Lazy Updates: An Efficient Technique to Continuously Monitoring Reverse kNN

Presented By: Ying ZhangJoint work with

Muhammad Aamir Cheema, Xuemin Lin, Wei Wang, Wenjie Zhang

University of New South Wales, Australia1

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 qp1 and p4 are the reverse nearest neighbors of q 2

Applications

• Location based services,

• Location based games,

• Army strategic planning …

Continuous Monitoring of RkNN

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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

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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

de

Static RNN query

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Unpruned area

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

de

Continuous RNN query [ICDE07]

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Unpruned area

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

8

Challenges

R

Q

?

Based on : Shortest pair ?Longest pair ?Combination of them ?

NO

Half-Space Pruning

qM N

p

mindist(x,MN) > dist(x,p)

x

Hp:M

Hp:N

x

x

– Any x on right side of LN :• mindist(x,MN) = dist(x,N)

– Hp:N : the half-space containing p and defined by the bisector between p and N

– Any x on left side of LM :• mindist(x,MN) = dist(x,M)

– Any x between LM and LN :• a parabola with mindist(x,

MN) = dist(x,p)

LMLN

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Challenges

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R

Q

?

qM N

p

Hp:M

Hp:N

– Frontier point Fp

– Moved to Fp, the intersection of the half-spaces correctly bounds the pruned area

– Hp:N passing Fp : normalized half-space H’p:N

Half-Space Pruning: Normalization

Fp

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H’p:M

H’p:N

Half-space Pruning: Pruning Rule 1

RM

N O

P

QD

A B

E

H’M:B

H’P:A

H’N:E

H’O:D

Fp

Pairs of antipodal corners are (B,M), (A,P), (E,N) and (D,O)

HM:B HP:A

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Dominance Pruning : pruning rule 2

RM

N O

P

QD

A B

C

H’M:B

H’P:A

H’N:C

H’O:D

Fp

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Trimming the rectangle

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Q

RF1RF2

R

Metric based pruning : pruning rule 3

Q

R

R’

maxdist(R,R’)

mindist(R’,Q)

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Order of pruning rules

Q

RF

R1

R2

R3

Fp

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

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Illustration of Filtering

O3

Q

O2

O1

O4

O6

O5

O7

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Illustration of Verification

O3

Q

O2

O1

O4

O6

O5

O7

O1

O2

O3

O5

O6

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Q

?

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

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Theoretical AnalysisCommunication Cost : |Q| x (M1 + M2 + 1) + M3

M1: # candidate objects

M2: #objects need exact location during the boolean range queries

M3: #objects that leave the safe regions

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N: Total number of objects w : width of the safe regionv: average speed of objects |Q| : The number of queries

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

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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 )

Experiments: Size of safe region

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Experiments: Number of objects

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IGERN : ICDE 2007 work for RNNSAC : Our algorithm

Experiments: Effect of data mobility

• Data mobility is the percentage of objects/queries that change their location within one time unit

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Experiments: RkNN queries

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CRkNN : MDM 2008 work for continuous monitoring RkNNSAC : Our algorithm

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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

Thanks

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