An Efficient and Scalable Approach to CNN Queries in a Road Network

Transcription

1 An Effiient and Salable Approah to CNN Queries in a Road Network Hyung-Ju Cho Chin-Wan Chung Dept. of Eletrial Engineering & Computer Siene Korea Advaned Institute of Siene and Tehnology 373- Kusong-dong, Yusong-gu, Taejon 35-7, Korea {hjho, hungw}@islab.kaist.a.kr Abstrat A ontinuous searh in a road network retrieves the objets whih satisfy a query ondition at any point on a path. For example, return the three nearest restaurants from all loations on my route from point s to point e. In this paper, we deal with NN queries as well as ontinuous NN queries in the ontext of moving objets databases. The performane of existing approahes based on the network distane suh as the shortest path length depends largely on the density of objets of interest. To overome this problem, we propose (a unique ontinuous searh algorithm) for NN queries and CNN queries performed on a network. We inorporate the use of preomputed NN lists into Dijkstra s algorithm for NN queries. A mathematial rationale is employed to produe the final results of CNN queries. Experimental results for reallife datasets of various sizes show that UNI- CONS outperforms its ompetitors by up to 3.5 times for NN queries and 5 times for CNN queries depending on the density of objets and the number of NNs required. Introdution Due to the advanes in mobile ommuniation and database tehnologies, diverse innovative mobile omputing appliations are emerging. The ability to support ontinuous queries from mobile lients on a road Permission to opy without fee all or part of this material is granted provided that the opies are not made or distributed for diret ommerial advantage, the VLDB opyright notie and the title of the publiation and its date appear, and notie is given that opying is by permission of the Very Large Data Base Endowment. To opy otherwise, or to republish, requires a fee and/or speial permission from the Endowment. Proeedings of the 3st VLDB Conferene, Trondheim, Norway, 25 network is essential for a lass of mobile appliations. In this paper, we investigate ontinuous nearest neighbor (CNN) searhes under the following two onditions: (i) Moving objets suh as ars or people run on a road network and stati objets suh as gas stations or restaurants are loated on the road network. (ii) The distane measure is defined as the shortest path length (network distane) on the network. CNN searhes on a road network are essential for emerging loation-based servies and many real-life GIS appliations. Cars move aording to a given path on a road. In addition, with the development of mobile devies suh as PDAs and ellular phones, it is pratially possible to trak ars and people in real-time. Suh loation-aware devies enable loation-based servies that provide users with a variety of useful information based on their urrent positions. Furthermore, CNN searhes on a road network onstitute interesting and intuitive problems from the pratial as well as theoretial point of view. Nevertheless, there is limited previous work in the literature. We present the following example of the CNN query with real-life semanti on the map in Figure, where it is assumed that a, b,, d, and e are gas stations : Find the 2 losest gas stations from all points on the path P from n to n 6 (i.e., P = {n, n 2, n 3, n 4, n 5, n 6 }). b a n 4 n n 3 2 n d Figure : Example ontinuous searh Unlike the output at a query point, the result of a ontinuous searh for a given path P ontains a set of (I, R I ) tuples, where I P is a valid interval over whih the same query result is generated and R I is n 5 n 6 e 865

2 a set of objets satisfying the query ondition at any point on I. Two key requirements for good ontinuous searh algorithms are as follows: (i) Use as few stati queries required to answer the ontinuous searh as possible. (ii) Redue the omputational overhead in determining valid intervals. The valid intervals speify the loations that the k NNs of a moving query objet remains the same. Existing approahes are largely affeted by the density of objets of interest. Voronoi-based Network Nearest Neighbor (VN 3 ) [7] and Upper Bound Algorithm () [8] developed by Kolahdouzan et al. for NN queries and CNN queries, respetively, are effiient for the ase where objets are sparsely distributed in the network. Hene, their approahes are negatively affeted by the inrease in the density of objets. Conversely, Inremental Network Expansion (INE) [9] presented by Papadias et al. for NN queries suffers from poor performane when objets are not densely distributed. To overome this problem, we propose an effiient ontinuous searh algorithm alled whih deals with NN queries as well as CNN queries. We improve the performane of NN queries with the introdution of preomputed NN lists in using Dijkstra s shortest path algorithm [3]. The tehnique diretly ontributes to the redution of the NN query ost of. For CNN queries, the advantage of over is that only two NN queries are performed between adjaent nodes, independent of the density of objets and the number of NNs required. The mathematial rationale is presented to show how we produe the final result (i.e., a set of (I, R I ) tuples). An advantage of is the ease of implementing it with an existing spatial aess method, suh as the R*-tree [], and the adjaeny list struture to represent the graph struture both of whih are disk-based. Our ontributions are summarized as follows: We propose an effiient ontinuous searh algorithm alled for NN queries as well as CNN queries in a road network. We inorporate the preomputed NN lists in using Dijkstra s algorithm [3] for NN queries and introdue tehniques whih avoid unneessary disk I/Os. outperforms VN 3 [7], whih is urrently regarded as the best approah, by up to 3.5 times in a realisti experimental environment. signifiantly redues the number of disk I/Os in proessing CNN queries sine stati queries are issued only at the intersetion points on a query path. outperforms its ompetitor, alled [8], in query response time by up to 5 times depending on the number of NNs required and the density of objets of interest. The rest of the paper is strutured as follows: Setion 2 gives the survey of previous methods for NN and CNN queries. Setion 3 presents the improved algorithms for NN queries based on the network distane. Setion 4 presents algorithms for the CNN searh. Setion 5 evaluates the proposed tehniques with omprehensive experiments. Finally, Setion 6 onludes the paper. 