The
Omega
Test:
programming
a fast
algorithm
for
William
Dept.
of Computer
Science
Univ.
and
integer
dependence
analysis
*
Pugh
and Institute
of Maryland,
practical
for Advanced
College
Park,
Computer
MD
Studies
20742
pugh@cs .nrad. edu, (301)-405-2705
Abstract
equalities
The Omega test is an integer programming
algorithm
can determine
whether
a dependence
exists between
The
that
two
gramming
j
array references, and if so, under what conditions.
Conventional wisdom holds that integer programming
techniques
are far too expensive
to be used for dependence
analysis,
except as a method of last resort for situations
that cannot
be decided by simpler methods.
We present evidence that
suggests this wisdom is wrong, and that the Omega test is
refer
to
write
1 to
= i
A[i,
100
to
j+l]
be
present
evidence
describe
is an
ties
the
array
data
dependence
LYZ89,
LY90,
focused
on approximate
fast
but
only
occurring)
for
the
Omega
Data
whether
*This
cases.
In
may
that
results
other
they
accurately
report
spurious
are
there
an integer
is supported
also
found
quired
to
indicate
scan
grant
to
a set
$01.50
the
of the
the
that
suggests
this
is wrong.
which
an
wisdom
determines
arbitrary
describe
set
of
average
the
more
equali-
that
suggest
required
vectors
subscripts
the
test
Omega
twice
and
Omega
by
for
an
workstation.
by the
than
We
there
linear
time
direction
required
We
whether
experiments
the
time
At
Omega
the
loop
test
to
time
re-
bounds.
is suitable
a more
a number
ynique
This
for
use
in
detailed
keys,
programming
7 that
methods
polynomial
determining
Omega
of
program-
test
such
produce
analysis
as loop
the
is
process
(e.g.,
often
incorpo-
as comput-
substantial
speed
for
each
(dependence
validity
must
problem,
time
the
in
Omega
structured
see [BC86,
vectors
test
as
other
ha.
10W-
a decision
this
[WO182]
of an exponenti.d
In
way,
may
re-
number
vectors
complex
to short-cut
GKT91]).
We
which
Treated
are used
transformations
To be competitive,
be able
the
complexity.
directions
of certain
and
complexity.
situations
yes or no.
direction
interchange).
inethod
time
answer
test
vectors
analysis
many
are accurate,
simply
a dependence
of direction
worst-case
in
dependence
such
the
methods
extension
integer
details,
that
worst-case
tests
of linear
to
an
is a NP-Complete
has exponential
Section
to determine
level,
new
with
in practice.
test
deciding
combines
elimination
of implementation
constraint
in
test
constraints
variable
Dependence
CCR-8908900.
dependence
for situations
methods.
array
the
quire
resort
compilers.
problem:
actuaI
programming
for
special-case
the
equality
order
of last
is rarely
that
(polynomial)
has
4
@ 1991 ACM 0-89791-459-7/91/0004
that
integer
to be used
500 ,usecs on a 12 MIPS
eliminating
show
dependence.
to
time
values
is performed.
that
determine
Conceptually,
ming.
to be
all
to
Fourier-Motzkin
(commonly
report
the
the
simpler,
to
a problem
would
for
W0189,
work
at
to
z and
<loo
holds
We
is less than
production
approximate
equivalent
solution
by NSF
this
certain
situations,
problems
exists
Ban88,
of
read
all programs,
test
pair
analyze
for deciding
are guaranteed
in
dependency
work
but
AK87,
Much
methods
exact
are conservative:
dependence)
BC86,
refer
pro-
example,
<loo
test,
solution
almost
Integer
MHL91].
compute
special
methods
[A1183,
GKT91,
that
inequalities.
that,
We
j]
of methods
the
this
variables
j’
expensive
by
Omega
integer
and
too
decided
improvements
study
time
In
loop
and
as a method
cannot
ing
has been extensive
the
i’
wisdom
except
Omega
There
at the
are far
analysis,
do
= A[lOO,
of
and
Conventional
techniques
do
100
values
below.
j+l=j’
rates
j
the
variables
array
=
shown
programming.
as an integer
a = 1O(I
analysis step in a parallelizing
compiler
(as
well as many other software tools) is data dependence analysis for arrays: deciding
if two references to an array can
refer to the same element and if so, under what conditions.
