KIS. 5B6. Russ Miller$. Andrew. Rau-Chaplin~. Department of Computer. Science. School of Computer. Science. State ... Kirkpatrick hierarchical search. DAGs).
Multisearch
Techniques
for Implementing
on a Mesh-Connected (Preliminary
Mikhail
J.
Department
of
West
IN
Russ Department State
Frank School
Science
47907,
NY
14260,
USA.
of Comp. Chiayi,
Sci.
Taiwan
*Research
search path associated with a search that the paths are determined “on-
partially
supported
by the Office
in
of Naval
Re-
search uuder
Contracts
NOO014-84-K-0502
and NOOO14-86-
K-0689,
Air
Office
Research
the
Force
Grant
AFOSR-9O-O1O7,
under
Grant
the
DCR-S451393,
Medicine under Grant t Research partially
of Scientific National
Science
and
National
and Engineering Research Council ~Research partially supported Foundation
under
$Research
Grant
partially
and Engineering
fice of the and
the
Scientific National
under
Foundation Library
Natural
of
and
Inform.
62107,
+ rfi)
1
Science
time
supported Council
supported
by the
by the Office
NOO014-84-K-0502, Research
Science
Natural
Sciences
Given
of Canada.
under
the
Grant
Foundation
of Naval Air
Force
Re-
on
Of-
Grant
that
structure
on that
DCR-
nodes,
structure,
performing
AFOSR-9O-O1O7,
under
a search
constant-degree
the
w fast
x
mesh-connected
fi
for Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republisb, requires a fee and/or specific permission.
structure.
The
0897914384/91/0007/0204
order,
in parallel
each.
trace
However,
at (say) should
$1.50 204
the
node visit
‘(on-line”:
as a graph O(n)
mrdtisearch
com-
problem
and
need
not
for
example,
pat h that ahead
is that
only
when
be pro-
one
processor query
and must a search which
(it does so by comparing
of
processes
be processed
a search
of time,
n
processes
can simultaneously
v of G can it determine next
G with
search
all of the search
searches
by using,
in G is not known
be determined
modeled and given
as possible
in any particular
ACM
on a &
Introduction
cessed
1991
Eng.
ROC.
Sciences
of Canada. by the National
8451393.
@
5B6.
IRI-S800514.
Research
WResearch partially search under Contract
the
RO1-LM05118. supported by the
KIS
puter. For most data structures, the search path traversed when answering one search query has length r = O(log n). For these cases, our algorithm processes O(n) snch queries O(4. The classes of in asymptotically optimsl time, graphs considered contain most of the important data structures that arise in practice (ranging from simple trees to Kirkpatrick hierarchical search DAGs). Multisearch is a useful abstraction that models many specific problems and can be used to implement parallel data structures on a mesh. Applications include interval trees and the related multiple interval intersection search, as well as hierarchical representations of polyhedra and its many applications (e.g., lines-polyhedron intersection queries, multiple tangent plane determination, intersecting convex polyhedra, and three-dimensional convex hull).
The rnrdtiseurch problem consists of efficiently performing O(n) search processes on a data structure modeled as a Denote by r the graph G with n constant-degree nodes.
problem
Canada
University
O(W
the multisearch
Science
University
Tsayf Cheng
Abstract
we solve
5B6.
of Computer
Ottawa,
Chung
KIS
Rau-Chaplin~
Carleton
at Buffalo
Jyh-Jong Inst.
Canada
School
Science
York
Science
University
Andrew
Miller$
of New
Computer
Ottawa,
USA.
National
length of the longest process, and assume line”. In this paper,
of
Dehnet
Carleton
of Computer
University Buffalo,
Version)
University
Lafayette,
Structures
Computer
Atallah*
Computer
Purdue
Data
will
instead query
node
is
of G it
its own search
key to the information information on the
and
specific
problem
problems
problem
being
for
on this
since
single
of G, creating
node
the added of time
many
how much
the nodes model
used
in the
graph
tant
to keep
topology that
cessor
not
necessarily
the
to the
start
processor
list.
search
most)
which
the
v.
case that
node
of
lated
search
G stored
EREW-PRAM,
“exclusive
read”
at least
apparently,
given
An
by
tions
on
afford
The
the
in
network
process
(say)
node
neously
something
In such but
for
more
than
more
is not
time
to
can
be
the
different
ERE W-PRAM, only
a processor
would
by
one
The
main
contribution
@
x &
length
of
problem
of in
mesh-connected the
longest
this O(@
paper
computer, search
path
where associated
r
on is with
on the
its myriads intersectic,n three-
convex
polyh-
are of considerable
im-
computational etc.
In additiolm,
problem
in parallel
the
that
that
it
we have
databases
broad
very
and
re-
mesh
in order
queries
to
an approach to obtain
by one step
the needs
o,ptime to be
network. the multisearch
different
our
all
Such
on multisearch,
queries
syn-
proportional
to move
are also very
terms,
queries
time
paths.
since,
of the
a judicious
technique
the search
required
based
are very
muitiproces-
hypercube
we use to solve
they
are
hypercube
network
of all
mesh
In
for That
search
diameter
and
.
