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USC Computer Science Technical Reports, no. 843 (2005)

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USC Computer Science Technical Reports, no. 843 (2005)

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Content Location-based Spatial Queries with Data Sharing in Mobile Environments Wei-Shinn Ku, Roger Zimmermann, and Chi-Ngai Wan Computer Science Department University of Southern California Los Angeles, California 90089 [wku, rzimmerm, cwan]@usc.edu July 6, 2005 Abstract Mobile clients feature increasingly sophisticated wireless networking support that enables real- time information exchange with remote databases. Location-dependent queries, such as determining the proximity of stationary objects (e.g., restaurants and gas stations) are an important class of inquiries. We present a novel approach to support nearest-neighbor queries from mobile hosts by leveraging the sharing capabilities of wireless ad-hoc networks. We illustrate how previous query results cached in the local storage of neighboring mobile peers can be leveraged to either fully or partiallycomputeandverifyspatialqueriesatalocalhost. Thefeasibilityandappealofourtechnique is illustrated through extensive simulation results that indicate a considerable reduction of the query load on the remote database. Furthermore, the scalability of our approach is excellent because a higher density of mobile hosts increases its effectiveness. 1 Introduction Location-basedqueriesareofinterestinanumberofapplications, forexample, geographicalinformation systems. An example query might be “find the nearest gas station” or “find the three nearest Italian restaurants.” Increasingly such queries are issued from mobile clients. In this study we propose an approach that leverages short-range, ad-hoc networks to share informa- tion in a peer-to-peer (P2P) manner among mobile clients to answer location-based nearest neighbor queries. Theefficiency of our approach is derived from theobservation that the results of spatial queries often exhibit spatial locality. For example, if two mobile hosts are close to each other, the result sets of their kNN queries for a specific object type may overlap significantly. Through mobile cooperative caching [3] of the result sets, query results can be efficiently shared among mobile clients. Figure1showsanexample. AttimeT themobilequerypointQcanestablishcontact withtwoother mobile hosts within its communication range: P ′ 1 and P ′ 2 . Both of these clients in the past executed a 1NN query for the nearest gas station when they were located at P 1 and P 2 , respectively 1 . The results that they obtained and cached were <n2,P 1 > and <n4,P 2 >. These two tuples represent candidate solutions for Q’s own 1NN query. Through a local verification process Q can determine whether one of the solutions obtained from its neighbors is indeed its own nearest gas station. Note that the current 1 In our notation we use the object identifier to represent its position coordinates. 1 1NN Candidates 1NN 1NN Communication range Q P ′ 1 P 1 P ′ 2 P 2 n1 n2 n3 n4 Figure 1: Example of 1NN peer-to-peer result sharing. location of the neighboring hosts, P ′ 1 and P ′ 2 , has no specific significance, as long as they are within the communication range of Q. Thecontributionsofthisstudyareasfollows. Inourmethodologywefirstidentifyasetofcharacter- istics that enable the development of effective sharing methods. We then introduce a set of algorithms that aid in the decision process within this distributed environment to verify whether the data items received from neighboring clients provide a complete, partial, or irrelevant answer to the posed query. Our initial method verifies results from a single neighbor, and we then extend it to work with multiple neighboring clients. Finally, through extensive simulation experiments we explore the benefits of our approach under different parameter sets (e.g., mobile host density, wireless transmission range). The rest of this paper is organized as follows. Section 2 introduces the related work for processing NN andkNN queries. Our approach is detailed in Section 3 and the experimental results obtained from our simulation model are contained in Section 4. Finally, Section 5 concludes the paper and outlines future research directions. 2 Related Work The existing work relevant to our approach can broadly be classified into two areas, namely nearest neighbor query processing and cache management in mobile environments. Nearest Neighbor Query Processing One of the most fundamental operations performed on spa- tial data sets is to find the nearest neighbors of a query object – either one (1NN) or a specific number k (kNN). R-trees [9] and their derivatives have been a prevalent method to index spatial data and increase query performance. To find nearest neighbors, Roussopoulos et al. [15] proposed a branch-and-bound algorithm that searches an R-tree in a depth-first manner. The best-first NN algorithm proposed by G´ ısli et al. [10] retains a heap with the entries of the nodes visited so far and it always expands the first entry in the heap. The algorithm is optimal since it visits only the minimally necessary nodes and it reports nearest neighbors in ascending order according to their distance to the query point. It can be applied without a-priori knowledge of the number k of queried nearest neighbors. Both the depth-first and best-first algorithms are designed for stationary objects and query points. They may be used when moving objects infrequently issue nearest neighbor queries (single-step search). Withtheemergenceofmobiledevicesattention hasfocusedontheproblemofcontinuouslyfindingk nearest neighbors for moving query points (k-NNMP). A naive approach might beto continuously issue kNN queries along the route of a moving object (multi-step search). This solution results in repeated server accesses and nearest neighbor computations and is therefore inefficient. One method to reduce 2 the computational complexity is to sample the trajectory instead of treating it as a continuous curve. Song and Roussopoulos [18] utilize partial results from queries launched at previous sampled positions to reduce the accesses to the server. Tao et al. [19] investigated the problem of finding the continuous nearest neighbor of every point along a segment, assuming that the trajectory is known and can be described in closed form with a function of reasonable complexity. Their approach pre-computes so-called split-points that divide the objectpathintosegmentsduringwhichthenearestneighborresultsetsremainunchanged. Thismethod is efficient, but requires knowledge of the object path (or at least an approximation). In the early work, nearest neighbor searches were based on the Euclidean distance between the query object and the sites of interest. However, in many applications objects cannot move freely in space but are constrained by a network (e.g., cars on roads, trains