Continuous Reverse Nearest Neighbor Monitoring Authors’ info: Tian Xia and Donghui Zhang College of Computer and Information Science Northeastern University Presenter: Kamiru, U

Outline Background Problem Definition Related Work Solution Straightforward solution Incremental solution Experiments

So, updates are very frequent in these kinds of applications Background Evolutional technologies on hardware enable new kind of data management applications to monitor continuous processes Obtaining amounts of state samples via sensors (Data Stream) and store into database So, updates are very frequent in these kinds of applications It is also a problem when monitor on the continuous spatial data / queries

Traditional Spatial Queries There are many existing algorithms to solve different kinds of spatial queries based on R*-tree, such as range queries k nearest neighbors (kNN) k closest pairs (kCP) reverse nearest neighbors (RNN)

Traditional Spatial Queries (Cont’) Most of them are efficient only on the static objects and queries If they try to monitor on the moving objects or queries, it is necessary to execute some high cost operations, such as deletion, insertion and update, to maintain the data in R-tree Many variations are derived from R-tree to monitor the moving objects TPR-tree [Saltenis 00] STAR-tree [Procopiuc 02] REXP-tree [Saltenis 02] FUR-tree [Lee 03]

Reverse Nearest Neighbors (RNN) RNN definition: an object o is considered as a query point q’s reverse nearest neighbor, if there does not exist another object o’ such that dist(o, o’) < dist(o, q) Example: o1 is the q0’s RNN o2 o0 o1 d2 d0 d1 q

Reverse Nearest Neighbors (RNN) (Cont’) Previous works on finding RNNs focused either on the static query [Stanoi 00, Tao 04], or the predictive query [Benetis 02] The predictive query is based on the assumption of knowing the trajectory information It uses trajectory-based TPR-tree to predict the result, but it is too expensive to maintain for the CRNN query Unlike the static RNN query, the CRNN query requires updating the result set efficiently to reflect the recent motion of objects and queries So they are inefficient or inapplicable in the CRNN monitoring problem

Problem Definition Given a set of objects O and a query set Q, all being static or moving, the CRNN query monitors the exact reverse nearest neighbors (RNN) of each query point over time

To all, I’m going to help Soldier A. Because he is my RNN. Application of CRNN Soldier C One example of CRNN’s application is in the battlefield, where a soldier registers a CRNN query to monitor the other soldiers who might need help from him. Help To all, I’m going to help Soldier A. Because he is my RNN. Time 1 Time 0 Soldier A Soldier A Soldier B Assume that Soldier B and C have registered Soldier A on their CRNN list

Related Work Stanoi et al. [Stanoi 00] proposed a method (SAE) that divides the space centered at the query q into six equal partitions of 60o For a given 2-dimensional dataset, RNN(q) will return at most six data points for any query point q [Smid 97, Korn 99] And the number of data points that satisfy RNN(q) is still a constant in higher dimensions.

SAE’s filter-refinement framework Finds six constrained NNs in each region as the candidates For each candidate, it performs the NN search to see whether the candidate really considers q as NN (filter out the false positives) S0 cand5 cand1 nn_cand5 60o S5 S1 q nn_cand1 S2 The candidates of q’s RNNs are o1, o2, o4, o5, o6, o7 The RNN(s) of q are o7 S3 S4

Continuous Spatial Queries Monitoring Continuous Nearest Neighbor (CNN) query was recently studied in [Xiong 05, Yu 05, Mouratidis 05] and three methods (denoted as SEA-CNN, YPK-CNN, CPM-CNN, respectively) All of them use a monitoring region (grid) for each query point to handle the updates In this paper, they use the conceptual space partitioning from CPM-CNN to monitor the update region

CPM-CNN CPM-CNN partitions the space into grid that organize the cells into conceptual rectangles The rectangles are denoted by the Direction (Up, Down, Left or Right) Level (i.e. the number of rectangles between q and itself) U3 U2 U1 L3 L2 L1 L0 U0 q D0 R0 R1 R2 R3 D1 D2 D3

Frequent updates R-tree (FUR-tree) Most R-tree variants (TPR-tree, STAR-tree, REXP-tree) process updates as combinations of separate top-down deletion and insertion operations Top-down update is inherently inefficient In R-tree, objects are stored into the leaf of the tree The root is the starting point of updates So FUR-tree propose a new concept of updating R-tree, which is bottom-up approach

Bottom-up approach is to access the leaf of an object’s entry directly FUR-tree (Cont’) Bottom-up approach is to access the leaf of an object’s entry directly It requires a secondary index on object IDs Hash Table E8 E7 E9 E5 E4 E6 E2 E1 E3 b a c e d f g h i …

Straightforward solution Straightforward solution indexes the objects using an FUR-tree and compute the RNNs of every query point at each time stamp using TPL TPL [Tao 04] method is currently the best approach for computing RNNs in the static case FUR-tree is the optimized for frequent updates of objects

CRNN Framework CRNN consider two situations on monitoring the RNNs Queries update When an existing query q moves to a new location, CRNN treats the update as deleting q with the old location re-compute q with the new location Although it is not the best way to handle it, re-computing a moving query is more efficient than updating from the old query result [Mouratidis 05] Objects update Uses pie-regions and circ-regions to monitor the object update Proposes two optimizations Lazy-update Partial-insert

CRNN Query Initialization If the pop-up element e is a rectangle Push the next level rectangle of same direction into H (heap), and for each cell c in e If e is a cell, for each object o in e For all candi != null update nn_canni and d(nn_candi, candi) if necessary candj is the nearest candidate to o If o in Sk update candk and dnnk = d(q, candk) if necessary update nn_candk and d(nn_candk, candk), where nn_candk is either q or candj which is closer L0 D0 U0 R0 D0 U0 R0 L1 C2, 5 … L2 D2 U2 R2 C8, 8 C3, 4 q C0, 0 S0 S1 S5 ... S4 S2 S3 (1) (2) ... (n)

