns them to S and B, respectively; finally, the current ISR is computed. To determine the correct half plane for each bisector in B, b is used as the reference point. Readers may notice that symbol z is eplained differently here from Table 2. Object z is used to determine the nown region and it is the ( + ) th NN of b when is at b. When deletion is taen into account, if the ( + ) th NN of b gets deleted, we still use the deleted object to determine the nown region. Therefore, we mae it more accurate here than in Table 2 when we did not consider deletions. The continuous processing part of V*-NN is event driven. It basically maintains the ISR as moves. An event is triggered when eits the current ISR. There are two types of events with this regard, RanUpdate and ReliabilityUpdate. These events are generated by a separate inepensive process that constantly checs the current position of against the ISR. When an event is generated, it is associated with a timestamp and the corresponding uery position. Algorithm 2: Compute-V*( b,, ) L BF-NN( b, + ) 2 z L.Item( + ) 3 p L.Item() 4 S S( b, z, p ) 5 B CreateBisectorList(L) 6 ISR ConstructISR(S,B, b) 7 return (L, z, S, B, ISR) Given that the uery trajectory is unnown, the uery positions are updated discretely and checing for new events has to be done based on these discrete updates. To provide accurate answers, the checing should be performed at a high freuency, which is acceptable because of the low cost. Note that this is different from processing the uery based on sampling, which reuires freuent tree searches instead of event checing. The answer we provide is continuous, not based on sampled locations. Since the uery positions are updated discretely, the events could happen anytime between two consecutive uery updates. To compute the eact time (position) the events happen, we assume a linear trajectory between two consecutive uery positions. We describe the two event types, RanUpdate and ReliabilityUpdate below and discuss how to handle them with reference to Algorithm (in the algorithm, is used to denote the position of the uery point when an event happens). RanUpdate This event is triggered when eits the current F(L), that is, crossing a ran-adjacent bisector. Besides the timestamp and uery position, a RanUpdate event also contains the information of the bisector crossed by (line 8). For this event, the rans of the two objects corresponding to the bisector are swapped (line 9) and the bisector list B is updated accordingly (line ) as eplained in the IRU algorithm (Section 3.2). If the event affects the ran of any of the NNs (line ), then the new NNs are reported. Moveover, if the ran update changes p (line 3), S also needs to be updated (lines 4 to 5). Changes are reported to the user if at least one of the nearest objects is affected. ReliabilityUpdate This event is triggered when is leaving S (that is, on the boundary of S ). It means that the th NN is about to become unreliable and hence the number of reliable objects is about to become less than. Therefore, the ReliabilityUpdate event calls Compute-V* using the event position,, to obtain new auiliary objects so that all the (+) maintained objects are reliable again. The new ISR is constructed accordingly. This event does not cause result update because neither the NN set nor their ordering changes. If another object really becomes nearer than the th NN, a RanUpdate will be triggered, which will update the result. Net, we give a running eample of the algorithm. Recall the eample in Figure 6. At the starting point, 4 NNs are retrieved in the order of a, c, b, f. The ISR is F a, c, b, f S(, f, c). On the trajectory of (starting at ), two events happen when (a) F c, a, b, f S(, f, a) (b) F c, a, b, e S(γ 2, e, a) Figure 7: Eample for Algorithm, ( = 2, = 2) crosses B ac at γ and eits S(, f, a) at γ 2. Figure 7 shows the effects of these two events. Figure 7(a) shows how the ISR changes after crosses B ac. At the instant that is crossing B ac at γ, a RanUpdate event is triggered, which causes a and c to swap their rans. The list L becomes c, a, b, f, and this causes both F(L) and S to change. Now a becomes p (2 nd NN), and hence the ISR becomes F c, a, b, f S(, f, a) (the grey region). The current 2 NNs, c and a, are reported to the user in that order. Figure 7(b) shows how the ISR changes after eits S(, f, a). At the instant that is eiting S(, f, a) at γ 2, a ReliabilityUpdate event is triggered, which calls Compute-V* to retrieve more objects. The new ( +) NNs are c, a, b, e, with the corresponding ISR, F c, a, b, e S(γ 2, e, a) (the grey region). 