2 Related Work Several algorithms have been developed using the network distane. Shahabi et al. proposed an embedding tehnique to transform a road network into a higher dimensional spae in order to utilize omputationally simple metris []. The main disadvantage of this method is that it provides only an approximation of the atual distane. Jensen et al. [6] propose a data model and definition of abstrat funtionality required for NN queries in spatial network databases. They use algorithms similar to Dijkstra s algorithm in order to perform online alulations of the shortest distane from a query point to an objet. Shekhar et al. [] present four alternative tehniques for finding the first nearest neighbor to a moving query objet on a given path. Papadias et al. [9] presented a solution alled INE for NN queries in spatial network databases by introduing an arhiteture that integrates network and Eulidean information and aptures pragmati onstraints. Sine the number of links and nodes that need to be retrieved and examined is inversely proportional to the ardinality ratio of objets, the main disadvantage of this approah is a dramati degradation in performane when the above ardinality ratio is very small, whih is a usual ase for real world senarios. In addition, it does not optimize primitive operations to failitate an effiient network searh sine it is designed to support both onventional spatial queries based on the Eulidean distane and queries based on the network distane. Kolahdouzan et al. proposed a new approah for NN queries in spatial network databases [7]. Their approah, alled VN 3, prealulates the network Voronoi polygons (NVPs) and some network distanes. VN 3 is based on the properties of the Network Voronoi diagrams. Their experiments with several real-world data sets showed that VN 3 outperforms INE [9] by up to an order of magnitude. The intuition is that the NVPs an diretly be used to find the first nearest neighbor of a query objet q. Subsequently, the adjaeny information of NVPs an be utilized to provide a andidate set for other nearest neighbors of q. However, in the ase where the number of NNs required inreases and the number of objets of interest inreases, VN 3 suffers from the omputational overhead of prealulating NVPs. Consequently, the performane of VN 3 degenerates onsiderably for high densities of objets. Feng et al. [4, 5] provide a solution for CNN queries in road networks. Their solution is based on finding 866

3 the loations on a path where a NN query must be performed. The main shortoming of this approah is that it only addresses the problem when the first nearest neighbor is requested (i.e., ontinuous -NN) and does not address the problem for ontinuous k NN queries. Finally, Kolahdouzan et al. presented a solution alled for CNN queries in spatial network databases [8]. restrits the omputation of NN queries to only the loations where they are required and hene, provides better performane by reduing the number of NN omputations. takes advantage of VN 3 for obtaining (k+) NNs at a stati loation. For this goal, retrieves (k+) NNs of a query objet q to ompute the minimum distane during whih the query objet an move without requiring a new NN query to be issued. However, still requires a large number of NN queries to find split points. The split points are the loations on the path at whih the k NNs of a moving query objet hange. Consequently, the exeution time inreases sharply in proportion to the number of NNs required and the density of objets. 3 NN searh These algorithms enhane previous researh results and they are the basis of our CNN searh algorithms. To larify the meanings of the terms used, we formally define an edge, a node, a path, a query point, and an objet. Definition 3. A road network onsists of a set of nodes, together with a set of edges, eah of whih diretly onnets two nodes. A path is a sequene of suessively neighboring edges where the terminating points are distint. A query point (e.g., the urrent position of a user) is a loation of interest on the road network. An objet (e.g., shool or gas station) is a point of interest whih is loated in the road network. We briefly explain the network distane used in a road network. Edges provide the onnetivity of the original road network. An edge onneting two nodes, n i and n j, has the network distane d(n i, n j ). If n i and n j are not adjaent nodes, d(n i, n j ) denotes the shortest path length from n i to n j. The same onept is applied to the network distane between a node and an objet and the distane between two objets. Sometimes, travel time may be used as the network distane, whih is very useful in many ases. Suppose that there exists an edge between n i and n j. The travel time from n i to n j following a narrow street may be larger than that of a path going through some intermediate nodes. In addition, the travel time may dynamially hange depending on traffi ongestion and road onditions. Network distanes between objets depend on their network onnetivity and are omputationally expensive to alulate. Therefore, our approah for NN searhes introdues the utilization of preomputed NNs. Lemma states a simple but important fat for our NN searhes. Lemma Consider a query point q on an edge n i n j. Let R q be the set of objets satisfying the query ondition, O nin j be the set of objets on n i n j, and R ni and R nj be the sets of objets satisfying the same query ondition at n i and n j, respetively. Then, R q (O nin j R ni R nj ) Proof) Lemma is self-evident, so its proof is omitted. If eah node on a graph has k preomputed NNs of its own, a k NN query result at any point on the graph