Reverse Nearest Neighbor Queries for Dynamic Databases

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Reverse Nearest Neighbor Queries for Dynamic
Databases
SIGMOD 2000

• SHOU Yu Tao
• Jan. 10th, 2003

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Outline of the Presentation

• Background
• Nearest neighbor (NN) search algorithm RKV95
• Reverse nearest neighbor (RNN) search algorithm
  SAA00
• Other NN related problems CNN, RNNa, etc.
• Conclusions
• References
• Q A

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Background

• RNN(q) returns a set of data points that have
  the query point q as the nearest neighbor.
• Receives much interests during recent years due
  to its increased importance in advanced database
  applications
• fixed wireless telephone access application
  load detection problemcount how many users
  are currently using a specific base station q ?
  if qs load is too heavy ? activating an inactive
  base station to lighten the load of that over
  loaded base station

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Nonsymmetrical property of RNN queries

• NN(q) p NN(p) q
• If p is the nearest neighbor of q, then q need
  not be the nearest neighbor of p (in this case
  the nearest neighbor of p is r).
• those efficient NN algorithms cannot directly
  applied to solve the RNN problems. Algorithms for
  RNN problems are needed.
• A straight forward solution-- check for each
  point whether it has q as its nearest neighbor
  -- not suitable for large data set!

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Two versions of RNN problem

• monochromatic version
• -- the data points are of two categories, say red
  and blue. The RNN query point q is in one of the
  categories, say blue. So RNN(q) must determine
  the red points which have the query point q as
  the closest blue point.
• -- e.g. fixed wireless telephone access
  application clients/red (e.g. call initiation
  or termination)
• servers/blue (e.g. fixed wireless base stations)
• bichromatic version
• -- all points are of the same color is the
  monochromatic version.
• Static vs Dynamic
• -- whether insertions or deletions of the data
  points are allowed.

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RNN problem this paper concerns

• Monochromatic case
• Dynamic case
• Whole Algorithm is based on(1). Geometric
  observations ? enable a reduction of the RNN
  problem to the NN problem.
• (2). NN search algorithm RKV95.
• Both RNN(q) and NN(q) are sets of points in
  the databases, while query point q may or may not
  correspond to an actual data point in the data
  base.

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Geometric Observations
Let the space around a query point q be divided
into six equal regions Si (1lines intersecting q. Si therefore is the space
between two space dividing lines. Proposition
1 For a given 2-dimensional dataset, RNN(q)
will return at most six data points. And they are
must be on the same circle centered at q.
L3
L2
s2
s1
s3
q
L1
s4
s6
s5
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Geometric Observations

• Proposition 2
• In each region Si
• (1). There exists at most two RNN points
• (2). If there exist exactly two RNN points in a
  region Si, then each point must be on one of the
  space dividing lines through q delimiting Si.
• Proposition 3
• In each region Si, let p NN(q) in Si,
  if p is not on a space dividing line, then either
  NN(p) q (and then RNN(q) p) or RNN(q) null.

p
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Important result from Observations

• Implications In a region Si, if the number of
  results of NN(q) is(1) one point only If
  NN(q) is not on the space dividing lines either
  the nearest neighbor is also the reverse nearest
  neighbor, or there is no RNN(q) in Si. (2) more
  than one point, (but the NN(q) of each region
  will return at most two points for each
  region) These two points must be on the two
  dividing lines and on the same circle centered at
  q.
• Allow us to have a criterion for limiting the
  choice of RNN(q) to one or two points in each of
  the six regions Si.
• The RNN query has been reduced to the NN query

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Basic NN Search Algorithm

• This is based on MINDIST metric only
• return single NN(q) result only

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Algorithms in RKV95

• Two metrics introduced effectively directing
  and pruning the NN search
• MINDIST (optimistic)
• MINMAXDIST (pessimistic)
• DFS Search

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MINDIST(Optimistic)

• MINDIST(RECT,q) the shortest distance from RECT
  to query point q
• This provides a lower bound for distance from q
  to objects in RECT
• MINDIST guarantees that all points in the RECT
  have at least MINDIST distance from the query
  point q.