5. On the number of auiliary objects Auiliary objects are an important part of the V*-Diagram techniue. They allow to move away from b while retaining the current NNs by providing the nowledge beyond the coverage of the search sphere of the original NNs. This maes it possible to continuously evaluate the MNN uery. In this subsection, we discuss possible values of, the number of auiliary objects. Generally, we find that should not assume the values of 0 and, which are eplained below. Having eual to 0 implies that z and the th object in L, p, are the same object. According to Corollary, in Euclidean space, the safe region with regard to z (which is S ) is the line segment bz. Unless moves along bz, eits S as it starts moving. Probabilistically, it is highly unliely that moves along bz since bz is just one direction among the infinite possible directions can move towards. Therefore, it is probable that always eits S as it moves and it triggers the ReliabilityUpdate event infinitely. As a result, should not be set to 0 for Euclidean space. When is a positive integer, the problem of infinitely triggering the ReliabilityUpdate event does not happen ecept under the coincidence described in the net paragraph. Therefore, any integer greater than is a valid value for. The effect of the value of on performance is further investigated in Sections 7 and 8. In theory, the problem of S being a line segment may happen with any value of when the last ( + ) objects in L have the same distance to b, which is dist( b, z). In this case, dist( b, p ) euals to dist( b, z). By definition, S = { : dist(, p ) + dist( b, ) dist( b, z)}. If we replace dist( b, z) by dist( b, p ) in the ineuality of the definition, we

8 get S = { : dist(, p ) + dist( b, ) dist( b, p )}, which is a line segment in Euclidean space. To completely avoid this problem, we can chec whether dist( b, p ) is eual to dist( b, z) after we retrieve ( + ) objects by a BF-NN call. If they are eual, then we increase the value of until dist( b, p ) is different from dist( b, z). In general, a larger value of provides a larger S, and hence the less freuent we need to retrieve new objects from the database. Having the value of too small will result in highly freuent BFNN calls. On the other hand, a too large value also incurs the overhead of retrieving more objects in every BF-NN call and more computation for maintaining them. 5.2 Insertions and deletions of objects Algorithm 3: DatasetUpdate(,p,Operation) if p W( b, z) then 2 if Operation = Insertion then 3 L Insert(L,p,) else L Delete(L,p) B.Update(L) if > L.Length() then b (L, z, S, B, ISR) Compute-V*( b,, ) ReportResult(L.Head()) else if dist(, p) dist(, p ) then p L.Item() S S( b, z, p ) if / S then b (L, z, S, B, ISR) Compute-V*( b,, ) else ISR ConstructISR(S,B,) ReportResult(L.Head()) else ISR ConstructISR(S,B,) In this subsection, we describe the algorithm to perform updates (that is, insertions and deletions) to the dataset for V*-NN. The algorithm is called DatasetUpdate and is presented in Algorithm 3. In this algorithm, denotes the position of the uery point when the update happens and it is passed in as an input. Let p be the object to be inserted or deleted. First, the algorithm checs whether the p is in W( b, z). If not, the update can be safely ignored because it cannot affect the ISR. Otherwise, an insertion/deletion of p into/from L is performed and B is updated accordingly (lines 2 to 6): insertion of p needs for computing distances from and the maintained objects to find the correct insertion slot in L; deleting p from L reuires only a simple looup operation. After the bisector update, the ISR and L could be in one of the following three cases: (i) The length of L becomes smaller than as a result of a deletion (line 7). In this case, b is set to and Compute-V* is called to retrieve more objects and compute the new ISR accordingly (lines 8 and 9). The new result is reported (line ). (ii) The length of L is still greater than but the update affects the NN set (line ). We update p and S (lines 2-3) and chec if is inside the new S (line 4). If is not inside the