This information
is used to determine
allowable
program
transformations
and optimizations.
For example,
we can
decide that in the following
program,
no location
of the array is both read and written.
Therefore,
the writes can be
done in any order or in parallel.
i
problem
of integer
be formulated
I<i<j
A fundamental
for
would
l<i’<j’
Introduction
for
a form
problem
is performed
loop
competitive
with approximate
algorithms
used in practice
and suitable for use in production
compilers.
The Omega test is based on an extension
of FourierMotzkin
variable elimination
to integer programming,
and
has worst-case exponential
time complexity.
However, we
show that for many situations
in which other (polynomial)
methods are accurate,
the Omega test has low order polynomial time complexity.
The Omega test can be used to simplify
integer programming problems,
rather than just deciding
them.
This has
many applications,
including
accurately
and efficiently
computing dependence direction
and distance vectors.
1
and inequalities,
above
a dependence
this
Section
enumeration
4, we show
how
the Omega
gramming
test
this
in hand,
that
precisely
and
distance
rectly
in deciding
5.1.
2
The
Omega
Omega
tion
to an arbitrary
test
to
a set
use ~O<,<n
variables
2.1
being
this
the
ing
di-
the
in
Sec-
Omega
a,x,
the
the
zo
c)
is
=
and
of
constraint
tightening
lutions,
and
(not
includ-
not
integer
integer
do not
we compute
al,
evenly
the
. . . . an.
constant
a.
a.,
greatest
the
it
a problem
easier
to
Given
a new
solutions
teger
the
The
if and
Of
problem
UO
not
the
integer
without
solutions,
rational
so-
P
no
that
has
and
equality
inequality
con-
constraints,
pro-
if the
original
problem
in the
process
we might
decide
regardless
of the
no integer
better
simpler
exists
the
constraint
into
all other
~
solve
S@(fIk)(at
result
in
all
substitution
this
mod
rn)zt
constraints.
In
the
original
produces:
la~ I = m – 1, this
~
Since
all
((a,
is equal
to
– (a, m~dm))
+ m(a,
m~dm))z,
= O
terms
are
now
divisible
by
m,
normalizing
the
produces:
+
~
(([at/m+
$
+ (a, m~dm))z,
.
,Cv—{k}
In the
ficient
original
of a i;
the
plicative
need
same
For
of
perform
a unit
the
as the
all
absolute
factor
only
before
constraint.
the
of z k.
changes
inequality
GCD
additional
if there
Since
coefficient
inequali-
of
constraints
solutions
test
[Banerjee88]
constraints.
approach
eliminate
We
= O
is
only
course,
has
integer
somewhat
which
all the
problem
Generalized
lowing
+
the
this
‘lakl~
that
at
other
most
I/m
step
value
value
variables,
of
coefficient
absolute
absolute
value
this
and
suited
solving
constraint.
that
for
our
a,z,
needs,
for
be added
~,=v
be
we found
appropriate
may
equality
a j # O such
by
toward
more
equalities
the
can
However,
in-
used
the
to
fol-
since
it is
situations
in
later.
=
O, we first
check
IaJ I = 1. If so, we eliminate
Xj
and
substitute
the
example,
each
of the
of the
this
1/2,
Since
O(log(max~et,
and
by
a multi-
m
>
_{O}
we can
coef-
original
substitution
coefficient
+
appears
consider
result
the
3, we
la, I ) ) times
eliminate
the
constraints:
7x + 12y + 312
3z+5rJ+14z
has
had
constraints.
eliminate
substitute
con-
we replace
determine
equality
eliminate
solutions.
inequality
and
constraint,
For
involving
we first
integer
–sign(ak).
teV—{k}
constraints
a problem
ducing
=
com-
constraint
tightens
but
m
constraint.