on tracks). Therefore, in a realistic environment the nearest neighbor computation must be based on the spatial network distance, which is more expensive to compute. A number of techniques have been proposed to manage the complexity of this problem [12, 13, 17]. Cache Management in Mobile Environments Caching is a key technique to improve data re- trieval performance in widely distributed environments. Leveraging the combined resources of several cooperating caches has been proposed to improve file system [5] and Web performance [20]. In conven- tional mobile environments, wireless connections are treated as extensions of the wired infrastructure. Hence, mobile clients retrieve information from database servers via intermediate base-stations. With the increasing deployment of new peer-to-peer wireless communication technologies (e.g., IEEE 802.11x and Bluetooth) there exists a new information sharing alternative known as peer-to-peer cooperative caching. With this technique mobile hosts communicate with neighboring peers in an ad-hoc manner to share information rather than having to rely on the communication link to the remote information sources. Peer-to-peer cooperative caching can bring about several distinctive benefits to a mobile system: improving access latency, reducing server workload and alleviating point-to-point channel congestion. As a disadvantage, it may increase the communication overheads among mobile hosts. Recently, a number of techniques have been proposed to address caching in ad-hoc peer-to-peer networks. TheCOoperativeCAching(COCA)[3]schemeinvestigates theeffects ofclient activity levels, data replication, and cache size. The benefits of clustering mobile clients into groups are investigated in [4]. Semantic caching stores additional information such as querydescriptions with thedata items in the local cache. Themetadata is used to determine whether a new query is fully answerable from the cache. In that case no communication with the server is required. If the query can only partially be answered, then it may be trimmed and sent to the server. In either case, the database processing complexity and the amount of data transferred can be significantly reduced [14]. One form of cachable metadata is the data access path which may be used to redirect future accesses to nearby cache nodes [21]. Another possibility is to cache the index that supports the objects. The cached index enables the objects to be reused for common types of queries [11]. Addressing specifically the nearest neighbor search for stationary objects, Zhenget al. [22] proposed an index based on Voronoi diagrams in conjunction with a semantic cache to enhance the efficiency of the search. 3 Sharing Based Nearest Neighbor Queries The fundamental idea of our methodology is to leverage the cached results from previous queries at reachable mobile hosts for answering spatial queries at the local host. In some circumstances the query 3 results obtained from peer hosts can only provide a partial answer or no answer at all. To achieve scalability it is imperative that a mobile client can locally determine whether the result set from its neighbors provides a full, partial or no answer. As a novel component in our methodology we present a verification algorithm that can certify if a result object is part of the solution set. We term such an object certain. If the object is not guaranteed to be part of the result set, we call it uncertain. The first variant of our verification procedure validates object certainty from a single peer. We then extend the process to multiple peers. If no full set of certain objects can be retrieved from neighboring peers, the query is forwarded to a spatial database server including the acquired partial result. The database search efficiency can be improved by utilizing the partial query results from clients. Inthisstudyweusethek-nearestneighbor(kNN)searchasanexampleofspatialqueriesandexplain in detail how they are processed by cooperating mobile hosts. Specifically, in Section 3.1 we introduce theinfrastructurethatweassumeforourwork. Next, Section3.2presentsalgorithmsforverifyingquery results from neighboring peers and the exact ranking of verified Points Of Interest (POI). Section 3.3 explains how to use our metrics to decrease the server load for processing kNN queries. Section 3.4 illustrates the extension of our algorithms for solving spatial network kNN queries. 3.1 Assumed Infrastructure Figure2depictsouroperatingenvironmentwithtwomainentities: remotespatialdatabasesandwireless mobile hosts. We are considering mobile clients, such as cars, that are instrumented with a global positioning system (GPS) for continuous position information. Furthermore, we assume that two-tiers of wireless connections are available on future automobiles. Traditional, cellular-based networks (such as utilized by the OnStar service) allow medium range connections to base-stations that interface with the wired Internet infrastructure. A second type of short-range networks allow ad-hoc connections with neighboring mobile clients. Technologies that enable short range communication include, for example, IEEE 802.11x. Benefiting from the power capacities of vehicles, we assume that each mobile host has a significant transmission range and virtually unlimited lifetime. The architecture can also support hand-held mobile devices. However, then power consumption becomes an additional parameter which we are not currently considering. Data Station Mobile Host Mobile Host Mobile Host Mobile Host Mobile Host Service Area of a Data Station Spatial Database Mobile Host Transmission Range Mobile Host Mobile Host Mobile Host Peer-to-Peer Channel Point-to-Point Channel Figure 2: Example system environment. 4 3.2 Euclidean Distance Nearest Neighbor Queries With our assumed infrastructure, a mobile host Q has two choices to find the solution to a kNN query. First, itcansendthequerydirectly tothedatabaseserverforexecution. However, sincespatialsearches are expensive to compute, the server may become a bottleneck when the number of mobile hosts grows. An alternative solution is to collect NN data from peers and harvest these existing query results for completingQ’squery. WetermthelatterapproachaSharing-based Euclidean distance Nearest Neighbor (SENN) query. We will subsequently extend the technique to make use of the network distance. We propose two approaches to process NN information obtained from peers for fulfilling NN queries of a mobile host Q. The single peer NN verification, also called kNN single , attempts to verify k certain objects by sequentially verifying the returned NN data from each peer. If this is not possible, then the multiple peer NN verification process, kNN multiple , attempts to complete the verification process with several peers. 