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Reverse
k
NN Search in Arbitrary Dimensionality
Yufei Tao
§
Dimitris Papadias
†
Xiang Lian
†
§
Department of Computer Science
City University of Hong Kong
Tat Chee Avenue, Hong Kong
taoyf@cs.cityu.edu.hk
†
Department of Computer Science
Hong Kong University of Science and Technology
Clear Water Bay, Hong Kong
{
dimitris,
xlian
}
@cs.ust.hk
Abstract
Given a point
q
, a
reverse k nearest neighbor
(R
k
NN)
query retrieves all the data points that have
q
as one of
their
k
nearest neighbors. Existing methods for processing
such queries have at least one of the following
deficiencies: (i) they do not support arbitrary values of
k
(ii) they cannot deal efficiently with database updates, (iii)
they are applicable only to 2D data (but not to higher
dimensionality), and (iv) they retrieve only approximate
results. Motivated by these shortcomings, we develop
algorithms for
exact
processing of R
k
NN with
arbitrary
values of
k
on
dynamic multidimensional
datasets. Our
methods utilize a conventional datapartitioning index on
the dataset and do not require any precomputation. In
addition to their flexibility, we experimentally verify that
the proposed algorithms outperform the existing ones
even in their restricted focus.
1. INTRODUCTION
Given a multidimensional dataset
P
and a point
q
, a
reverse nearest neighbor
(RNN) query retrieves all the
points
p
∈
P
that have
q
as their nearest neighbor.
Formally, RNN(
q
) = {
p
∈
P

¬∃
p'
∈
P
such that
dist
(
p
,
p'
) <
dist
(
p
,
q
)}, where
dist
is a distance metric (in this paper we
assume Euclidean distance). Although the problem was
proposed recently [KM00], it has already received
considerable attention due to its importance in several
applications
involving
decision
support,
resource
allocation, profilebased marketing, etc. Other versions of
the problem include (i)
continuous
RNN [BJKS02], where
P
contains linearly moving objects with fixed velocities,
and the goal is to retrieve all RNNs of
q
for a future
interval; (ii)
bichromatic
RNN [SRAA01] where, given a
set
Q
of queries, the goal is to find the objects
p
∈
P
that
are closer to some
q
∈
Q
than any other point of
Q
; (iii)
stream
RNN [KMS02], where data arrive in the form of
streams, and the goal is to report aggregate results over
the RNNs of a set of query points.
This
paper
focuses
on
conventional
(i.e.,
monochromatic
) reverse nearest neighbor queries. In
addition to single RNN search, we deal with
reverse k
nearest neighbor
(R
k
NN) queries, which retrieve all the
points
p
∈
P
that have
q
as one of their
k
nearest neighbors.
Specifically, R
k
NN(
q
) = {
p
∈
P

dist
(
p
,
q
)
≤
dist
(
p
,
p
k
),
where
p
k
is the
k
th farthest NN of
p
}. Figure 1.1 shows
four 2D points, where each point
p
is associated with a
circle covering its two nearest neighbors For example, the
two NNs of
p
4
(
p
2
,
p
3
) are in the circle centered at
p
4
.
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 Fall '07
 CYRUSSHAHABI
 Computer Science

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