Equality
straints,
ma
—
divide
solutions.
2.2
then
m)z,
~k
xk = ‘Si@I(ak)??ZCI
constraint,
by g (i.e,
term
rational
produce
making
mod
constraint:
tcV—{k}
any
We then
is an inequality
in the
P has
–
a,
the
coef-
produce
is an equality
divide
constraint
dividing
P may
thus
integer
but
to produce
here
coefficients
when
If a problem
produce
we
).
floors
a and
b
b) – b fi
con-
are integers
rational
constraint
not
variable
a mod
(a mod
standard
A normalized
by g. If the constraint
floor
lao/g]
ak
for
constraint
If the
the
Taking
a new
then
else
constraints
coefficients
with
a constraint,
g does
<b/2
and
1
I O < i s n}).
any
coefficients
described
g of the
unsatisfiable.
with
that
constraint
our
coefficients).
coefficients
we take
the
algorithms
the
as follows:
and
set of indices
V = {i
that
of the
=
> 0 as our
tightening)
divisor
we scale
divisor
straint
We create
Note
test
simplify
we define
normalized.
all
with
(k # O) and
solu-
ineqnalities,
To
c). –
(i.e.,
a constraint
To normalize
all the
am~db=ifamodb
‘Iaklma+
(the
non-integer
To
and
<Z<n a, x,
z
we assume
in which
m~d
be quickly
is an integer
to
as ~1
~o<,<n
been
common
coefficients,
ficients
that
used
is 1.
ao)
ties.
input
ai~t
(and
If we are given
mon
there
equalities
(such
paper,
variable
value
:Gv
we use V–to-denote
has
in one
greatest
m = Iak I + 1. We define
-=z(
manipulated
manipulating
straint
described
al~or;thms),
Normalizing
Throughout
are
can
are
whether
= O and
and
of the
absolute
depen-
be
vectors
The
our
a,z,
represen~a~ions,
k be the index
has the smallest
transformations,
distance
as ~l<t<n
(and
can
let
let
that
test
equalities
presentation
information
set of linear
(such
possible
of program
determines
of linear
coefficient
With
a set of constraints
all
Otherwise,
pro-
them.
techniques
as a problem.
inequalities
the
These
integer
deciding
produce
This
and
it.
to simplify
just
describes
validity
direction
The
referred
efficiently
the
from
than
concisely
vectors.
or standard
computed
tion
rather
we can
dency
be modified
can
problems,
The
methods
Figure
of
example,
these
straints,
term
not
these
constraint
no contradictions
constraints
so we
constraints.
ity
resolve
17
=
7
constraints
as shown
in
1.
In this
ing
above
=
know
there
A contradiction
with
or an equality
divisible
and
by the
zero
arose
are
during
no
integer
arises
if we encounter
with
and
the
solutions
conto
the
an equal-
a non-zero
a constant
of the coefficients
solv-
inequality
exist
coefficients
constraint
GCD
there
constant
term
of the
that
variables.
is
This
substitution
resulting
x=–8cr-4y
-z-l
constraints
–7a–2y+3z=3
–24cr-7y+
y=cr+3p
18=–20–1
Figure
llz=
10
–3a–2/3+z=l
z=3cr+26+l
– 21p
2ff+b
=-1
+ 112 = 10
first
to
in
key observation
P
attempt
of equatity
that
We
do
not
and
lower
lower
constraints
2.3
Inequality
been
equality
the
process
can
we report
(such
constraint
the
than
one
has
solutions.
elimination
ply
an
a pair
bound
equality
of
made
6),
on
xk
cases
If neither
and
makes
have
very
2.3.1
use
slip
special
two
O, where
a~ >
x~
and
at
the
all
inequalities
each
psi;
example,
~z=v
so-
moduced
of upp;r
if ~,cv
a~x,
P’
~ O is a;
bound
a,z,
~ O is a
upper
bound
contains:
have
There
to
is staudard
the
This
can
if and
only
there
to
P.
important
is
Motzkin
related
solutions
solutions
if
Fourier-
directly
of integer
be very
integer
P’
More
lack
of integer
of P’
not
solution
P.