3.2.1 Single Peer NN Verification: kNN single The objective of the kNN single method is to verify if the point of interest data returned from each peer can be valid nearest neighbors of a mobile host Q. In order to verify if a returned POI object n i is one of the top k nearest neighbors of a mobile host Q, we utilize the spatial relationship between mobile hosts and their POIs as follows. Lemma 3.1 Let Q and P 1 be two mobile hosts, and let P 1 have k nearest neighbors, n 1 ,n 2 ,...,n k which are sorted in ascending order according to their distance to P 1 . If Dist(Q,n i )+δ > Dist(P 1 ,n k ) then n i cannot be verified as one of the top k-nearest neighbors of Q. In the above lemma, Dist(Q, n i ) is the Euclidean distance between Q and n i , δ is the Euclidean distance between Q and P 1 , and Dist(P 1 , n k ) is the Euclidean distance between P 1 and its cached farthest nearest neighbor n k . An illustration of Lemma 3.1 is shown in Figure 3. The nearest neighbor n 2 of mobile hostP 1 – which is a peer of mobile host Q – cannot be verified as one of the top k-nearest neighbors of Q. An uncertain area exists in the circle which takes Q as its center point and Dist(Q, n 2 ) as its radius. A point of interest n i may be located in that area with Dist(Q, n i )<Dist(Q, n 2 ). Therefore, we can only classify n 2 as an uncertain nearest neighbor. Lemma 3.2 Let Q and P 1 be two mobile hosts, and let P 1 have k nearest neighbors, n 1 ,n 2 ,...,n k which are sorted in ascending order according to their distance to P 1 . For any nearest neighbor n i of P 1 , if Dist(Q, n i ) + δ ≤ Dist(P 1 , n k ) then n i is one of the top k-nearest neighbors of Q. Proof Assume n i / ∈ kNN of Q. Then, there exist m 1 ,m 2 ,...,m k ∈ kNN of Q such that ∀m j ∈ {m 1 ,m 2 ,...,m k }, Dist(Q,m k ) < Dist(Q,n i ). We can identify a point M located at the intersection of the extension of the line from P 1 to Q and the circumference of the circle with center Q and radius Dist(Q,n i ) (identified as circle C Q in Figure 4). Then, ∀m i ∈ kNN of Q: Dist(P 1 ,m i )< Dist(P 1 ,M) (1) Recall that we assume the following inequality holds: Dist(Q,n i )+δ≤ Dist(P 1 ,n k ) (2) Because the circle C Q (with center Q and radius Dist(Q,M)) is fully covered by the circle C P1 (with center P 1 and radius Dist(P 1 ,M)) it follows that Dist(Q,n i )+δ =Dist(Q,M)+δ = Dist(P 1 ,M) (3) 5 Q P1 n1 n2 n3 Dist (Q, n2) + > Dist (P1, n3) Dist (Q, n2) Uncertain Area Figure 3: Verification of a point of interest with an uncertain area. ByEquations1,2and3,∀m i ∈kNNofQ,Dist(P 1 ,m i )< Dist(P 1 ,n k ). Thusn k / ∈kNNofP 1 . However, this contradicts the assumption that n k ∈ kNN of P 1 . Therefore, n i must be one of the top k-nearest neighbors of Q. Q P1 n1 n2 n3 Dist (Q, n2) + < Dist (P1, n3) M CQ CP1 Figure 4: Verification of a certain point of interest. Anillustration of Lemma3.2 isshownin Figure4. Thenearest neighborn 2 ofmobile hostP 1 , which is a peer of mobile host Q, can be verified as the nearest neighbors of Q and is termed a certain nearest neighbor. Because theEuclidean distancebetweenn 2 andQ plus theEuclidean distance betweenQ and P 1 , δ, is no greater than the Euclidean distance between P 1 and its presently cached farthest nearest neighbor, n 3 . We observe quite intuitively that the cached query locations which are located closer to the query host Q have a higher likelihood (but are not guaranteed) to be able to provide useful (i.e., adjacent) 6 points of interest. We therefore will use Heuristic 3.3 to guide the order in which the cached NN query results returned from neighboring peers are processed. Heuristic 3.3 LetNN set be a result set of nearest neighbor objects and their query points returned from the peers of the query mobile hostQ. SortingNN set in ascending order according to the distance between eachquery point tothe location ofQmay save computation time during processing by asubsequentsearch algorithm. Lemma 3.4 ∀ x, y, and Q, if Dist(x,Q) < Dist(y,Q) and if y ∈ kNN of Q, then x ∈ kNN of Q and Rank(x,Q) < Rank(y,Q). Since Dist(x,Q) < Dist(y,Q), we know that x is closer to Q than y. According to the definition of nearest neighbor and y ∈ kNN of Q, we conclude that x ∈ kNN of Q and Rank(x,Q) < Rank(y,Q). Lemma 3.5 ∀ x, y, and Q, if Dist(x,Q) 6= Dist(y,Q), then x 6= y. This follows from the definition of Euclidean distance – since we have Dist(x,Q) 6= Dist(y,Q), x 6= y. Lemma 3.6 ∀P andQ, Dist(P,Q)=δ,∀n i ,n j ∈kNNofP, ifDist(P,n i )<Dist(P,n j )andDist(Q,n i ) + δ ≤ Dist(P,n j ), then Rank(n i ,Q) < k and n i ∈ kNN of Q. Proof According to Lemma 3.2 and the assumed conditions, n i must be one of the top k nearest neighbors of Q. Since we know that n j ∈ kNN of P, there are at most k−1 POIs within circle C P (with center P and radius Dist(P,n j )). Therefore, we can verify at most k−1 POIs for Q by utilizing n j and it follows that Rank(n i ,Q) < k. Lemma 3.7 ∀ P and Q, Dist(P,Q) = δ. Given n 1 ,n 2 ,...,n k ∈ kNN of P and n i sorted in ascending order to their distance to Q: Dist(Q,n 1 ) < Dist(Q,n 2 ) < ... < Dist(Q,n k ). If Dist(Q,n i ) + δ ≤ Dist(P,n j ) and Dist(P,n i ) < Dist(P,n j ), then n i is the i th nearest neighbor of Q and Rank(n i ,Q) =i. Proof Since Dist(Q,n i )+δ ≤ Dist(P,n j ) and Dist(P,n i ) < Dist(P,n j ), by Lemma 3.6 we observe: Rank(n i ,Q)<k (4) Because we are given that Dist(Q,n 1 )< Dist(Q,n 2 )<...< Dist(Q,n k ), we deduce that there exist i−1 points (n 1 ,n 2 ,...,n i−1 ) which are closer to Q than n i . Therefore, we derive: Rank(n i ,Q)≥i (5) From Equations (4) and (5), we conclude: i≤ Rank(n i ,Q)<k (6) In the next step, we prove Rank(n i ,Q) = i by contradiction. Suppose Rank(n i ,Q) > i, then there must exist a NN n p , such that n p / ∈ {n 1 ,n 2 ,...,n i−1 } and Dist(n p ,Q) < Dist(n i ,Q). Therefore, ∀n x ∈{n i ,n i+1 ,...,n k }, Dist(n p ,Q) < Dist(n x ,Q). By Lemma 3.5, we observe n p / ∈{n i ,n i+1 ,...,n k }. Consequently, n p / ∈ {n 1 ,n 2 ,...,n k } and n p / ∈ kNN of P. As shown in Figure 5, n p is inside the circle 7 with Q as its center point and Dist(Q,n i ) as its radius. According to the characteristics of triangles we derive: Dist(n p ,P)<Dist(n p ,Q)+Dist(Q,P) (7) and Dist(n p ,Q)+Dist(Q,P)< Dist(n i ,Q)+Dist(Q,P) (8) Sincewe aregiven Dist(n i ,Q)+Dist(Q,P)< Dist(P,n j ) andEquations (7) and(8), itfollows that: Dist(n p ,P)< Dist(P,n j ) (9) By Lemma 3.4 and Equation (9) we deduce that n p ∈ kNN of P, which contradicts what we concluded before, namely that n p / ∈ {n 1 ,n 2 ,...,n k } and n p / ∈ kNN of P. Therefore, the assumption Rank(n i ,Q) >i cannot hold and it follows that Rank(n i ,Q)≤i. Since we know that Rank(n i ,Q)≥i from Equation (5), we conclude that Rank(n i ,Q) =i. Q P np ni nj Figure 5: The spatial relationships between n i , n p , Q, and P. The kNN single method maintains a heap H with the entries of certain and uncertain points of interest discovered so far (illustrated in Table 1). The size of H is determined by the total number of queried interest objects Q k . Initially H is empty and the kNN single method processes the sorted NN P in sequence. The heap H is updated according to the distance from the location of Q to a POI object and its certainty. If there exist uncertain nearest neighbor objects in H, a newly discovered certain NN object will replace an uncertain object and H maintains the certain objects in an ascending order of their Euclidean distance to Q. Uncertain objects exist inH only if the number of certain objects is less than Q k . These uncertain objects are also stored in ascending distance order. Consider the following example to illustrate the operation of kNN single . Figure 6 illustrates the location of Q and its two closest mobile hosts, P 1 and P 2 . Because the cached query location of mobile host P 1 is the closest to Q, kNN single starts its NN verification process from P 1 . The single peer NN verification rule follows Lemma 3.2. Assuming that Q searches for four nearest neighbors, then after processing P 1 and P 2 the content of the heap H is as shown in Table 1. Based on the set NN P of both peer P 1 and P 2 , Q can retrieve two certain NNs, n 2−P1 and n 1−P1 , and two uncertain NNs, n 3−P1 and n 3−P2 . 