new S, then Compute-V* is called (line 6). Otherwise, the ISR is updated to reflect the changes in B and S. Since the NNs have changed, the new result is reported to the user (line 9). (iii) The update has no effects to the NN set (line 20). Only the ISR is updated to reflect the change in B. 5.3 MNN with dynamically changing values The ability to gracefully handle changes to the value of is crucial for the distance browsing functionality [6]. For static NN ueries, distance browsing is a feature that allows NNs to be incrementally retrieved without having to specify the value of in advance. In this paper, we allow the value of to be changed without incurring heavy computation. Algorithm KUpdate (Algorithm 4) shows how V*-NN handles dynamically changing values. The algorithm has two inputs: the current location of the uery point and the new value. We first chec if the new is greater than the length of L. If yes, b is set to and Compute-V* is called. Otherwise, p and S are updated for the new (lines 5 and 6). If is not inside the new S, b is set to and Compute-V* is called. Otherwise, only the ISR has to be updated to incorporate the new S (line ). Finally, the new NNs are reported (line 2). As we can see, the V*-NN algorithm can easily accommodate dynamically changing values due to its incremental nature. Algorithm 4: KUpdate(,) if > L.Length() then 2 b 3 (L, z, S, B, ISR) Compute-V*( b,, ) 4 else 5 p L.Item() 6 S S(, z, p ) 7 if / S then 8 b 9 (L, z, S, B, ISR) Compute-V*( b,, ) 2 else ISR ConstructISR(S,B,) ReportResult(L.Head()) 6. THE V*-DIAGRAM IN SPATIAL NET- WORKS When the movement of is constrained by networ connectivity, NN problems should be solved based on the networ distance. For eample, a car is travelling on a road and it eeps trac of the nearest gas stations based on the road networ distance. In this section, we show how the V*-Diagram techniue can be applied to the domain of spatial networs, which also satisfies our metric space assumption. Due to the space limitation, we could not fully elaborate the application of V*-Diagram in the spatialnetwor model, but only present the essential components of the techniue. The essence of the V*-Diagram is the ISR, which consists of two ey components: safe regions with regard to data objects and the fied-ran region (FRR). Hence, we focus our discussions on how to determine these two components in spatial networs, while the algorithms to compute the ISR and process MNN ueries can be reused. To avoid an ehaustive discussion, we use an eample to illustrate the main idea. A spatial networ is usually represented as a set of vertices and a set of edges, where an edge is defined by two vertices. Given points p and p 2 in the networ, dist(p, p 2) is the length of the shortest path between p and p 2. Figure 8 shows a spatial networ of eight nodes, a to h. Distance dist( b, p) is 2 using the path via b.

9 We use pnt(p, p 2, l) to denote a point on the edge (p, p 2) with distance l to p, and seg(p, p 2, l) to denote a section on the edge (p, p 2) with the starting point p and length l. For eample, z is at pnt(a, e, 3), and all points between a and z can be represented as seg(a, e, 3). Consider an MNN uery with =. Suppose is 2 for the V*-Diagram and b is at pnt(a, b, 3). The 3 NNs for b are: p, r and z. We can use any NN techniue for spatial networs (such as those described in [8, 9, 5, 7]) to retrieve the ( + ) NNs at b. The algorithm also returns the edges in the range of dist( b, z) to b (that is, W( b, z)), (a, b), (a, c), (a, e), (b, d), (b, f), (d, c), (d, g), and the distances from the vertices of these edges to b. Figure 8: V*-Diagram in a spatial networ ( = and = 2) Net, we determine the safe region for p (which is p in this eample). The safe region S( b, z, p) is the region where for any point in the region, the sum of dist( b, ) and dist(p, ) is less than or eual to dist( b, z), which is 6. We need to eplore all the edges in S( b, z, p) to identify its boundary. Since W( b, z) encloses S( b, z, p), and we already now the edges that are in W( b, z) via the NN uery eecuted at b, we only need to consider those edges. We use edge (b, f) as an eample. For a point on (b, f) to be in S( b, z, p), dist( b, ) + dist(p, ) has to be less than 6 according to the definition. Since dist( b, ) + dist(p, ) is eual to dist( b, b)+dist(p, b)+2dist(b, ), and both