[DE73].
hand,
existence
to
to
P’
the
rules
Tightening
the
in verifying
that
P’
solutions.
is an integer
solution
for
Xk to
more
if and
variable
elim-
and
that
arising
true,
can
there
in
a new
we report
involves
practice,
problem
problem
disprove
the
is not
confirm
integer
of
solutions
solutions
but
not
tests
of the
are decisive,
fact
the
that
crack
the
we then
only
between
integer
the
two
upper
bound
jla~
bound
between
Our
is to
must
~t=v
second
The
a,x,
first
is an upper
~ O and
~,cv
a~x,
of these
is a lower
bound.
Resolving
a multiple
(j+
the
is greater
exist
l)la~a~
a j
I and
yet
and
there
— la~ I and
since
jla~a~
the
is
the
the
bound
is
lower
of
bound
Therefore,
and
lower
between
this,
the
that
is a multiple
I + “k.
upper
distance
such
the lower
bound
than
upper
upper
of Iak a~ I between
must
between
the
the
and
the upper
(~ +l)laka~l
If
where
them.
there
a~ 1. Since
distance
exist
case
as possible,
of a~ it is at least
the
a multiple
the
bound
is
upper
and
of “k a~ must
them.
adaptation
produce
of
Fourier-Motzkln
a second
inequality
in
combining
each
to produce
0 > a~.
exist
is less than
than
lower
solutions
the
apart
bound,
laka~ I – la~ I – “k.
a test
tests
as far
doesn’t
lower
maximum
in
forms.
inequalities
the
and
is a multiple
exact
apply
there
upper
of [a~a~ I between
Consider
of I“k a~ I between
ak it is at most
we have
an integer
is a multiple
are
a multiple
greater
we ap-
if there
bounds.
bounds
Since
has
existence
exact,
that
not
has in-
method
guarantees
lower
no
this
that
original
integer
are
only
lower
P that
does
pair
a new
variable
problem
P“,
not
which
involve
of upper
and
x~
lower
elimination
contains
and
the
bounds
every
result
of
on Xk in
P
inequality
~
bound
these
of x~ produces:
If
there
integer
is an integer
value
an integer
When
are the
only
through
by
–aj
and
ak respectively
gives:
of z~
solution
the
same,
if there
as an ezact
Multiplying
of integer
an-
details
Consider
on
existence
x + 2y ~ 5, the
variable,
Fourier-Motzkln
have
through
The
in terms
elimination
we
by
to
variable
does
constraints.
rednndant
prove
if the
test
that
of these
can
and
solution
constraints
eliminations.
that
For
problem
problem
out
we
constraints,
moblem
elimination
reports
This
to
cannot
the
of the
if it
one
produces
test
and
exists.
most
cases
only
If this
that,
of the
it
if and
an adaptation
that
z L. and
above)
on z~.
is a rational
a rational
of tight
constraint
Fourier-Motzkln
most
emxcti
solutions
most
with
attempt
In
solutions.
solution
involve
bound
There
a
than
3X + 2zJ ~
equality
If the
we apply
in
is
at
solutions.
solutions
devised
and
are
If we find
efficiently
z + 2y ~ O and
involves
variable,
integer
teger
the
in-
(e. g.,
has no solutions.
pairs
that
0).
if we find
dealing
two
another
more
contradictory
integer
[DE73]
integer
problem
appropriate
(e. g., given
ination
integer
disprove
is redundant).
problem
it
the
3X + 2y
constraints
constraint
that
elim-
constraints
one
constraints
for
for
eliminate
If
the
equality
to see if anv
3X + 5y s
Therefore,
methods
checking
other
that
as 6 ~
with
to our
W bile
all
check
contradict
equality
constraints.