8 Certain/Uncertain C C UC UC Points of Interest n 2−P1 n 1−P1 n 3−P1 n 3−P2 Distance to Q √ 2 √ 3 √ 5 √ 8 Table 1: The data structure of the heap H. kNN single is executed iteratively with each peer in thenearest neighbor data setNN P . Ifk elements inH are certain, thekNN query is fulfilled andH will remember the topk NN in sequence. Otherwise, we need to perform kNN multiple to expand the search space to include more candidate certain interest objects. Q P1 n1-P1 n2-P1 n3-P1 P2 n1-P2 n2-P2 n3-P2 Q P1 n1-P1 n2-P1 n3-P1 Fig. 6a. Mobile host Q retrieves two cer- tain nearest neighbors from the NN set of peer P 1 . Fig. 6b. Mobile host Q can only retrieve uncertain nearest neighbors from peer P 2 . Figure 6: The mobile host Q and its two closest peers, P 1 and P 2 . 3.2.2 Multiple Peer NN Verification: kNN multiple Under some conditions the kNN single method may not be able to verify all k nearest neighbors. There- fore, we extend the verification process to include results from multiple peers simultaneously. Figure 7 demonstrates an example in which a point of interest (n 2−P3 ) cannot be verified by the NN data set of a single peer; neither with peer P 3 nor with peer P 4 . The kNN multiple method combines the area of all the peers, each bounded by the outermost NN circle, into a certain region R c (the shaded area in Figure 7c). It is expensive to compute an exact solution for R c . Therefore, we adopt a polygonization technique that transforms all the certain area circles into polygons to closely approximate the certain area reported by each peer. After this transformation the polygons can be merged together as a certain region R c by performing the MapOverlay algorithm [6]. The kNN multiple verification technique is exe- cuted based on R c similarly to kNN single . Lemma 3.8 provides the rules for verifying nearest neighbors with multiple peers. Lemma 3.8 If the nearest neighbor data set NN P is composed of data from j peers, the certain region R c can be represented as: R c =P 1−area ∪P 2−area ∪···∪P j−area . For any interest object n i in R c , the distance between Q and n i is used as a radius to draw a circle C ni . If C ni is fully covered by R c , then n i is a certain NN of Q. 9 P3 n1-P3 n2-P3 Q n3-P3 CP3 P4 n2-P3 n2-P4 P4 n1-P4 Q CP4 P3 P3 n2-P3 n2-P4 P4 Q n3-P3 and n1-P4 n1-P3 CP3 CP4 Fig. 7a. Mobile hostQ cannot verify POI n 2−P3 as a certain NN with peer P 3 . Fig. 7b. Mobile hostQ cannot verify POI n 2−P3 as a certain NN with peer P 4 , either. Fig.7c. Aftermergingthecer- tain regions of peerP 3 andP 4 , POIn 2−P3 can beverified as a certain object. Figure7: An exampleof multiplepeersNNverification. Thepointofinterestn 2−P3 can only beverified as a NN of Q based on the region of both peer P 3 and peer P 4 . 3.3 Nearest Neighbor Query Pruning Bounds We assume that the spatial database server executes an efficient k-nearest neighbor search algorithm basedonR-treeindexing[10]forsolvingkNNqueries. TheNNsearchissupportedwithapriorityqueue containing the nodes visited so far. Initially the priority queue contains the entries of the R-tree root sorted according to their minimum distance (MINDIST) to the query point Q. In general most of the moving objects have executed either one or bothkNN single andkNN multiple processes before forwarding kNN search queries to the server. Hence, it is worthwhile to calculate branch expanding upper and lower bounds from the entries in heap H to speed up the NN search process at the server. The heap H is in one of six different states after a mobile host has executed both the kNN single and kNN muitiple mechanisms without retrieving k certain objects: • State 1: H is full and it contains both certain and uncertain entries. • State 2: H is full and it contains only uncertain entries. • State 3: H is not full and it contains both certain and uncertain entries. • State 4: H is not full and it contains only certain entries. • State 5: H is not full and it contains only uncertain entries. • State 6: H contains no entry. In State 1 there may exist some POIs which are closer to Q compared with the last element in H. Hence, we can consider the last entry of H as the final candidate nearest neighbor in the NN search and forward its distance attribute to the server as the branch expanding upper bound. In addition, the distanceattributeD ct ofthelastcertainentrycanbeanotherbound,thebranchexpandinglowerbound. Because we are certain about the POIs within the circle region C r with radius D ct and center point Q, the NN search algorithm executed in the server does not need to expand any minimum bounding rectangle which is completely covered by C r . Conversely, when H is full and contains just uncertain entries, we can infer only the upper bound (State 2). In States 3 and 4 after the mobile host performed the two algorithms, there have merely less than k interest objects been found. Therefore, we can only 10 inferthelower boundfromthedistanceattributeof thelast certain element inH. Inthelast twostates, H is not full and contains only uncertain entries or no entry at all. Consequently we cannot infer any bounds from them. To take advantage of the two new bounds for kNN queries, we slightly modified the kNN best- first search algorithm such that it calculates one more metric, the maximum distance (MAXDIST), for pruning R-tree branches. MAXDIST indicates which MBRs are totally covered in region C r and the algorithm does not need to expand them. Furthermore, we added two new MBR pruning strategies for the kNN search as follows: 1. Any MBR M with MAXDIST(Q, M) smaller than the branch expanding lower bound is pruned (downward pruning). 