an be immediately obtained from objets on an edge ontaining the query point and the k NNs for the two nodes assoiated with the orresponding edge. However, it is very diffiult to maintain the k NNs for all nodes when the value of k is very large or the number of nodes is very large. For this reason, our approah maintains preomputed NNs for only a small portion of the nodes. We formally define intersetions and ondensing points as follows: Definition 3.2 A node where three or more edges meet is alled an intersetion point and an intersetion point whih maintains preomputed NNs is alled a ondensing point. Eah ondensing point stores m ( ) NNs where the value of m is provided as a parameter. The performane of our approah improves with an inrease in the number of ondensing points and m, the size of the preomputed NN lists. Naturally, our approah shows the best performane if the number of ondensing points points equals that of intersetion points and the value of m is greater than or equal to the value of k, where k is the number of NNs required in a query. In ase k > m, we an dynamially ompute k NNs starting from already preomputed m NNs. Query results an be obtained from a ombination of the preomputed information and the Dijkstra s searh method. Partiularly, in the ase where objets are sparsely distributed, our approah is expeted to show muh better performane than INE [9] whih is simply based on Dijkstra s algorithm. This is beause our approah an ompute the result of an NN query without visiting the NNs diretly. In addition, our algorithms an avoid unneessary disk I/Os by introduing two additional data strutures, E visited and E empty as seen in Figures 3 and 5 respetively, whih keep useful information from previously visited edges. Figure 2 shows the algorithm for the NN searh. If the query loation q on a network is given, our NN searh algorithm starts with find edges (q) on line 4 to disover the set E q of edges ontaining q. For example, in the ase where q is a node, adjaent edges of 867

4 the node belong to E q. Originally, our guess for the network distane kth dist from q to the kth nearest objet o kth is infinity. Next, we explore the objets on the edge n q n q2 to retrieve the qualifying objets whose network distanes from q are within kth dist, whih is performed by explore (n q n q2, q, kth dist) on line 6. Objets on the edges whih ontain q are first investigated in explore (n q n q2, q, kth dist) and their distanes from q are omputed in this funtion. For a network expansion, we push (n q, d(q, n q )) and (n q2, d(q, n q2 )) to the priority queue PQ. If PQ is empty or the network distane d(q, n ) of the element in PQ is not less than kth dist, the algorithm breaks out of the while loop and terminates. Otherwise, the adjaent edges of n are explored repeatedly. Note that PQ is a priority queue, so d(q, n ) on line 2, whih is popped from PQ is less than or equal to the distane from q to any other node in PQ. pro NN Searh (q, k, kth dist) /* q is a query point, k is the number of nearest neighbors to be retrieved, and kth dist is an initial distane between the kth objet and q. */ begin. /* R keeps objets satisfying the query prediate */ 2. R := 3. /* find the set E q of edges ontaining q */ 4. E q := find edges (q) 5. for eah n q n q2 E q do 6. explore (n q n q2, q, kth dist) 7. PQ.push (n q, d(q, n q )) 8. PQ.push (n q2, d(q, n q2 )) 9. end-for. while PQ is not empty do. (n, d(q, n )) := PQ.pop () 2. if d(q, n ) kth dist then 3. Searh Node (n, d(q, n ), kth dist) 4. else 5. exit while loop 6. end-while end Figure 2: NN searh algorithm The Searh Node algorithm shown in Figure 3 explores the preomputed NNs and the adjaent edges of node n with an offset d(q, n ). If node n is a ondensing point, the algorithm first srutinizes eah on i th of n where on i th is the i-th NN of n. If d(q, n ) + d(n, on i th ) is less than kth dist, a tuple (on i th, d(q, n ) + d(n, on i th )) is added to R and kth dist is updated to the network distane d(q, o kth ) of the kth nearest objet in R. If node n is a ondensing point and d(q, n ) + d(n, o m th n ) is greater than or equal to kth dist, the Searh Node ends without searhing adjaent edges. Otherwise, the algorithm should srutinize eah adjaent edge n n i of node n. To avoid unneessary visit of the same edge, E visited of line maintains a set of (n v n v2, d max (q, n v n v2 )) tuples where n v n v2 is a previously visited edge and d max (q, n v n v2 ) = min{d(q, n v ) + d(n v, n v2 ), d(q, n v2 ) + d(n v2, n v )}. pro Searh Node (n, d(q, n ), kth dist) /* n is a node popped from PQ, d(q, n ) and kth dist are as previously defined */ begin. /* let P n = {o st n,..., o m th n } 2. be the set of preomputed NNs of n */ 3. if n is a ondensing point then 4. for eah on i th P n do 5. if d(q, n ) + d(n, on i th ) < kth dist then 6. R := R {(on i th, d(q, n ) + d(n, o i th 7. kth dist := d(q, o kth ) where o kth R 8. end-for 9.. if n is not a ondensing point or. d(q, n ) + d(n, o m th n n ))} ) < kth dist then 2. for eah adjaent edge n n i of n do 3. if (n n i E visited and d(q, n ) d max (q, n n i )) then 4. skip the visit of this edge 5. else 6. Searh Edge (n n i, d(q, n ), kth dist) 7. PQ.push (n i, d(q, n ) + d(n, n i )) 8. E visited := E visited {(n n i, d(q, n ) + d(n, n i )} 9. end-for end Figure 3: Searh Node algorithm Figure 4 shows that E visited plays an important role in avoiding unneessary dupliate aesses to previously visited edges. Given a query point q, whih is node n, we first explore adjaent edges, n n 2 and n n 3. Then, E visited = {(n n 2, 3), (n n 3, 4)}. Next, a searh is exeuted with (n 2, 3) popped from PQ={(n 2, 3), (n 3, 4)}. Hene, n 2 n and n 2 n 3 will be visited. However, n 2 n is not visited sine d(q, n 2 ) = 3 is not less than d max (q, n n 2 ) = 3 in E visited. Edge n 2 n 3 is visited sine this edge does not belong to E visited, and then (n 2 n 3, 8) is added to E visited. As a result, E visited = {(n n 2, 3), (n n 3, 4), (n 2 n 3, 8)}, d(q, a) = d(q, n 2 ) + d(n 2, a) = 4 and d(q, b) = d(q, n 2 ) + d(q 2, b) = 7. Finally, n 3 is explored. n 3 n is not visited due to the same reason that n 2 n was not visited. However, n 3 n 2 should be visited beause d(q, n 3 ) = 4 is less than d max (q, n 2 n 3 ) = 8 in E visited. Therefore, d(q, b) = 7 is updated to d(q, b) = d(q, n 3 ) + d(n 3, b) = 5. The Searh Edge algorithm shown in Figure 5 inspets objets on the edge n n i, where d(q, n i ) = d(q, n )+d(n, n i ). To avoid dupliate aesses to edges where no objets are loated, E empty keeps the 868