CRNN Query Initialization (Cont’) When we pop up C2,5 candj = null because all candi == null Set cand1 = o7 nn_cand1 = q S1 is checked C8, 8 C3, 4 q C0, 0 CHECKED After we find all candidates candi in each Si For all nn_candi == q, perform NN search on candi update nn_candi and d(nn_candi, candi) Output candi if nn_candi == q

Monitoring region of CRNN query In order to enable the possibility of incremental processing, it is necessary to maintain the monitoring region for the continuous query, such that guarantee the query results are unaffected as long as no update happens inside the region Straightforward proposal might consider the union of every circle Center is some RNN objects Radius is its distance to the query point q But it does not work o1 o3 q o2 o4 o5 o5’

Monitoring region of CRNN query (Cont’) Pie-region Given a query point q, the space is divided into 6 partitions (Si) Pie-region in Si is a pie centered at q and having the constrained NN in Si on the perimeter Circ-region Circ-region in Si is a circle centered at the candidate in Si and having either q or an object closer than q on the perimeter cand0 nn_cand0 cand1 S0 S1 S5 q nn_cand1 S2 S4 S3

Handling Updates in Pie-regions The pie-region information is stored in each cell Updating the pie-region Some object(s) (o4, o8) move into pie-region (Si) Set candi = o and dnni = dist(o, q) Candidate(s) (o4) leave a pie-region (Si) Perform a constrained NN serach in Si to determine the new pie-region Candidate(s) (o6, o7) moves in the same pie-region (Si) Update dnni Finally, use updateCand (performing the NN serach) to update the circ-region

Handling Updates in Circ-regions We cannot store circ-regions by associating every cell that intersects with them, because it is expensive for the following reasons: Circ-region is not always changed incrementally Circ-region may change frequently This paper use FUR-tree to store the circ-region that correspond to each candidate

Handling Updates in Circ-regions (Cont’) FUR-tree maintain the following: the radius of circ-region and the candidate store to the leaf it stores the max radius for all candidates in the sub-tree each candidate will also store the queries it belongs to The hash table store the set of nn_candi the pointers to their corresponding candidates in the leaf

Optimization Lazy-Update Partial-Insert The NN search is performed only when the enlarged circ-region cover q Partial-Insert FUR-tree stores the candidates whose circ-regions’ radii are larger than a threshold Other candidate cand and the corresponding nn_cand are stored in a hash table

Comparison with the straightforward solution

Varying the data size

Varying the percentage of moving data per time stamp

References [Saltenis 00] S. Saltenis, C.S. Jensen, S.T. Leutenegger, and M.A. Lopez. Indexing the Positions of Continuously Moving Objects. In Proc. of ACM SIGMOD, 2000. [Procopiuc 02] C. Procopiuc, P. Agarwal, and S. Har-Peled. Star-Tree: An Efficient Self-Adjusting Index for Moving Objects. In Proc. of ICDE (poster), 2002. [Saltenis 02] S. Saltenis and C.S. Jensen. Indexing of Moving Objects for Location-Based Services. In Proc. of ICDE, 2002. [Lee 03] Mong-Li Lee, Wynne Hsu, Christian S. Jensen, Bin Cui, and Keng Lik Teo. Supporting frequent updates in r-trees: A bottom-up approach. In VLDB, pages 608–619, 2003. [Stanoi 00] Ioana Stanoi, Divyakant Agrawal, and Amr El Abbadi. Reverse nearest neighbor queries for dynamic databases. In ACM SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery, pages 44–53, 2000. [Tao 04] Yufei Tao, Dimitris Papadias, and Xiang Lian. Reverse knn search in arbitrary dimensionality. In VLDB, pages 744–755, 2004. [Benetis 02] Rimantas Benetis, Christian S. Jensen, Gytis Karciauskas, and Simonas Saltenis. Nearest neighbor and reverse nearest neighbor queries for moving objects. In IDEAS, pages 44–53, 2002. [Smid 97] M. Smid. Closest point problems in computational geometry. In Handbook on computational Geometry, Elsevier Science Publiching, 1997. [Korn 99] F. Korn and S. Muthukrishnan. Influence sets based on reverse nearest neighbor queries. Technical report, AT&T Labs Research, http://www.research.att.com/resources/trs/,1999. [Mouratidis 05] Kyriakos Mouratidis, Dimitris Papadias, and Marios Hadjieleftheriou. Conceptual partitioning: An efficient method for continuous nearest neighbor monitoring. In SIGMOD Conference, pages 634–645, 2005.

References (Cont’) [Xiong 05] Xiaopeng Xiong, Mohamed F. Mokbel, and Walid G. Aref. Sea-cnn: Scalable processing of continuous k-nearest neighbor queries in spatio-temporal databases. In ICDE, pages 643–654, 2005. [Yu 05] Xiaohui Yu, Ken Q. Pu, and Nick Koudas. Monitoring k-nearest neighbor queries over moving objects. In ICDE, pages 631–642, 2005.

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Properties of monitoring region Appendix A Properties of monitoring region The region usually has a regular shape The region only contains the result objects The region does not rely on the distances between objects

Appendix B d3 3 d1 == d, o1 and o4 are RNNs of q d2 < d, only o2 is RNN of q d3 > d+k > d, only o4 is RNN of q k 4 d1 1 d2 2 d d 60o q d d d d