Approximate parallel
scheduling. Part I: The basic
technique
with
applications
to optimal parallel list ranking in
logarithmic
time
by
Richard Colet
Uzi Vishkint
Ultracomputer Note
#110
Computer Science
Department Technical Report
#244
October, 1986
Ultracomputer
Research
Laboratory
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New
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Courant
Institute of
Mathematical Sciences
Division
of Computer
Science
251
Mercer
Street, New York, NY 10012
Approximate parallel
scheduling.
Part I:
The
basic
technique
with
applications to
optimal
parallel list
ranking
in
logarithmic
time
by
Richard Colef
Uzi
Vishkint
Ultracomputer
Note
#110
Computer
Science
Department
Technical
Report
#244
October,
1986
t
This
research
was
supported in
part by NSF grant DCR-84-01633 and
by an IBM
Faculty
Development
Award.
iThis
research
was
supported in
part by
NSF grants NSF-DCR-8318874 and
NSF-DCR-
8413359,
ONR grant
N00014-85-K-0046
and by the Applied Mathematical
Science
subpro-
gram
of the
office of
Energy
Research,
U.S. Department of Energy under
contract
number
DE-AC02-76ER03077.
ABSTRACT
We
define a
novel
scheduling
problem; it is
solved in
parallel by
repeated,
rapid,
approximate reschedulings. This
leads to the
first
optimal
logarithmic time
PRAM
algorithm for list ranking.
Companion
papers
show how to
apply these results to
obtain
improved
PRAM upper
bounds for a
variety
of
problems on graphs, includ-
ing:
connectivity, biconnectivity,
minimum
spanning tree,
Euler
tour and st-
numbering, and
a
number of
problems on trees.
1. Introduction
The model of
parallel computation
used in this
paper is a
member of the
parallel ran-
dom access machine (PRAM)
family. A
PRAM employs
p
synchronous processors all
having access to a common memory.
In this paper
we
use
an
exclusive-read
exclusive-write
(EREW)
PRAM. The
EREW PRAM does
not
allow simultaneous access by more
than
one processor to the same
memory location for read or write
purposes. See
[Vi-83]
for a
survey
of results concerning
PRAMs.
Let Seq(n) be
the
fastest
known worst-case
running time of
a sequential algorithm,
where
n
is the
length of
the input for the problem at hand.
Obviously, the best upper
bound
on the parallel time
achievable
using
p
processors,
without improving the sequential
result, is of
the form 0(Seq(n)/p). A parallel algorithm
that achieves this running time is
said to
have optimal speed-up or more simply to be
optimal. A primary goal in parallel
computation is to design
optimal algorithms that also run as
fast as possible.
Most
of the problems
we
consider can be
solved
by parallel algorithms that obey the
following framework.
Given an input
of
size
n
the parallel
algorithm employs a
reducing
procedure to
produce
a
smaller instance of the same problem (of size
^
n/2, say).
The
smaller problem is
solved recursively
until this brings us below some threshold for the
size
of
the problem.
An
alternative procedure is
then used to complete the
parallel algorithm.
We refer the reader
to [CV-86d]
where this algorithmic
technique,
which is
called
accelerating
cascades, is
discussed.
Typically, we need
to reschedule the
processors
in
order
to apply the
reducing
procedure efficiently to the smaller sized problem.
Suppose
the input for a
problem of
size n
is
given
in an array of size n.
A
natural
approach
is
to
compress the
smaller
problem
instance into a smaller array, of size
s
n/2.
This
is
often
done
using a prefix
sum
algorithm
(it
takes 0(log
n) time on n/log n
processors to
compute
the
prefix sums
for n
inputs
stored
in an array). Thus if we need to
reschedule
the