2. Any MBR M with MINDIST(Q, M) greater than the Euclidean distance from Q to the k th element in H is discarded (upward pruning). The complete algorithm of the sharing based kNN query is described in Algorithm 1. Algorithm 1 SENN: Sharing-based Euclidean distance Nearest Neighbor query (Q, k) 1: /* Q is the query mobile host */ 2: Query moving object peers within the communication range 3: Sort the peer query results NN P according to their last query locations 4: for each element e of NN P do 5: kNN single (Q, k) 6: end for 7: if a certain kNN set is found by kNN single then 8: return kNN set 9: else 10: kNN multiple (Q, k) 11: end if 12: if a certain kNN set is found by kNN muitiple then 13: return kNN set 14: end if 15: if the heap H is full and an uncertain kNN set is acceptable by the mobile host then 16: return the uncertain kNN set 17: else 18: /* The heap H is not full or an uncertain kNN set is not acceptable */ 19: Query the server with pruning upper bound and lower bound, if available 20: /* Forward the branch expanding upper bound and the branch expanding lower bound found by kNN muitiple */ 21: return the kNN set 22: end if 3.4 Spatial Network Nearest Neighbor Queries In the real world, mobile objects often move on pre-definednetworks (e.g., roads, railways, etc.). In this scenario, the spatial network distance provides a more exact estimation of the travel distance between two objects. Papadias et al. [13] have proposed a technique to solve spatial network nearest neighbor queries. However, to the best of our knowledge, sharing based spatial network nearest neighbor queries 11 Euclidean Distance ED (Q, n1) Network Distance ND(Q, n1) Q Network Distance ND(Q, n1) n1 Network Distance ND(Q, n1) Q Network Distance ND(Q, n2) n1 n2 Euclidean Distance ED(Q, n2) n3 Network Distance ND(Q, n2) Fig. 8a. The 1 st Euclidean NN. Fig. 8b.The 2 nd Euclidean NN. Figure 8: Nearest neighbor search in a spatial network environment with the IER algorithm. have not been studied before. Here, we extend our sharing based Euclidean distance nearest neighbor method to support applications in spatial network environments. Leveraging existing methods,weassumeadigitization processthatgenerates amodelinggraphfrom an input spatial network. The modeling graph contains three categories of graph nodes: the network junctions,thestart/endpointsofaroadsegment, andotherauxiliarypoints. Theshortestpathbetween two nodes can be computed with Dijkstra’s algorithm [7], which is leveraged as the basis for computing the network distance between any two arbitrary points. For two nodes i and j, we observe that their Euclidean distance ED(n i ,n j ) always provides a lower bound on their network distance ND(n i ,n j ). We refer to this fact as the Euclidean lower bound property. Papadias et al. proposed two algorithms for nearest neighbor queries in spatial network databases: the Incremental Euclidean Restriction (IER) and the Incremental Network Expansion (INE). Here we extend the IER algorithm to utilize cached NN query results in a P2P sharing infrastructure. IncrementalEuclideanRestrictionTheIERalgorithmisbasedonthemulti-stepkNNtechniques[8, 16]. To execute a nearest neighbor search for query point Q, IER first retrieves the Euclidean distance nearest neighbor n 1 of Q and computes the Euclidean distance ED(Q,n 1 ). Next it calculates the network distance from Q to n 1 , ND(Q,I1). Subsequently we can use Q as the center to draw two concentric circles with radii ED(Q,n 1 ) and ND(Q,n 1 ). Due to the Euclidean lower bound property, objects closer toQ thann 1 in the network must bewithin the network distance,ND(Q,n 1 ). Therefore, the search space becomes the ring area between the two circles as shown in Figure 8a. In the next iteration, the second closest object n 2 is retrieved (by Euclidean distance). Since in our example ND(Q,n 2 ) < ND(Q,n 1 ), n 2 becomes the current candidate for spatial network nearest neighbor and the search upper bound becomes ND(Q,n 2 ). This procedure is repeated until the next Euclidean nearest neighbor is located beyond the search region (as n 3 is in Figure 8b). The extension of IER to a kNN search is straightforward. In our P2P environment, we assume that each mobile host retains the data of the local spatial network modeling graph. A mobile host Q executes the SENN algorithm first to obtain k certain nearest neighbors (by Euclidean distance) and then calculates the network distance of the k objects based on its local spatial network modeling graph. The resulting objects are sorted in ascending order of their network distance to Q and the Euclidean distance of ND(Q,I k ) between Q and the k th object becomes the search upper bound, S bound . Next, Q retrieves the subsequent Euclidean distance nearest 12 Algorithm 2 SNNN: Sharing-based Network distance Nearest Neighbor query(Q, k) 1: /* Q is the query mobile host */ 2: Execute the Sharing-based Euclidean distance Nearest Neighbor (SENN) query algorithm for re- trieving k nearest neighbors {n 1 ,...,n k } 3: for each object n i do 4: Compute its network distance to Q 5: end for 6: Sort {n 1 ,...,n k } in ascending order according to their network distance to Q 7: Set ND(Q, n k ) as the upper search bound S bound 8: i = 1 9: repeat 10: n next = the object with the longest distance to Q which is returned by SENN(Q, k+i) 11: if ND(Q, n next )<ND(Q, n k ) then 12: /* The next Euclidean NN is closer than the k th NN */ 13: Replace n k with n next 14: Sort {n 1 ,...,n k } in ascending order according to their network distance to Q 15: Set S bound =ND(Q, n k ) 16: end if 17: i = i+1 18: until ED(Q, n next )>S bound neighborsincrementally fromitspeersorthespatialdatabaseserverandkeepsupdatingthek candidate spatial network NNs until the next Euclidean NN falls beyond the search region. This Sharing-based Network distance Nearest Neighbor (SNNN) algorithm is formalized in Algorithm 2. 4 Experimental Validation To evaluate the performance of our approach we have implemented our sharing based spatial query algorithms within a simulator. The main objective of our peer-to-peer design is to increase scalability in two dimensions. First, the server access workload can be reduced as queries are answered directly by peers. Second, for the remaining queries that must be sent to the server our technique diminishes the search overhead by providing pruning-bounds for the R-tree algorithm. Consequently, the focus of our simulation is on quantifying the server load variations as a function of two main parameters, the spatial query request rate (SQRR) and page access rate (PAR). SQRR is for quantifying how many percent of the total client spatial queries are required to be processed by the spatial database server and PAR is for evaluating server side memory (primary and secondary) access rate for a sequence of spatial queries. We have performed our experiments with both synthetic and real-world parameter sets. 4.1 Simulator Implementation Our simulator consists of two main modules, the mobile host module and the server module. The objective ofthemobilehostmoduleistogenerateandcontrolthemovements andquerylaunchpatterns ofallmobilehosts. Eachmobilehostisanindependentobjectwhichdecidesitsmovementautonomously. Theservermoduleprocessesspatialsearches andisresponsibleforestimatingtheI/Oloadofthespatial database server. Spatial data indexing is provided with the well known R*-tree algorithm [1]. We implemented our SENN query algorithm in the mobile host module. 