dist( b, b) and dist(p, b) are, is in S( b, z, p) when dist(b, ) is less than or eual to 2. Conseuently, pnt(b, f, 2) forms part of the boundary of S( b, z, p). The boundary points on other edges can be computed in a similar manner. After obtaining all boundary points, we get S( b, z, p), which consists of edge (b, d), seg(b, a, 3) and seg(b, f, 2), plotted as grey thic segments in the figure. As discussed in Section 4.2, the FRR is determined by the ranadjacent bisectors the maintained objects. In a spatial networ, a bisector reduces to points on edges. For eample, B pr has two bisecting points, one on pnt(b, d, 2.5) and the other on pnt(a, c, 0.5). The FRR F p, r, z is the region with B pr and B rz as the boundary and containing b. It consists of (b, f), seg(b, a,.5) and seg(b, d, 2.5), shown as the segments in the dashed triangle. As a result, the ISR (S( b, z, p) F p, r, z ) consists of seg(b, a,.5), seg(b, d, 2.5) and seg(b, f, 2). It is shown as the segments in the grey region. According to Algorithm, eiting the ISR via pnt(b, a,.5) ( B rz) or seg(b, d, 2.5) ( B pr) triggers a RanUpdate event, and eiting the ISR via pnt(b, f, 2) triggers a ReliabilityUpdate event. 7. PERFORMANCE ANALYSIS Among all techniues listed in Table, only RIS-NN by Zhang et al. [23] and V*-NN provide continuous answers for the MNN uery with unnown uery trajectory and without accessing all the data. Therefore, this section focuses on a comparative performance analysis on RIS-NN and V*-NN in terms of tree node accesses (that is, IO cost). We assume a 2D space, although the analysis can be etended to higher dimensional spaces. We also assume that the data objects are uniformly distributed. V*-NN only has node accesses in the calls to BF-NN, which are triggered by the ReliabilityUpdate event (line 9 of Algorithm ). We analyze the freuency of ReliabilityUpdate events, f b, as follows. A ReliabilityUpdate event happens when eits S( b, z, p ). We denote the point of eit as e. The ReliabilityUpdate event happens only once during the process of moving away from b until reaching e. Then a BF-NN is performed and e becomes the new b. A ReliabilityUpdate event will happen again when the net time eits S( b, z, p ). Therefore, f b is inversely proportional to the distance travels from b to e. In the worst case, moves in a straight line, and f b is inversely proportional to dist( b, e). The epected value of dist( b, e) is obtained as follows. When moves to e, p is on the boundary of the reliable region (with regard to e). For a better understanding, imagine in Figure 5, p is p, is e and it is on the boundary of S( b, z, p). We can see that dist( b, e) euals dist( b, χ) dist( e, χ). Note that dist( b, χ) is the radius of W(, z), which is a sphere that contains (+) points; dist( e, χ) is the radius of the reliable region with regard to e, which is a sphere that contains points. According to [2], the distance between the uery point b and the th NN is 2/C v( p /n), where C v is the vicinity constant [2]. Therefore dist( b, e), which is dist( b, χ) dist( e, χ), can be epressed as 2 ( C v p /n p ( + )/n). Since f b is inversely proportional to dist( b, e), O(f b ) = O(/( p /n p ( + )/n)). The epression of O(f b ) can be relaed as follows: /n (+)/n 2 /n = (+)/n /n /n+ (+)/n (+)/n /n 2 (+)/n /n (+)/n+ /n (+)/n+ /n (+)/n = 2 = 2 4. (+)/n /n /n /n (+)n Therefore O(f b ) is O( ). Typically, is comparable to 2 and ( + ) is much smaller than. Thus, we obtain that f b is n O( ). Let Cnn be the cost of a BF-NN call. Then we obtain n the total IO cost of V*-NN, C nno( ). The RIS-NN processes the MNN uery as follows. Every time eits the current VD cell, RIS-NN is eecuted to obtain the new VD cell and the corresponding NN. The total cost depends on the freuency of crossing VD cells and the cost of each RISNN run. In the worst case, moves along a straight line. The freuency of crossing VD cells is proportional to the average linear density 4 of the VD cells. The number of the VD cells is O(n ) in 2D space [4]. We assume that is much smaller than n. Thus, the density of VD cells is O(n), which corresponds to a linear density of O( n). Each RIS-NN run reuires 2 TPNN ueries on average [23]. Let C tpnn be the cost of a TPNN call. Then the total IO cost of RIS-NN for the MNN uery is 2C tpnno( n). 4 The number of VD cells crossed per unit length along a straight line.