5 3 n q 4 Then, R path = O path R ni R ni+ R nj n 2 a 3 b n 3 Figure 4: d(q, a) = 4 and d(q, b) = 5 set of the previously visited edges without objets. It is expeted that E empty is very effetive when the objets are sparsely loated. If d(q, o i ) < kth dist, a tuple (o i, d(q, o i )) is added to R and kth dist is updated to the network distane d(q, o kth ) of the kth nearest objet in R. When the NN searh algorithm ends, the k NNs in R are returned. pro Searh Edge (n n i, d(q, n ), kth dist) /* n n i is an edge to be visited, d(q, n ) and kth dist are as previously defined. */ begin. if n n i E empty then 2. return 3. /* let O nni be the set of objets on the edge n n i */ 4. if O nn i is empty then 5. E empty := E empty {n n i } 6. return 7. for eah objet o i O nn i 8. if d(q, n ) + d(n, o i ) < kth dist then 9. R := R {(o i, d(q, n ) + d(n, o i ))}. kth dist := d(q, o kth ) where o kth R. end-for end Figure 5: Searh Edge algorithm 4 CNN searh We start with two basi onepts for ontinuous searh in Setion 4.. Setion 4.2 presents algorithms for CNN searh. 4. Basi Ideas We extend Lemma to Lemma 2. Lemma 2 states that the union of the set of objets on the query path and the sets of objets satisfying the query prediate at nodes on the query path is equivalent to the union of sets of query results at all points on the query path. This is the first basi onept of. Lemma 2 To perform a ontinuous searh along a path={n i, n i+,..., n j }, it is suffiient to retrieve objets on the query path and to run a stati query at eah node n k (i k j). Let R path be the set of objets satisfying the ontinuous query ondition at some point on the query path, O path be the set of objets on the query path, and R nk (i k j) be the set of objets satisfying the query ondition at n k (i k j). Proof) For any query point q path, R q is defined as the set of objets satisfying the query ondition at q. Without loss of generality, we an assume that q belongs to an edge n k n k+ (i k j ). Thus, R q an be represented by Lemma as follows: Thus, R path = R q O nk n k+ R nk R nk+ q path R q O path R ni R ni+ R nj () The following inverse formula whih exhanges the left side with the right of the equation () an be trivially proved using the ontradition method. O path R ni R ni+ R nj R path (2) From the above equations () and (2), R path = O path R ni R ni+ R nj Lemma 2 provides an insight into how we an ompute the ontinuous query result by ombining objets on the query path and stati query results at nodes on the query path. Figure 6 shows the hange in network distane between a dynami query point q and three stati objets a, b, and when q moves on the query path P = {n, n 2, n 3, n 4, n 5 }. Let x be the total movement length of q while q moves along the query path. For instane, in Figure 6(a), x = when q is loated at n, x = when q reahes a, and so forth. The path an be viewed as a line segment with a length of 6. As q moves from n to n 5 along the query path, x hanges from to 6. Then, for x [, 6], d(q, a) = x as depited in Figure 6(b). q reahes b via n 3. Therefore, d(q, b) = d(q, n 3 )+d(n 3, b) = x 3 +. d(q, ) an be omputed in the same way as d(q, a) in that both a and are on the query path. Hene, d(q, ) = x 5. In this way, the hange in the network distane d(q, obj) between a moving query point q and a stati objet obj on a network an be expressed as a pieewise linear equation. This is the seond basi onept of our ontinuous searh algorithms. Therefore, without relying on issuing queries, mathematial analysis an be used in determining NNs. We an extend the seond onept to paths with one way traffi. Suppose that the same path (i.e., {n, n 2, n 3, n 4, n 5 }) in Figure 6(a) is given for the network of Figure 7(a). The differene from Figure 6(a) is that the two edges n 2 n 3 and n 3 n 4 an be traversed 869

6 n a n 6 b n 2 n 3 n 4 (a) path = {n, n 2, n 3, n 4, n 5 } d(q,obj) (,) b a (b) d(q, a)= x, d(q, b)= x 3 +, d(q, )= x 5 Figure 6: d(q, a), d(q, b), and d(q, ) n a n 6 b n 2 n 3 n 4 (a) Path = {n, n 2, n 3, n 4, n 5 } d(q,obj) one-way 2 b a x (,) (b) d(q, a), d(q, b), and d(q, ) Figure 7: d(q, obj) for the query path with one-way traffi in only one diretion. That is, d(n 2, n 3 ) = d(n 3, n 4 ) = and d(n 3, n 2 ) = d(n 4, n 3 ) =. Figure 7(b) illustrates the impat of the query path with one-way traffi on d(q, a), d(q, b), and d(q, ). Unlike the query path with bi-diretional traffi in Figure 6, a path with one-way traffi should be managed arefully sine the diretion of movement is fixed in a one-way road. For instane, there is no way to move from n 3 to a and from n 4 to b. Thus, d(n 3, a) = and d(n 4, b) = in Figure 7(a). In the ase objet a is loated before the beginning of a one-way road, it is not possible for q to return to a after q has passed node n 2. That is, d(q, a) = for x [2,6]. Similarly, after q passes n 3, there is no way for q to reah b. Thus, d(q, b) = for x [3,6]. Sine is loated after the one-way edges, it is not affeted by n 2 n 3 and n 3 n 4. After q goes past, it is possible to return to. As shown in Figure 7(b), d(q, a) = x for x [,2], x n 5 n 5 d(q, a) = for x (2,6], d(q, b) = x 3 + for x [,3], d(q, b) = for x (3,6], and d(q, ) = x 5 for x [,6]. 