13 Parameter Description POI Number The number of point of interest in the system MH Number The number of mobile host in the simulation area C Size The cache capacity of each mobile host M Percentage The mobile host movement percentage M Velocity The mobile host movement velocity (mph) λ Query The mean number of queries per minute Tx Range The transmission range of queries λ kNN The mean number of queried nearest neighbors T execution The length of a simulation run Table 2: Parameters for the simulation environment. Each mobile host is implemented as an independent object that encapsulates all its related param- eters such as the movement velocity Move Velocity , the cache capacity C Size , the wireless transmission range Tx Range , etc. All mobile hosts move inside a geographical area and the measure can be decided by users (in our experiment, we adopt a 2 miles by 2 miles area and a 30 miles by 30 miles area). Additionally, user adjustable parameters are provided for the simulation such as execution length, the number of mobile hosts and their query frequency, the number of POIs, etc. Table 2 lists all of the simulation parameters. The simulation is initialized by randomly choosing a starting location for each MH within the simulation area. The movement generator then generates trajectories with two different modes, the free movement mode and the road network mode. In the former mode, each mobile host moves obstacle-free within the environment and the movement velocity is fixed. The road network mode is more realistic since MHs follow an underlying road network and their travel speed s is determined by the speed limit on the corresponding road segment. We employed the random waypoint model [2] as our mobility model. Each MH selects a random destination inside the simulation area and progresses towards it. When reaching that location, it pauses for a random interval and decides on a new destination for the next travel period. This process repeats for all MHs until the end of the simulation. Every simulation has numerous intervals (whose lengths are Poisson distributed) and at the end of such an interval, the simulator selects a random subset of the mobile hosts to launch spatial queries. The subset size is controlled via the λ Query parameter (e.g., 500 queries per minute). These hosts then execute the SENN algorithm by interacting with their peers. A mobile host will first attempt to answer each spatial query from its local cache and via the SENN algorithm. If this is unsuccessful, the query will be forwarded to the remote database server. Each mobile host manages its local NN query result cache with a combination of the following two policies: 1. A MH only stores the query location (the coordinates where it launched the query) and all the certain nearest neighbors of the most recent query. 2. IfakNNquerymustbesent totheserver, theMHwillqueryfor as manyNNas its cache capacity allows (e.g., if the cache capacity is 10, the query will be for 10-NNs). The single peer nearest neighbor verification process is implemented according to the algorithm detailed in Section 3.2.1. A MH sequentially verifies the candidate points of interest starting with the results obtained from its closest peer query location (using Euclidean distance). In the multiple peer verification algorithm of Section 3.2.2, multiple, potentially overlapping circles must be combined to provide the verification area. We utilize a polygonization technique that transforms all the peer certain area circles into polygons and then sequentially merges them into a certain region R c by performing the 14 MapOverlay algorithm. Afterwards, a MH can verify POIs with the kNN multiple verification technique based on R c . 4.1.1 Simulation Parameter Sets Toobtainresultsthatcloselycorrespondtorealworldconditionsweobtainedoursimulation parameters from data sets that report, for example, car and gas station densities in urban areas. We term the two parameters sets based on these real-world statistics the Los Angeles County parameter set and the Riverside County parameter set. • Points of Interest: We obtained information about the density of interest objects (e.g., gas sta- tions,restaurants,hospitals,etc.) intheGreaterLosAngelesareafromtwoonlinesites: GasPrice- Watch.com 2 and CNN/Money 3 . Because gas stations are commonly the target of kNN queries, we use them as the sample POI type for our simulations. The server load variations of other POI types are expected to be very similar. • Mobile Hosts: We collected vehicle statistics of the Greater Los Angeles area from the Federal Statistics web site 4 . The data provide the number of registered vehicles in the Los Angeles and RiversideCounties(5,498,554 and944,645, respectively). Inoursimulationsweassumethatabout 10% of these vehicles are on the road during non-peak hours according to the traffic information from Caltrans 5 . We further obtained the land area of each county to compute the average vehicle density per square mile. Parameter Los Angeles Riverside Synthetic Units County County Suburbia POI Number 16 5 11 MH Number 463 50 257 C Size 10 10 10 M Percentage 80 80 80 % M Velocity 30 30 30 mph λ Query 23 2.5 13 min −1 Tx Range 200 200 200 m λ kNN 3 3 3 T execution 1 1 1 hr Table 3: The simulation parameter sets for the Los Angeles, the Riverside Counties, and the synthetic suburbia of a 2 miles by 2 miles area. The Los Angeles and the Riverside County parameter sets represent very dense, urban area and a low-density, more rural area. Hence, for comparison purposes we blended the two real parameter sets together to generate a third, synthetic parameter set. The synthetic data set demonstrates vehicle and interest object densities in-between Los Angeles County and Riverside County, representing a suburban area. Table 3 lists the three parameter sets of a 2 miles by 2 miles area and Table 4 demonstrates the parameter sets of a 30 miles by 30 miles area. 2 http://www.gaspricewatch.com 3 http://money.cnn.com/ 4 http://www.fedstats.gov/ 5 http://www.dot.ca.gov/hq/traffops/saferesr/trafdata/ 15 Parameter Los Angeles Riverside Synthetic Units County County Suburbia POI Number 4050 2160 3105 MH Number 121500 11700 66600 C Size 20 20 20 M Percentage 80 80 80 % M Velocity 30 30 30 mph λ Query 8100 780 4440 min −1 Tx Range 200 200 200 m λ kNN 5 5 5 T execution 5 5 5 hr Table 4: The simulation parameter sets for the Los Angeles, the Riverside County, and the synthetic suburbia of a 30 miles by 30 miles area. 4.1.2 Road Network Generation WegeneratedourroadnetworkfromtheTIGER/LINEstreetvectordataavailablefromtheU.S.Census Bureau 6 . Theroad segments belongto several differentcategories, suchas primaryhighways, secondary and connecting roads, and rural roads. The segments associated with a different road classes are associated with different maximum driving speeds. Each mobile hosts monitors the speed limit on the road that it is currently traveling on and adjusts its velocity accordingly. One of the challenges when integrating road segments into a complete road network is to isolate intersecting paths and determine if they are indeed intersections. For example, freeways generally project many intersections in two- dimensional space, but many of them are over-passes or bridges. Our solution is to detect intersection points with the help of their endpoint coordinates. In addition, differing road classes let us distinguish over-passes from intersections. 