A Decomposition
of Multidimensional
Point
Sets with
Applications
to
kNearestNeighbors
and
nBody
Potential
Fields
PAUL
B.
CALLAHAN
AND
S.
RAO
Johns Hopkins
University,
Baltimore,
Maiyland
KOSARAJU
Abstract.
We define the notion
of a wellseparated
pair
decomposition
of points in ddimensional
space. We
then
develop
efficient
sequential
and
parallel
algorithms
for
computing
such
a
decomposition.
We apply the resulting
decomposition
to the efficient
computation
of knearest
neighbors and nbody potential
fields.
Categories
and Subject
Descriptors:
F.2.2 [Analysis
of Algorithms
and
Problem
Complexity]:
Nonnumerical
Algorithms
and Problems—geometrical problems
and
computations
F.1.2
[Compu
tation
by Abstract
Devices]: Modes of Computation—parallelism
and concurrency
General Terms: Algorithms,
Theory
Additional
Key Words and phrdses: Afl nearest neighbors, fast multipole
method
1.
Introduction
We
define
the
notion
of
a
wellseparated
pair
decomposition
of
a
set
P
of
n
points
in
d dimensions.
This
consists
of
a bina~
tree
whose
leaves
are
points
in
P,
with
internal
nodes
corresponding
to
subsets
of
P
in
the
natural
way,
and
a
list
of
pairs
of
nodes,
such
that
the
sets
corresponding
to
each
node
are
geometrically
separated
in
a manner
to
be
defined,
and
each
distinct
pair
of
points
is “covered”
by
exactly
one
of
the
pairs
of
nodes.
()
We
show
that
although
there
are
;
pairs
of
points,
we
can
always
find
a
wellseparated
pair
decomposition
using
O(n)
pairs
of
nodes.
Additionally,
we
show
that
such
a
decomposition
can
be
computed
in
O(n
log
n)
sequential
time,
which
we
prove
is
optimal,
and
in
0(log2n)
time
on
a
CREW
PRAM
using
O(n)
processors.
Using
this
decomposition,
we
show
that
the
knearest
neighbors
of
each
point
can
be
computed
in
O(n)
sequential
time,
and
in
O(log
n)
parallel
time
on
a CREW
PRAM
with
0(n)
processors,
for
any
fixed
k.
Note
that
this
gives
The work of both authors was supported by the National
Science Foundation
under grant CCR
9107293 and by NSF/DARPA
under grant CCR 8908092.
Authors’
address: Computer
Science Department,
Johns Hopkins
University,
Baltimore,
MD
21218.
Permission to copy without
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is granted provided that the copies are
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for direct commercial advantage, the ACM
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of the publication
and its date appear, and notice is given that copying is by permission of the
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To copy otherwise, or to republish, requires a fee and/or
specific permission.
Q 1995 ACM
00045411/95/01000067
$03.50
Journalof the Associationfor ComputingMachinery,Vol 42,No, 1,Jmuay 1995,pp
6790,
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P. B.
CALLAHAN
AND
S. R.
KOSARAJU
O(logzn)
parallel
time
for
the
whole
problem,
including
the
construction
of
the
decomposition.
This
is the
first
parallel
algorithm
for
this
problem
that
runs
in
deterministic
O(logcn)
parallel
time
with
O(rz ) processors
for
a c
that
is not
a
function
of
d.]
The
composed
sequential
algorithm
has
a strong
similarity
to
that
of
Vaidya
[1986;
1989].
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 Fall '09
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 Algorithms, Big O notation, Parallel algorithm, P. B. CALLAHAN, fair split tree

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