Continuous Reverse Nearest Neighbor Monitoring

1
Continuous Reverse Nearest Neighbor Monitoring

• Authors info
• Tian Xia and Donghui Zhang
• College of Computer and Information Science
• Northeastern University
• Presenter
• Kamiru, U

2
Outline

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

3
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

4
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)

5
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

6
Reverse Nearest Neighbors (RNN)

• RNN definition
• an object o is considered as a query point qs
  reverse nearest neighbor, if there does not exist
  another object o such that dist(o, o) lt dist(o,
  q)
• Example

o1 is the q0s RNN
o2
o0
o1
d2
d0
d1
q
7
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

8
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

9
Application of CRNN
Soldier C

• One example of CRNNs 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, Im 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
10
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.

11
SAE

• SAEs 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 qs RNNs are o1, o2, o4, o5,
  o6, o7
• The RNN(s) of q are o7

S3
S4
12
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

13
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)

14
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

15
FUR-tree (Cont)

• Bottom-up approach is to access the leaf of an
  objects entry directly
• It requires a secondary index on object IDs

Hash Table
16
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

17
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

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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

S0
S1
S5
...
S4
S2
S3
(1)
(2)
(n)
...
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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

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

20
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
21
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
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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

23
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

24
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

25
Optimization

• Lazy-Update
• 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

26
Comparison with the straightforwardsolution
27
Varying the data size
28
Varying the percentage of movingdata per time
stamp
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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 608619, 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 4453, 2000.
• Tao 04 Yufei Tao, Dimitris Papadias, and Xiang
  Lian. Reverse knn search in arbitrary
  dimensionality. In VLDB, pages 744755, 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
  4453, 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, ATT 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 634645, 2005.

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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
  643654, 2005.
• Yu 05 Xiaohui Yu, Ken Q. Pu, and Nick Koudas.
  Monitoring k-nearest neighbor queries over moving
  objects. In ICDE, pages 631642, 2005.

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The END

• Thank you for your attendance

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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

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Appendix B
d3
3
d1 d, o1 and o4 are RNNs of q d2 lt d, only o2
is RNN of q d3 gt dk gt d, only o4 is RNN of q
k
4
1
d1
d2
2
d
d
60o
q
d
d
d
d