4.2 Algorithms Step : Divide query path into subpaths Step 2: Determine valid intervals of eah subpath Step 3: Merge valid intervals of adjaent subpaths Figure 8: Sketh of CNN Searh algorithm As shown in Figure 8, our CNN searh algorithm onsists primarily of three subtasks. In Step, we divide a query path into multiple subpaths, where intersetion points on the query path beome the start and end points of eah subpath. In Step 2, the algorithm omputes valid intervals for the subpaths obtained in Step. The details are explained in Figure 9. Finally, in Step 3, the algorithm ombines the valid intervals and the query results for their adjaent subpaths to obtain the final result. Sine Step and Step 3 an be onduted without diffiulty, we fous on Step 2. Among intersetion points, the points with larger fanouts are seleted and used as ondensing points. The reason for this is beause, as the fanouts inrease, the number of edges to be visited also inreases. Let s SP and e SP be the start and the end points of subpath SP, respetively. Let S SP and E SP be the sets of k NNs at the two points s SP and e SP, respetively. Let O SP be the set of objets on subpath SP. The use of (k+) NNs at s SP makes it possible to quikly determine whether S SP is equal to E SP. Lemma 3 shows this fat. Lemma 3 Let o k and o k+ be the k-th and (k+)-th NNs from point s SP, respetively. Let L SP be the length of the subpath SP. Then, the following is satisfied. d(s SP, o k+ ) d(s SP, o k ) 2 L SP S SP = E SP and O SP S SP Proof) Let o i ( i N) be the i-th NN from point s SP where N is the total number of objets. Then, S SP = {o, o 2,...,o k }. Let MIN(d(a, b)) and MAX(d(a, b)) be the minimum and maximum values of d(a, b), respetively. For d(e SP, o i ), MIN(d(e SP, o i )) is d(s SP, o i ) L SP and MAX(d(e SP, o i )) is d(s SP, o i ) + L SP. We first show that S SP = E SP. It suffies to show that MIN(d(e SP, o k+ )) MAX(d(e SP, o k )) due to the fat that MIN(d(e SP, o i )) MAX(d(e SP, o i )), MIN(d(e SP, o i )) MIN(d(e SP, o i+ )), and 87

7 MAX(d(e SP, o i )) MAX(d(e SP, o i+ )). d(s SP, o k+ ) d(s SP, o k ) 2 L SP {d(s SP, o k+ ) L SP )} {L SP + d(s SP, o k )} MIN(d(e SP, o k+ )) MAX(d(e SP, o k )). Therefore, MIN(d(e SP, o k+ )) MAX(d(e SP, o k )). Consequently, E SP = {o, o 2,..., o k }. Next, we show that O SP S SP. Sine o j (k + j N) annot be on SP under the given ondition, O SP is a subset of S SP. Lemma 4 enables the algorithm to determine whether a subpath has two or more valid intervals. Lemma 4 S SP is equal to E SP and O SP is a subset of S SP if and only if there is no split point in SP. Proof) Consider a query point q on subpath SP. Aording to Lemma, R q (S SP O SP E SP ) If S SP is equal to E SP and O SP is a subset of S SP, the union of S SP, O SP, and E SP has k elements. Therefore, for any point q SP, R q is fored to have the same k NNs. That is, there is no split point in SP. Conversely, if there is no split point in SP, R q has the same k NNs for all points on SP. That is, the union of S SP, O SP, and E SP has k elements. To satisfy this ondition, S SP equals E SP and O SP is a subset of S SP. Lemmas 3 and 4 are expeted to be more effetive when objets are populated sparsely in the network. This is beause the distanes among objets in a low density data set are longer than those in a high density data set. Unlike NN queries, CNN queries require the exeution of subsequent NN queries at adjaent loations on the query path. The k NNs obtained from s SP an be used as initial andidate NNs at e SP. Let o i ( i k) be the i-th NN at s SP. Suppose that d(s SP, o i ) is speified. Then, d(e SP, o i ) is determined as follows: If o i is on the subpath SP, d(e SP, o i ) is L SP d(s SP, o i ). If the shortest path from s SP to o i goes through e SP, d(e SP, o i ) is d(s SP, o i ) L SP. Otherwise, d(e SP, o i ) is d(s SP, o i ) + L SP. Other andidates an be found by expanding from e SP Figure 9 presents the Step 2 of the CNN searh algorithm in Figure 8 whih determines valid intervals for a subpath. The inputs to the Step 2 of the CNN searh algorithm are k and subpath, where the names are self desriptive. Algorithms for Filter Tuples and Find Split Points are desribed in detail in Figures and 2, respetively. The result R in this algorithm is the set of (obj, x, y) tuples where obj is a qualifying objet for the ontinuous searh. Then, x and y are alulated to determine the points of a hart suh as the one shown in Figure 6 aording to the following onditions: pro CNN Searh (k, subpath SP) /* k is the number of NNs requested and SP is a query subpath */ begin. /* R keeps objets satisfying the NN query */ 2. /* Step 2.. sanning the subpath */ 3. O SP := San Subpath (SP) 4. R := O SP /* Step 2.2. issuing two NN queries */ 7. S SP := NN Searh (s SP, k, kth dist) 8. /* Lemma 3 */ 9. if d(s SP, o k+ ) d(s SP, o k ) 2 L SP then. return /* no split point is on SP */. else 2. E SP := NN Searh (e SP, k, kth dist) 3. R := R S SP E SP /* Lemma 4 */ 6. if S SP = E SP and O SP E SP then 7. return /* no split point is on SP */ /* Step 2.3. filtering tuples */ 2. Filter Tuples (k, R) /* Step 2.4. finding split points on SP */ 23. Find Split Points (k, R) end Figure 9: CNN searh algorithm for the subpath. If obj is an objet on a given subpath, x := d(s SP, obj) and y :=. 2. If obj is an objet satisfying the query prediate at s SP, x := and y := d(s SP, obj). 3. If obj is an objet satisfying the query prediate at e SP, x := L SP and y := d(e SP, obj). In the following, we provide further details of the four steps of the CNN searh algorithm for a subpath. Step 2.. Sanning Subpath from s SP toward e SPIn the first step, the algorithm retrieves the objets on the subpath and adds their orresponding tuples in the form of (o i, x oi, ) to R where o i is an objet on the query path, and x oi := d(s SP, obj). Step 2.2. Issuing two NN Queries at s SP and e SPIn the seond step, the CNN searh algorithm issues two NN queries at s SP and e SP on the subpath. This step is onduted using the NN searh algorithm of Figure 2 and Lemmas 3 and 4. In line 3, the k NNs obtained at s SP and e SP are added to R. In the ase of the k NNs obtained at s SP, a tuple 87

