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
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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
p2are
is the
of q
p1 and p4
thenearest
reverseneighbor
nearest neighbors
of q
2
Applications
• Location based services,
• Location based games,
• Army strategic planning …
Continuous Monitoring of RkNN
3
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
Preliminaries
• Half-space Pruning [VLDB04]
– Objects in the half-space
containing a can be pruned
Static RNN query
• Filtering
Unpruned area
– Repeat until no objects in
unpruned area
b
c
• Verification
a
– p is RNN iff no object p’ s.t.
q
dist(p,p’) < dist( p,q)
d
e
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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
Continuous RNN query [ICDE07]
Unpruned area
b
c
a
q
d
e
6
Proposed Framework
• Assign rectangular safe regions to
objects and queries
•
Prune objects using safe regions
b
c
•
Advantages
a
– Low Computation Cost
q
– Low Communication Cost
d
e
7
Challenges
Based on :
Shortest pair ?
NO pair ?
Longest
Combination of them ?
R
?
Q
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Half-Space Pruning
– 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
mindist(x,MN) > dist(x,p)
LN
LM
x
– Any x on left side of LM :
• mindist(x,MN) = dist(x,M)
p
x
x
Hp:N
– Any x between LM and LN :
• a parabola with mindist(x,
MN) = dist(x,p)
M
q
Hp:M
N
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Challenges
R
?
Q
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Half-Space Pruning: Normalization
– Frontier point Fp
– Moved to Fp, the intersection
of the half-spaces correctly
bounds the pruned area
Hp:M
– Hp:N passing Fp :
normalized half-space H’p:N
p
Fp
H’p:N
Hp:N
H’p:M
M
q
N
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Half-space Pruning: Pruning Rule 1
HM:B
HP:A
N
R
M
Fp
B
A
Q
D
E
O
P
H’N:E
H’M:B
H’O:D
H’P:A
Pairs of antipodal corners are (B,M), (A,P), (E,N) and (D,O)
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Dominance Pruning : pruning rule 2
O
N
R
M
P
Fp
H’N:C
A
B
Q
D
C
H’M:B
H’O:D
H’P:A
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Trimming the rectangle
R
RF2
RF1
Q
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Metric based pruning : pruning rule 3
R’
mindist(R’,Q)
R
maxdist(R,R’)
Q
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Order of pruning rules
R3
R1
RF
R2
Fp
Q
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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
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Illustration of Filtering
O7
O5
O6
O3
O2
Q
O1
O4
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Illustration of Verification
O7
O6
O5 O 5
O6
?
O3
O3
O2
O2
Q
Q
O1
O1
O4
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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
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Theoretical Analysis
Communication 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
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
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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 )
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Experiments: Size of safe region
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Experiments: Number of objects
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
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Experiments: RkNN queries
CRkNN : MDM 2008 work for continuous monitoring RkNN
SAC : 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
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Thanks
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