Partitioning these
solving
time
of the
techniques
for
of search
is in
+ r%)
re-
as hi-
determination,
of moving
algorithms
[DR90],
●
multisearch
com-
from
those
different
from
techniques
for
combination
pro’bused
solving
of the
in
[PVS83]. the
following
ideas:
to simulta-
number
in
the
as well
applications
G, and
in their
the
problem
containing
be unable
a constant
lem
data
later
and
and
a fundamental
problem
advancement
The
a
to be permuted to
accessed
since
data
queries.
the
viable mesh
less than
is not
trees
modeling,
in [D R90].
through
nodes
per
over
(and
of implement-
recognition,
(perhaps
diameter
next
challenging
models,
access
similarly
a time,
the
the
which
is distributed
considerable
v‘s information
store
keys
shall
search,
problems
additional
on the idea
chronously to
We
interval
and solid
multisearch
was bused
we
ab-
problems
and intersecting
pattern
explored
the
applica-
compu-
areas).
timai
is even
processors
location at
search
yet
was
which
to
that
is a useful
problem
plane
hulll,
has many
The
a lin-
trees
DAG
parallel
of polyhedra
is such
SOTS was studied
for
assume
we also consider
memory,
different
memory
query
[PVS83]
keys,
since
of processors.
Furthermore,
each
problem
they
search
and requires
items.
Wagener
problem
a shared
allow
this
and
lines-polyhedron
these
vision,
probably
v‘s
simple
search
intersection
in robotics
multisearching
not
are cases
on a mesh-connected
tangent
portance
can be used).
multisearch
stored
multiple
Note
lated
the
including
that
the
solve
interval
that
are,
optimid
considered
from
them).
include
edra.
access
(although
multidimensional
ordering
for networks
around
and
to do here
involving
no linear
way
to
does
processes
length
algorithm
important
specific
solve
structures
convex
if k processes
search
has
our
multisearch many
representations
queries,
v‘s information,
k search
it
multiple
pro-
the
of the
sequential
to
Applications
of applications
from
to simultaneously
G is a 2-3 tree
ordering
cannot
unable
Vishkin
comes
of the model:
to these
elegant
Paul,
case where
difficulty
access node
assigned
information.
to
restriction
to simultaneously
k processors
ear
the
graphs
(ranging
models
data
geometry,
In the
of most
hierarchical
mentioned,
dimensional
in
cases
the
in asymptotically
classes
in both
use
erarchical
he search
these
contain
be used
in a
processor).
were
paper parallel
network’s
Each
can
the
puter.
one oft
a query
in practice
that
ing
is impor-
be stored
containing
(at
(in
not
answering
Kirkpatrick
already
hence
G, so
structures,
geometry).
As
of pro-
structure
arise
tational
one
data
queries
The
is so important
network
It
such
They
powerful
straction
containing
computational
one
at
in the
below.
parallel
is a network stored
O(W.
listed
only
most
when
O(n)
time,
the
since
When
v in G need
to be processed
at a node,
for
traversed
of G that
ahead
start).
as the
also cent ains initially
queries
even tally
occur
a
(with
that,
O(lOg~). Th&t is, for
=
processes
of
visit
paths,
the
of node
to
problem
adjacency
same
adjacent
want
search
each
that
networks
the full
problem
node’s
is not the
processor
they
with
in mind
a neighbour
will
of time
the
that
for
might
G is initially
way,
of G and
and
we cannot
congestion
ahead
natural
node
that
to solve
the
It is a challenging
a “congestion”
of G at which
cessors,
can be used to solve
Note
path r
multisearch
that
searches
complication
we do not know
The
query.
depend
later).
EREW-PRAMS
processors,
of this
performed
solved).
abstraction
(more
both
at v — the nature
comparison
problem
is a useful
many
stored
of the
G into
in sequence,
Making
many
pieces others
copies
and
processing
some
of
in parallel.
of some
pieces
of G (the
“bck-
1The 3-D convex hull problem has optimal mesh solutions, recently obtained independently of ours and using
a
the
different, tisearch
a
205
purely method
geometric
approaches
we use [LPJC90,
rather
H190].
than
the mul-
t leneck”
ones,
trying
to
these
copies
then Of
i.e.,
go
making
some
the
of the
copies
take
time
store
but
there
all
these
copies
to
store
0(1)
space
each
would
time,
since
is not
even
enough
of G (there copies
U
to
G
some
portions
pieces
of the
of G
mesh
(not
into
suitably
necessarily
the
has
n
f(start,
shaped
rectangular
(v, .f(v,
q))
for ~ E.
more,
f(v,
q and
can
q)
cations, we
and
do
paths of
mappings
not
develop
above-mentioned
yet
(in
how
the
problem
fact
time).
must
even
The
instead
through
G.
can
ideas,
The
The
above
cal
directed
short),
tions
first
one
some
exact
some
p >
handle
G
]Li