4.2 Experimental Results with the Road Network Mode Weusedallthreeinputparameter sets–LosAngeles County, RiversideCounty, andSyntheticSuburbia – to simulate our peer sharing techniques in conjunction with the road network mode. We varied the following parameters toobserve their effect on thesystem performance: thewireless transmissionrange, thecache capacity, the movement velocity, andthenearestneighbor numberk. Theperformancemetric in the mobile host module was SQRR. The primary difference between the three different parameter sets is the vehicle and the POI density. Hence, we utilized the simulation to verify the applicability of our design to different geographical and urban areas. All simulation results were recorded after the system reached steady state. 4.2.1 Effect of the Transmission Range In our first experiment we varied the mobile host wireless transmission range from 10 meters to 200 meters, with all other parameters unchanged. We chose 200 meters as a practical upper limit on the transmission range of the IEEE 802.11 technology. Because of obstacles such as buildings, this range could diminish to 100 meters or less in urban areas. Figure 9 and 10 illustrate percentage of the queries thatcanberesolvedbyonepeer,multiplepeersortheserverwiththeLosAngelesCounty,theSynthetic Suburbia, and the Riverside County parameter sets, respectively in the two simulation regions. As the 6 http://www.census.gov/geo/www/tiger/ 16 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer Fig. 9a. Los Angeles County. Fig. 9b. Synthetic Suburbia. Fig. 9c. Riverside County. Figure 9: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the wireless transmission range of a 2 miles by 2 miles area. 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 20 40 60 80 100 120 140 160 180 200 Transmission Range (Meters) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer Fig. 10a. Los Angeles County. Fig. 10b. Synthetic Suburbia. Fig. 10c. Riverside County. Figure 10: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the wireless transmission range of a 30 miles by 30 miles area. transmission range extends, an increased number of queries can be answered by surrounding peers. As expected, the effect is most pronounced in Los Angeles County, because of its high vehicle density. At a transmission range of 200 m only around 20%∼30% of the queries must be sent to the server. 4.2.2 Effect of the MH Cache Capacity Next we varied the mobile host cache capacity. The cache capacity denotes how many nearest neighbor objects a mobile host can store. Figure 11 and 12 illustrate cache capacity changes from 1 to 9 and 4 to 20respectively with thethreeparameter sets. InFigure12a, even though thenumberofinterest objects is much larger than the maximum capacity of the cached NN query results, we can find a remarkable server workload decrease with a higher MH cache capacity. In Figure 11c, however, because of the sparseness of interest objects, the server workload becomes stabilized after a cache capacity of five. 4.2.3 Effect of the MH Movement Velocity We studied the effect of host movement velocity by changing the MH velocity from 10 miles per hour (mph)to50mphandtheresultsareshowninFigure13and14. Weobservethatthemovement velocity has a stronger effect on the server workload in areas with a lower vehicle and interest object density. However, the effect is quite gradual in all cases. 17 0 20 40 60 80 100 120 9 7 5 3 1 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 9 7 5 3 1 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 9 7 5 3 1 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method Fig. 11a. Los Angeles County. Fig. 11b. Synthetic Suburbia. Fig. 11c. Riverside County. Figure 11: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the mobile host cache capacity of a 2 miles by 2 miles area. 0 20 40 60 80 100 120 20 16 12 8 4 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 20 16 12 8 4 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 20 16 12 8 4 Number of Cached Items Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method Fig. 12a. Los Angeles County. Fig. 12b. Synthetic Suburbia. Fig. 12c. Riverside County. Figure 12: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the mobile host cache capacity of a 30 miles by 30 miles area. 4.2.4 Effect of k We were also interested in the effect that varying the number of requested nearest neighbors, i.e., k, would have on the system performance. In our simulation we chose k randomly for each host and each query in the range from 1 to 9 and 3 to 15 respectively in the two regions. Figure 15 and 16 illustrate the results. The server workload of the Los Angeles County parameter set increases 68% and 29% when we raise the k from 1 to 9 and from 3 to 15 in the two regions. The server workload of the Riverside County parameter set increases by only 11% and 19%, because its starting level is much higher. Not surprisingly result sharing is much more effective for small values of k. 4.3 Experimental Results from the Free Movement Mode We executed the simulations in free movement mode with otherwise the same parameter settings as beforewiththefree movement mode. Weobservefromtheexperimentalresultsthattheserverworkload withtheLosAngelesCountyparametersetdecreasesbetween5%and8%(inthe2milesby2milesarea) and2%to5%(inthe30milesby30milesarea)underallconditions. Theresultsofthesyntheticandthe Riverside parameter sets are very close to their counterparts of the road network mode. Because mobile hosts do not have to follow any underlying road network for their movements, this change decreases the distance between mobile hosts compared with the road network mode and hence slightly increases the performance of our sharing algorithm. This effect is more evident in regions with a high vehicle density such as Los Angeles County. 18 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer Fig. 13a. Los Angeles County. Fig. 13b. Synthetic Suburbia. Fig. 13c. Riverside County. Figure 13: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the mobile host movement velocity of a 2 miles by 2 miles area. 