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MINMAXDIST(Pessimistic)

• MBR property Every face (edge in 2D, rectangle
  in 3D, hyper-face in high D) of any MBR contains
  at least one point of some spatial object in the
  DB.
• MINMAXDIST Calculate the maximum dist to each
  face, and choose the minimal.
• Upper bound of minimal distance
• MINMAXDIST guarantees that at least 1 object with
  distance less or equal to MINMAXDIST in the MBR

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Illustration of MINMAXDIST

(t1,t2)
MINDIST
(q1,q2)
(t1,p2)
MINMAXDIST
y
x
(s1,s2)
(t1,s2)
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Pruning

• Downward Pruning during the descending phase
• MINDIST(q, M) MINMAXDIST(q, M)
• M can be pruned
• Distance(q, O) MINMAXDIST(q, M)
• O can be discarded
• Upward Pruning when return from the recursion
• MINDIST(q, M) Distance(q, O)
• M can be pruned

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DFS Search on R-Tree

• Traversal DFS
• Expanding non-leaf node during the descending
  phase Order all its children by the metrics
  (MINDIST or MINMAXDIST) and sort them into an
  Active Branch List (ABL). Apply downward pruning
  techniques to the ABL to remove unnecessary
  branches.
• Expanding leaf node Compare objects to the
  nearest neighbor found so far. Replace it if the
  new object is closer.
• At the return from the recursion phase Using
  upward pruning tech.

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RNN Algorithm

• Algorithm Outline for RNN(q) query
• 1. Construct the space dividing lines so that
  space has been divided into 6 regions based on
  the query point q.
• 2. (a) Traverse R-tree and find one or two
  points in each region Si that satisfy the nearest
  neighbor condition NN(q). -- this part is also
  called conditional NN queries
• (b) The candidate points are tested for the
  condition whether their nearest neighbor is q and
  add to answer list if condition is fulfilled.
• 3. Eliminate duplicates in RNN(q)

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How to find NN(q) in Si
p2
Si
p1
q
p7
p3
p6
p4
p5

• Brute-force Algorithm
• finds all the nearest neighbors until there is
  one in the queried region Si.
• ?inefficient! (as shown in the figure)

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How to find NN(q) in Si

• The only difference between the NN algorithm
  proposed by RKV95 and conditional NN algorithm
  resides only in the metric used to sort and prune
  the list of candidate nodes.

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New MINMAXDIST definition
Mindist(q, M)
Mindist(q, M)
Minmaxdist(q, M)
Minmaxdist(q, M)
queried region S
queried region S
MINMAXDIST(q, M, Si) distance to furthest
vertex on closest face IN Si MINDIST(q, M, Si)
MINDIST(q, M)
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New metric definition
Mindist(q, M, Si) Mindist(q, M) Because
mindist(q, M) is valid for all case, since it
provides a definite lower bound on the location
of data points inside an MBR, although a little
bit looser.
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CNN/NN algorithm difference

• When expanding non-leaf node during the
  descending phase
• NN Search
• Order all its children by the metrics (MINDIST
  or MINMAXDIST) and sort them into an Active
  Branch List (ABL). Apply downward pruning
  techniques to the ABL to remove unnecessary
  branches.
• CNN Search-- build a set of lists
  branchListinodecard 0
  ? the list whose pointer points to the
  children of that node and overlaps with region
  (i1)
• i num_section ?the list contains the counter
  (for each child) the total number of sections
  overlaps with this child ? child with higher
  counter is visited first for I/O optimization.

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Other NN related researches

• NN and RNN for moving objects BJKS02
• CNN PTS02
• RNNA over data streams KMS02

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Conclusions

• The RNN algorithm proposed is based on using the
  underling indexing data structure (R-tree), also
  necessary to answer NN queries.
• By integrating RNN queries in the framework of
  already existing access structures, the approach
  developed in this paper is therefore algorithmic
  and independent of data structures constructed
  especially for a set of such queries.
• No additional data structures are necessary,
  therefore the space requirement does not
  increase.

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References

• RKV95 N. Roussopoulos, S. Kelley, and F.
  Vincent. Nearest neighbor queries. In SIGMOD,
  1995.
• SAA00 I. Stanoi, D. Agrawal, and A. El Abbadi.
  Reverse nearest neighbor queries for dynamic
  databases. In Proceedings of the ACM SIGMOD
  Workshop on Data Mining and Knowledge Discovery
  (DMKD), 2000.
• KM00 Korn, F. and Muthukrishnan, S., Influence
  Set Based on Reverse Nearest Neighbor Queries.
  SIGMOD, 2000.
• BJKS02 Benetis, R., Jensen, C., Karciauskas,
  G., Saltenis, S. Nearest Neighbor and Reverse
  Nearest Neighbor Queries for Moving Objects.
  IDEAS, 2002
• PTS02 Papadias, Tao, Y. and Shen, D.,
  Continuous Nearest Neighbor Search. VLDB, 2002.
• KMS02 Korn, F., Muthukrishnan, S. and
  Srivastava, D., Reverse nearest neighbor
  aggregates over data streams. VLDB, 2002.

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Questions and Answers

• Any Questions?

?
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Thank you for attending this presentation!