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Lazy Updates: An Efficient Technique to Continuously Monitoring Reverse kNN Presented By: Ying Zhang Joint work with Muhammad Aamir Cheema, Xuemin Lin, Wei Wang, Wenjie Zhang University of New South Wales, Australia 1

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Reverse Nearest Neighbor Nearest Neighbor Query (NN) –Find the object closest to q Reverse Nearest Neighbor Query (RNN) –Korn et. al. Sigmod 2000 –Find objects s.t. q is their NN Reverse k Nearest Neighbor Query (RkNN) –Find objects s.t. q is their kNN p2 is the nearest neighbor of q p1 and p4 are the reverse nearest neighbors of q 2

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Applications Location based services, Location based games, Army strategic planning … Continuous Monitoring of RkNN 3

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Related Work Continuous RNN and RkNN –Benetis et. al (IDEAS 2002) : motion patterns (e.g., speed, direction) of objects and query are known –Xia et. al (ICDE 2006) : continuous RNN without motion patterns –Kang et. al (ICDE 2007) : improve the ICDE 2006 techniques –Wu et. al (MDM 2008) : extend to RkNN monitoring 4

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Preliminaries Half-space Pruning [VLDB04] –Objects in the half-space containing a can be pruned Filtering –Repeat until no objects in unpruned area Verification –p is RNN iff no object p s.t. dist(p,p) < dist( p,q) q c b a d e Static RNN query 5 Unpruned area

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Preliminaries Do filtering starting from the existing candidates if –Query moves, or – Candidate objects move, or – An object moves to the unpruned area Do verification q c b a d e Continuous RNN query [ICDE07] 6 Unpruned area

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Proposed Framework Assign rectangular safe regions to objects and queries Prune objects using safe regions Advantages –Low Computation Cost –Low Communication Cost q c b a d e 7

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8 Challenges R Q ? Based on : Shortest pair ? Longest pair ? Combination of them ? NO

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Half-Space Pruning q MN p mindist(x,MN) > dist(x,p) x H p:M H p:N x x –Any x on right side of L N : mindist(x,MN) = dist(x,N) –H p:N : the half-space containing p and defined by the bisector between p and N –Any x on left side of L M : mindist(x,MN) = dist(x,M) –Any x between L M and L N : a parabola with mindist(x, MN) = dist(x,p) LMLM LNLN 9

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Challenges 10 R Q ?

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q MN p H p:M H p:N –Frontier point F p –Moved to F p, the intersection of the half-spaces correctly bounds the pruned area –H p:N passing F p : normalized half-space H p:N Half-Space Pruning: Normalization FpFp 11 H p:M H p:N

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Half-space Pruning: Pruning Rule 1 R M NO P Q D A B E H M:B H P:A H N:E H O:D FpFp Pairs of antipodal corners are (B,M), (A,P), (E,N) and (D,O) H M:B H P:A 12

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Dominance Pruning : pruning rule 2 R M N O P Q D A B C H M:B H P:A H N:C H O:D FpFp 13

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Trimming the rectangle 14 Q R F1 R F2 R

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Metric based pruning : pruning rule 3 Q R R maxdist(R,R) mindist(R,Q) 15

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16 Order of pruning rules Q RFRF R1R1 R2R2 R3R3 FpFp

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Algorithm Overview Initial Computation – Filtering: Determine candidates – Verification: Verify each candidate Continuous Monitoring –Update candidate objects (filtering) if Query or a candidate moves out of safe region, or An object enters the unpruned area –Verify all candidates 17

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Illustration of Filtering O 3 Q O 2 O 1 O 4 O 6 O 5 O 7 18

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Illustration of Verification O 3 Q O 2 O 1 O 4 O 6 O 5 O 7 O1O1 O2O2 O3O3 O5O5 O6O6 19 Q ?

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Data structure Query table : id, safe region, candidate objects Object table : id, safe region Use grid data structure to support update Each cell c of the grid : – Object list : objects whose safe regions overlap c – Influence list : queries whose unpruned area overlaps c 20

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Theoretical Analysis Communication Cost : |Q| x (M 1 + M 2 + 1) + M 3 M 1 : # candidate objects M 2 : #objects need exact location during the boolean range queries M 3 : #objects that leave the safe regions 21 N: Total number of objects w : width of the safe region v: average speed of objects |Q| : The number of queries

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Experiment settings Generate Moving objects and queries using Brinkhoff Generator [1] on road map of Texas Data space : 1000 Km X 1000 Km Our algorithm (SAC) is compared with IGERN [2] for RNN queries and CRkNN [3] for RkNN queries [1] T. Brinkhoff. A framework for generating network-based moving objects. GeoInformatica, 2002. [2] J. M. Kang, M. F. Mokbel, S. Shekhar, T. Xia, and D. Zhang. Continuous evaluation of monochromatic and bichromatic reverse nearest neighbors. ICDE, 2007. [3] W. Wu, F. Yang, C. Y. Chan, and K.-L. Tan. Continuous reverse k-nearest-neighbor monitoring. MDM, 2008 22

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23 Evaluation of pruning rules Avg. time for PR3 : 1.1 µs ( metric based pruning rule ) PR2 : 2.3 µs ( dominance pruning rule ) PR1 : 10.5 µs ( half space based pruning rule )

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Experiments: Size of safe region 24

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Experiments: Number of objects 25 IGERN : ICDE 2007 work for RNN SAC : Our algorithm

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Experiments: Effect of data mobility Data mobility is the percentage of objects/queries that change their location within one time unit 26

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Experiments: RkNN queries 27 CRkNN : MDM 2008 work for continuous monitoring RkNN SAC : Our algorithm

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28 Conclusion Study the problem of continuously monitoring reverse kNN. Propose new framework based on safe region Outperform previous algorithms in terms of computation cost and communication cost

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Thanks 29

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Conceptual Grid-tree Grid data structure is preferred for moving objects, so we also use a grid structure To efficiently search the objects in unpruned area, we treat grid as a conceptual tree 30

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Half-space Pruning: Pruning Rule 1 R M N O P Q D A B E Pairs of antipodal corners are (B,M), (A,P), (E,N) and (D,O) H M:B H P:A H N:E H O:D Any point that lies in intersection of normalized halfspaces between all pairs of antipodal corners of R and Q can be pruned (such point is closer to every point in R than any point in Q) X 31

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Continuous Monitoring The pruning remains valid unless at least one of the following two happens; 1.Query or a candidate object leave its safe region 2.An object enters into the unpruned region If any of the above two happens Repeat the filtering phase Verify all candidate objects 32

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Extensions Extension to RkNN queries –An entry e can pruned if it is pruned by at least k filtering objects –An object p is verified if there are less that k objects within range dist(p,q) Extension to Bichromatic queries –Let there be two set of objects O and P and query belongs to O –Repeat until no object of type O in unpruned region Find a nearby object of type O in the unpruned region Prune the space using this object –All objects of type P that lie in the unpruned region are the canidate objects –Verify them if there are less than k objects of type O within the range 33

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Theoretical Analysis The filtering phase is required if at least one of the following two happens; 1.The query or a candidate objects leaves its safe zone 2.Any other object enters in the unpruned region – Let p be the probability that at a given timestamp at least one of the above two conditions hold Computation Cost = p x C Fil + C ver – Where C Fil is the cost of filtering phase and C ver is the cost of the verification phase – Let C br be the cost of a boolean range query 34

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Experiments: Effect of Speed 35

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Experiments: Number of queries 36

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