8 (o s,, d(s SP, o s )) for o s is added to R. For the k NNs obtained at e SP, a tuple (o e, L SP, d(e SP, o e )) for o e is added to R. The k NNs obtained from s SP are used as andidate NNs of a urrent NN query. Note that e SP of the urrent subpath orresponds to s SP of the next subpath. If S SP is equal to E SP and O SP is a subset of E SP, the subpath is simply a valid interval and it has the same k NNs by Lemma 4. Step 2.3. Filtering Tuples In the third step, among the tuples obtained in the first and seond steps, some tuples with the same objet in R are removed using the over relationship. y 2 y d(q,obj) (,) x 2 t 2 = (obj, x 2, y 2 ) x t = (obj, x, y ) Figure : t overs t 2 iff y 2 x 2 x + y For two tuples t and t 2, let d(q, t.obj) and d(q, t 2.obj) be the network distanes from q to the objet obj in t and the objet obj in t 2, respetively. However, note t.obj and t 2.obj denote the same objet. Definition 4. For two tuples t and t 2 ontaining the same objet obj, t is said to over t 2 if d(q, t.obj) d(q, t 2.obj). For two tuples t = (obj, x, y ) and t 2 = (obj, x 2, y 2 ) in R, t overs t 2 if and only if y 2 x 2 x + y. Using the seond onept in Setion 4., the network distane from q to t.obj is d(q, t.obj) = x x + y and the distane to t 2.obj is d(q, t 2.obj) = x x 2 + y 2 as shown in Figure. To satisfy the over ondition, a point (x 2, y 2 ) should satisfy the inequality y x x +y as shown in Figure. That is, y 2 x 2 x + y. Sine d(q, t.obj) and d(q, t 2.obj) denote the different network distanes from q to obj, if d(q, t.obj) d(q, t 2.obj), the tuple t 2 is found to be unneessary and removed. Figure presents the algorithm whih filters redundant tuples from R. Step 2.4. Dividing the subpath into valid intervals In the fourth step, we first divide the subpath into valid intervals. This is based on the divide and onquer method. Next, to retrieve the k NNs for eah valid interval, all we have to do is to simply determine the k smallest d(q, t.obj) values at a point in the orresponding interval, where d(q, t.obj) denotes the x pro Filter Tuples (k, R) begin. while there are distint tuples t i, t j R do 2. if t i.obj = t j.obj and d(q, t i.obj) d(q, t j.obj) then 3. R := R {t j } 4. end-while end Figure : Filter Tuples algorithm network distane from q to obj in tuple t. Lemma shows that query result R q at any point q SP is a subset of the union of O SP, S SP, and E SP. Consequently, valid intervals of SP an be determined using objets whih belong to O SP, S SP, and E SP. To divide a subpath into valid intervals, the algorithm omputes all ross points of line segments drawn for d(q, t.obj)s where t.obj belongs to the union of O SP, S SP, and E SP. Then, it adds their x values to X, the set of x values for the ross points. The x values in X are sorted in asending order. Eah pair of neighboring values in X, x k and x k+, is used to identify a valid interval. Note that k NN queries are not issued to retrieve the k NNs for eah pair of neighboring values. In eah valid interval, the k NNs with the k smallest d(q, t.obj)s are the query result. If adjaent valid intervals have the same query result, they are merged into a single valid interval. Figure 2 presents the algorithm whih determines the split points on the subpath. Example : For an effetive explanation, we proeed with an exemplary CNN query whih is for stati objets a to e on the road network of Figure 3, ontinuously display the two losest objets from any position on the query path = {n 3, n 5, n 7, n 8 }. Step First, we divide the entire query path into multiple subpaths on the basis of intersetion points of the query path. Then, we obtain two subpaths SP = {n 3, n 5, n 7 } and SP 2 = {n 7, n 8 }. Step 2 We determine the valid intervals for the two subpaths. For simpliity, we fous on omputing valid intervals for SP = {n 3, n 5, n 7 }. Step 2. We san subpath SP to ollet the objets on SP. Let O SP be the set of (obj, x, y) tuples ontaining the objets on SP, where obj, x, and y are defined previously. Then, O SP = {(, 2, )}. This means that the distane to from the start point n 3 of SP is 2. Step 2.2 Let S SP and E SP be the sets of (obj, x, y) tuples for the objets satisfying the query ondition at the start point and the end point of subpath SP respetively, where obj, x, and y are defined previously. Two NN queries are exeuted at the start point n 3 and the end point n 7 of SP. Then, S SP = {(a,, ), (b,, )} and E SP = {(e, 4, ), (, 4, 2)}. Note that in E SP, the tuple (, 4, 2) may be replaed with 872

9 pro Find Split Points (k, R) begin. while there are distint tuples t i, t j R do 2. (x +, y + ) := Compute Cross Points(t i, t j ) 3. /* let X be the set of x values of ross points */ 4. X := X {x + } 5. end-while R SP = /* R SP is a set of (I, R I ) tuples */ 8. /* obtain k NNs for eah valid interval */ 9. /* let x i and x i+ be two adjaent x values. and R [a,b] be the set of k NNs in [a, b] */. for x i, x i+ (x i, x i+ X and x i x i+ ) do 2. {([x i, x i+ ], R [xi,x i+])} 3. := Compute k NN Objets (R, [x i, x i+ ]) 4. R SP = R SP {([x i, x i+ ], R [xi,x i+])} 5. end-for while adjaent intervals have the same k NNs do 8. if R [xi,x j] = R [xj,x k ] then 9. R [xi,x k ] := R [xi,x j] 2. R SP = R SP {([x i, x j ], R [xi,x j]), ([x j, x k ], R [xj,x k ])} 2. R SP = R SP {([x i, x k ], R [xi,x k ])} 22. end-while 23. return R SP end d(q,obj) e a,b (,) Figure 4: Plotting tuples in R on the hart Step 2.3 Some tuples in Figure 4 are removed by using the over relationship. Figure 5 shows the result after removing a redundant tuple {(, 4, 2)} in Figure 4. d(q,obj) e a,b (,) x x Figure 2: Find Split Points algorithm n n 4 n 2 n 3 n 5 n 8 n 6 b 2 a 2 e 2 n 7 d Figure 3: Example road network the tuple (d, 4, 2) sine d(n 7, ) is equal to d(n 7, d) and objet d an also be ontained in the query result at n 7 instead of objet. As a result, R beomes the union of O SP, S SP, and E SP as follows: R = {(obj, x, y) (,2,),(a,,),(b,,),(,4,2),(e,4,)}. Figure 4 shows the result of mapping 5 tuples in R on 5 points where the seond and third attributes of tuples are used as x and y oordinates of the points on a 2-dimensional hart. The x-axis represents the movement length of q from the start position n 3 of the CNN searh. The y-axis represents the network distane from q on the query path to all objets a, b,, d, and e in R. This an be explained by the seond basi onept of Setion 