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer 0 20 40 60 80 100 10 15 20 25 30 35 40 45 50 Mobile Host Speed (Mph) Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Queries Solved by Multi-Peer Fig. 14a. Los Angeles County. Fig. 14b. Synthetic Suburbia. Fig. 14c. Riverside County. Figure 14: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of the mobile host movement velocity of a 30 miles by 30 miles area. 4.4 Experimental Results from the Spatial Database Server In order to evaluate the nearest neighbor query pruning bounds of Section 3.3, we extended the R- tree incremental nearest neighbor (INN) algorithm [10] with one more metric, MAXDIST. For each incoming NN query from mobile hosts, the server module executes both the original INN algorithm and our extended INN algorithm with pruningbounds (denoted by EINN) to compare the performance improvement with respect to page accesses. We examined the behavior of the two algorithms as the number of k increases. We utilized the R*-tree for indexing the POI data set (gas station locations) in the server module. The R*-tree has an advantage in query response time over the conventional R-tree algorithm by utilizing more sophisticated insertion and node-splitting methods, which attempt to minimize a combination of overlap between bounding rectangles and the total area. The branching factor of both the index and leaf nodes was set to 30. Because NN queries are generated by randomly selected mobile hosts, query points are uniformlydistributed over thesimulation area. Theexperiments are executed sufficiently often to obtain consistent results. At the end of a spectrum there are two extreme I/O behaviors of the spatial database server: all requested memory pages are found in main memory or every I/O leads to disk activity. In the former case,becauseoffastmainmemoryaccess,wecannotdiscernasignificantperformancedifferencebetween INN and EINN. However, any reasonably large data set will not fit into main memory and more disk I/Os will be performed. Hence, the database I/O behavior is closer to the other end of the spectrum. Since the EINN usually requests fewer R*-tree nodes and objects than INN, we believe that the kNN search algorithm with query pruning bounds (EINN) will have good scalability with large data sets. 19 0 20 40 60 80 100 120 9 7 5 3 1 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 9 7 5 3 1 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 9 7 5 3 1 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method Fig. 15a. Los Angeles County. Fig. 15b. Synthetic Suburbia. Fig. 15c. Riverside County. Figure 15: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of k of a 2 miles by 2 miles area. 0 20 40 60 80 100 120 15 12 9 6 3 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 15 12 9 6 3 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method 0 20 40 60 80 100 120 15 12 9 6 3 Number of k Percentage of Total Queries Queries Solved by the Server Queries Solved by Single-Peer Method Queries Solved by Multi-Peer Method Fig. 16a. Los Angeles County. Fig. 16b. Synthetic Suburbia. Fig. 16c. Riverside County. Figure 16: The percentage of queries that are resolved by one peer, multiple peers and the server as a function of k of a 30 miles by 30 miles area. During the simulation process, the server module counts the number of R*-tree node (index nodes and data nodes) accesses which correspond to both main memory and disk I/Os. According to our observation, the number of node accesses provides a good predication of the actual NN query I/O cost. Next, we varied the number of k with all the three parameter sets with both the EINN and INN algorithms. The server module recorded the relevant R*-tree page access information (Section 4.2.4). As shown in Figure 17, the EINN algorithm performs consistently better than INN, while the rate of growth is similar for both. We conclude that the pruning bounds can always decrease the number of page accesses. We varied the number of k from 3 to 15 with the three parameter sets and the EINN algorithm accesses 10% to 21% fewer pages than INN. We conclude from all the performed experiments that the mobile host density has a considerable impact on the spatial query request rate. As a result, if more mobile hosts travel in a specific area, each MH has a higher opportunity to fulfill its kNN queries by peers. Furthermore, the nearest neighbor query pruning bounds also have a significant positive effect on the page access rate and successfully decrease the server load. 5 Conclusions and Future Work We have presented a novel approach for answering spatial nearest neighbor search queries by leveraging results from neighboring peers within a mobile environment. Significantly, our method allows a mobile peer to locally verify whether candidate objects received from neighbors are indeed part of its own 20 0 5 10 15 20 25 30 4 6 8 10 12 14 Number of of k R*-tree pages accessed EINN (LA) INN (LA) EINN (SYN) INN (SYN) EINN (RV) INN (RV) Figure 17: The page access comparison of EINN and INN as a function of k. nearest neighbor data set. Our simulation results indicate that the technique can reduce the access traffic to remote servers by a significant amount, for example up to 80% in a dense urban area. This is achieved with minimal caching at the peers. By virtue of its peer-to-peer architecture, the method exhibits great scalability: the higher the mobile peer density, the more queries can be answered by peers. Therefore, the load on the remote databases increases sub-linearly with the number of clients. We plan to extend our work to investigate other types of spatial queries, such as range and spatial join searches. 6 Acknowledgments This research has been funded in part by NSF grants EEC-9529152 (IMSC ERC), CMS-0219463 (ITR), and equipment gifts from the Intel Corporation, Hewlett-Packard, Sun Microsystems and Raptor Net- works Technology. References [1] N. Beckmann, H.-P. Kriegel, R. Schneider, and B. Seeger. The R*-tree: An Efficient and Robust Access Method for Points and Rectangles. 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Wei-Shinn Ku, Roger Zimmermann, Chi-Ngai. "Location-based spatial queries with data sharing in mobile environments." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 843 (2005).

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Creator Chi-Ngai (author), Ku, Wei-Shinn (author), Zimmermann, Roger (author)

Core Title USC Computer Science Technical Reports, no. 843 (2005)

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Rights Department of Computer Science (University of Southern California) and the author(s).

Internet Media Type application/pdf

Source 20180426-rozan-cstechreports-shoaf (batch), Computer Science Technical Report Archive (collection), University of Southern California. Department of Computer Science. Technical Reports (series)

Access Conditions The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Repository Email csdept@usc.edu

Inherited Values

Title Computer Science Technical Report Archive

Description Archive of computer science technical reports published by the USC Department of Computer Science from 1991 - 2017.

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)