4.. For instane, when x =, d(q, a) = d(q, b) =, d(q, ) = 2, d(q, ) = 6 and d(q, e) = 5, when x = 2, d(q, a) = d(q, b) = 3, d(q, ) =, d(q, ) = 4 and d(q, e) = 3, et. n 9 Figure 5: Filtering tuples Step 2.4 As stated previously, a tuple (obj, x, y ) on the hart gives the network distane from q to obj as follows: d(q, obj) = x x + y for x [, 4] where 4 is the length of the subpath. In Figure 5, sine d(q, a) = d(q, b) = x +, d(q, ) = x 2, and d(q, e) = x 4 +, Two NNs are a and b when x =, two NNs are and e when x = 3, and so forth. Therefore, to find two NNs at any point on the query path, we simply examine the hart to identify the k smallest d(q, obj) values at the orresponding point. This is similar to a well-known skyline problem [2] suh as drawing the skyline of a ity given the loations of the buildings in the ity. The first skyline to be drawn with the smallest d(q, obj) values at eah point beomes the set CNN st of the losest neighbors from any point on the query path. That is, CNN st = {(I, R I ) ([, 2 ], {a or b)}), ([ 2, 3 2 ], {}), ([3 2, 4], {e})}. Based on the divide and onquer method, the subpath is broken into valid intervals whih are determined by ross points of line segments on the hart. Cross points are easily obtained using linear equations. As shown in Figure 6, the query path is divided into 4 valid intervals as follows: I =[, 2 ], I 2=[ 2, 2], I 3=[2, 3 2 ], and I 4=[3 2, 4]. In eah valid interval, we just find the k objets with the k smallest d(q, obj) values. For example, for I = [, 2 ], a and b are the 2 nearest neighbors and their network distanes from q are d(q, a) = d(q, b) = x +. Similarly, for I 2 = [ 2,2], 873

10 and a are the two NNs and their network distanes are d(q, ) = x 2 and d(q, a) = x +. For all valid intervals, their query results are shown in Figure 7(a) and the final CNN searh result is shown in Figure 7(b) sine I 3 and I 4 are merged into I 3 = [2, 4] sine they all have the same qualifying objets and e. Hene, R SP = {(I, R I ) ([, 2 ], {a, b}), ([ 2, 2], {a, }), ([2, 4], {, e})}. d(q,obj) e a,b (,) I I 2 I 3 I 4 Figure 6: Dividing the query path into valid intervals I R I I =[, 2 ] {a, b} I 2 =[ 2, 2] {, a} I 3 =[2, 3 2 ] {, e} I 4 =[3 2, 4] {, e} (a) Initial result I R I I =[, 2 ] {a, b} I 2 =[ 2, 2] {a, } =[2, 4] {, e} I 3 x (b) Final result Figure 7: CNN searh result for SP = {n 3, n 5, n 7 } Step 3 In Step 2, we obtained the CNN query result for SP = {n 3, n 5, n 7 }. That is, R SP = {(I, R I ) ([, 2 ], {a, b}), ([ 2, 2], {a, }), ([2, 4], {, e})}. In the same way, if we ompute the CNN query result for SP 2 = {n 7, n 8 }, R SP2 = {(I, R I ) ([4, 7], {d, e})}. In Step 3, we simply merge R SP and R SP2 for SP and SP 2, respetively, in order to the final query result for the entire query path P = {n 3, n 5, n 7, n 8 }. Consequently, R P = {(I, R I ) ([, 2 ], {a, b}), ([ 2, 2], {a, }), ([2, 4], {, e}), ([4, 7], {d, e})}. 5 Performane Study In Setion 5., we experimentally ompare our algorithms and other algorithms for NN queries in terms of I/O ost using a system running Windows on a 2.7 GHz proessor and 52 MB memory. In Setion 5.2, we explore the performane of CNN queries in terms of disk I/O and exeution time using the same system. In most ases, the osts of NN queries are disk I/O bound and the omputational ost is onsidered to be trivial in omparison to the I/O ost while the omputational ost is non-trivial for CNN queries. To represent a road network, we use real road data for Wisonsin in the United States from Tiger/Line data [2]. We set the page size to 4 KB and employ an LRU buffer of 6 MB whih aommodates approximately % of the road data. The road data desription is as follows: N =,469,468 and E =,594,867, where N is the set of nodes and E is the set of edges. The working spae is fixed to the two-dimensional unit spae [,) 2. For simpliity, we onsider bidiretional edges. However, this does not affet the interpretability and value of the results. The number of intersetion points is 223,569. Among them, we have hosen 64,748 (about 4.4% of total number of nodes) nodes as ondensing points where four or more edges meet. Eah ondensing point maintains the NN list whih onsists of preomputed NNs. Naturally, if the size of the preomputed NN list is larger, the query ost dereases while the maintenane ost inreases. Due to the high maintenane ost, we do not employ more than preomputed NNs. We use real-world data sets also from Tiger/Line data that represent shopping enters, ampgrounds, parks, shools, and lakes or ponds in Wisonsin. The sets ontain 78, 423, 54, 2979, and 577 objets, respetively. In order to simulate a large dataset suh as that of restaurants, whih is not available in Tiger/Line data, we make a omposite data set whih onsists of the points from the sets for shopping enters, ampgrounds, parks, shools, and lakes or ponds. Aording to the response of the Wisonsin State Government to our inquiry, there are,25 restaurants in Wisonsin. To investigate the performane of NN queries, we exeute workloads of 2 queries whose loations are randomly seleted on the network. Similarly, we also perform workloads of CNN queries whose initial loations are also randomly distributed in a road network and next edges are seleted with even probability. 5. NN queries We ompare query osts of for NN queries with those of VN 3 [7], whih is urrently regarded as the best approah for NN queries on the road network. Figure 8 shows the number of disk I/Os inurred by eah of the two methods for the value of k ranging from to 64. As the dataset ardinality inreases, the number of page aesses drops quikly. When the value of k is, VN 3 generates the result set with onstant ost, regardless of the density of objets. This is expeted beause the implementation of VN 3 is based on the Voronoi diagram whih is effiient to find the first NN. shows performane as good as VN 3 for NN queries whih require less than NNs beause it maintains the NN lists of preomputed NNs at the ondensing points. As shown in Figures 8(a) and 8(b), VN 3 is more effiient than sine requires a larger portion of network to be retrieved due to the very low density of objets. VN 3 also requires preomputed values to be retrieved from 874

11 the database and the number of these preomputed values required inreases for lower densities and larger values of k. However, suffers more than VN 3 from the lower densities due to the inrease in the searh spae. On the other hand, as shown in Figures 8(), 8(d), 8(e), and 8(f), outperforms VN 3 with the inrease in the density of objets. Sine both methods proess queries by using preomputed information, the performane gap between and VN 3 is losely assoiated with the density of objets. In the ase where objets (e.g., shopping enters) are distributed sparsely, VN 3 has an advantage over UNI- CONS sine the number of NVPs is as small as that of objets. Conversely, in the ase where objets (e.g., omposite data) are distributed densely, is in a stronger position than VN 3 due to the redued searh spae and the help of ondensing points. Additionally, in, objets sharing the same edge have the same searh key and therefore, fewer disk I/Os are required. As a whole, the experimental results indiate that exept for the partiular ases with lower objet densities, outperforms VN 3 by up to a fator of 3.5 and the performane differene between the two approahes inreases with the density of objets and the value of k. 5.2 CNN queries We onduted several experiments to ompare the performane of with its ompetitor, the approah presented in [8]. We alulated the number of page aesses and the required times for different values of k (a) Shopping enters (78) () Parks (54) (b) Campgrounds (423) (d) Shools (2979) VN VN (a) Shopping enters (78) (b) Campgrounds (423) (e) Lakes or ponds (577) (f) Composite data (99) VN3 () Parks (54) VN3 (e) Lakes or ponds (577) VN3 (d) Shools (2979) VN3 (f) Composite data (99) Figure 8: Performane omparison for NN queries Figure 9: Page aesses for CNN queries Figure 9 depits the number of page aesses for and approahes when the length of query paths is fixed to.5 and the value of k varies from to 64. As shown in Figure 9, always outperforms. When the objets of interest are distributed densely in the network (e.g., omposite data), the performane of is up to 5 times better than that of. The reason for this is that requires a large number of NN queries. Suh a trend is striking, partiularly when the value of k is large. The advantage of over is minimal when the objets of interest are distributed sparsely (e.g., shopping enters). In these ases, an filter out several adjaent nodes from the omputation of NNs. Based on Lemmas 3 and 4, an also avoid the exeution of NN queries from intersetion points on the query path. The use of ondensing points is very helpful for answering CNN queries. This is due to the fat that the CNN searh algorithms 875

12 of require the exeution of onseutive NN queries at intersetion points only regardless of the density of objets and the value of k. Exeution Time (Se) Exeution Time (Se) Exeution Time (Se) (a) Shopping enters (78) () Parks (54) (e) Lakes or ponds (577) Exeution Time (Se) Exeution Time (Se) Exeution Time (Se) (b) Campgrounds (423) (d) Shools (2979) (f) Composite data (99) Figure 2: Exeution time for CNN queries Figure 2 illustrates the query exeution times for and when the length of query paths is fixed to.5 and the value of k varies from to 64. Naturally, the exeution time of inreases greatly with the value of k. The inrease in the value of k generates more split points and leads to the exeution of a large number of (k+) NN queries on the query path. However, issues NN queries at intersetion points on the query path regardless of the density of objets. The experiments for traveling paths of length between. and. show similar trends. The experimental results onfirm that is superior to and it is optimized for CNN queries. 6 Conlusions In this paper, we developed new ontinuous searh algorithms whih answer NN queries at any point of a given path. We also verified that our ontinuous searh algorithms require a small number of stati queries in produing the ontinuous searh result. Experimental results with TIGER/Line data demonstrated that outperforms its ompetitors for various numbers of NNs and data sets. 7 Aknowledgments This researh was supported in part by the Ageny for Defense Development, Korea, through the Image Information Researh Center at Korea Advaned Institute of Siene and Tehnology, and in part by the Ministry of Information and Communiations, Korea, under the Information Tehnology Researh Center (ITRC) Support Program. Referenes [] N. Bekmann, H. Kriegel, R. Shneider, and B. Seeger: The R*-Tree: An Effiient and Robust Aess Method for Points and Retangles. In Pro. of the ACM SIGMOD, 99. [2] S. Borzsonyi, D. Kossmann, and K. Stoker: The Skyline Operator. In Pro. of ICDE, 2. [3] E. Dijkstra: A Note on Two Problems in Connetion with Graphs. Numerihe Mathematik, Volume(), 959. [4] J. Feng and T. Watanabe: A Fast Method for Continuous Nearest Target Objets Query on Road Network. In Pro. of Virtual Systems and Multi- Media, 22. [5] J. Feng and T. Watanabe: Searh of Continuous Nearest Target Objets along Route on Large Hierarhial Road Network. In Pro. of the Data Engineering Workshop, 23. [6] C. Jensen, J. Kolarvr, T. Pedersen, and I. Timko: Nearest neighbor queries in road networks. In Pro. of ACM GIS, 23. [7] M. Kolahdouzan and C. Shahabi: Voronoi-Based K Nearest Neighbor Searh for Spatial Network Databases. In Pro. of VLDB, 24. [8] M. Kolahdouzan and C. Shahabi: Continuous K-Nearest Neighbor Queries in Spatial Network Databases. In Pro. of STDBM, 24. [9] D. Papadias, J. Zhang, N. Mamoulis, and Y. Tao: Query Proessing in Spatial Network Databases. In Pro. of VLDB, 23. [] C. Shahabi, M. Kolahdouzan, and M. Sharifzadeh: A road network embedding tehnique for k-nearest neighbor searh in moving objet databases. In Pro. of ACM GIS, 22. [] S. Shekhar and J. Yoo: Proessing in-route nearest neighbor queries: a omparison of alternative approahes. In Pro. of ACM GIS, 23. [2] US Bureau of the Census: Tehnial Doumentation. TIGER/Line Files