them
revert
directly
with
inequalities
replace
once
first
3X + 5y ~ 2 and
deal
inequality
first
We
constraints
contradiction,
also
is used
eliminated.
constraints
We
variable
constraints
following
have
to
(as shown
on Zk in P, the
The
of Fourier-Motzkln
do this by checking if there is an integer
P’ that contains
all inequalities
to a new problem
P.
by combining
I
of elimination
We
lutions
solution
–31cl
I
1: Example
is the
ination.
set
of integer
there
aO ~ O (where
6
points
we
know
solution
there
for
the
lower
ao is greater
P.
upper
is an
l’”
if the
to be
than
zero
and
P’
1,
elimination
constraints
in P“).
to
coefficients
of z~ are
The
inequalities
P“
if and
is referred
bound
coefficients
only
P’
to
This
to be exact.
Xk produce
by
solution
to
the
bound
to be exact
for
described
solution
If all
is maranteed
coefficients
P“,
that
be an integer
is an integer
is also guaranteed
non-unit
will
eiirninatzors.
elimination
to
extends
to P.
of z~ are —1. or all
the
solution
that
having
of the
In these
form
cases,
test
takes
advantage
have to consider
both
P’
the
Omega
What
if
integer
ger
P’
integer
has
solutions?
solution
solutions
Then
to
of the fact
we know
it is tightly
P,
that
P’
we don’t
and P“.
but
P“
that
if
nestled
of upper and lower bounds.
Let
coefficient
of x k in any inequality.
teger solution
exists, there must
~,ev
a,z, >0 on X, such that
does
there
not
have
is an inte-
between
some pair
m be the most negative
We know that if an inexist some lower bound
p,!
2: Eliminations
P’ and P“
We produce
integer
each lower
For ~ in the
a new
range
problem
equality
bound
O to
Pj
~,
~v a, z, ~ O on z~ in turn.
[(lmakl–lml
containing
–ak)/lmlJ,
the
of P and
the
constraint:
akzk
=
E
j+
negative
coefficient
consider
P
Since
,cV–{k}
teger
If
we find
an
integer
solution
to
some
lower
bound
just
considered
=
Pj,
report
we
the
we change
in P the
to
~
to
We
range
not
find
the
lower
7x ~ 9y – 8. The
This
9y – 8 + j
the
integer
means
for
all
disprove
O <
to
range
so we
of in-
lower
bound,
11x > 3 – 13y +j
=
~
[“’-~~-”]
=
any
PJ and
we have
on z, so we report
the
existence
next
PA
most
have
solutions,
Pj
j
for
we
J in
the
on to the
consider
solutions
bounds
to
we move
j
in
=
—11.
unable
P,
3 – 13y.
all
in Figure 2. Since P’ has
we consider each lower
does not,
z is
in example
= 5. None of these have
7X ~ 9y – 8 + 6 to P.
we are still
for
as shown
the lower bound
P A 7x
solutions
11x
problem.
performed
P“
of
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add a restriction
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bound in turn.
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we produce
constraints
solutions
unnormalized
comb mat mn
98>0
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62~0
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upper bound
121x <231 – 143y
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lower bound
(33143y)+ 100< 121z
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<490
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Figure
We consider
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combination
1982 (1
190y + 45 ~ o
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upper bound
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lower bound
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that
9.
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do
exhausted
P has
no integer
solutions.
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and
steps
expensive
and
check
if we can
now
solutions
to
P.
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solutions
to
P.
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bound
on Z,$. If we have
we report
that
2.3.2
disprove
we report
existence
there
we move
exhausted
no solution
Choosing
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that
on
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of integer
are
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no
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arise
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eliminate
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bounds.
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we choose
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sible.
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expect
arise
in the
used,
we illustrate
very
hard
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if any,
of typical
(and
an example
perform
test
show
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and
as close
inexact
to zero
as poswill
programs.
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of)
performed
force
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the techniques
the
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Consider
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test
to
to arise
the
type
2.4
Implement
and
time.
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Equalities
rithms
only
represent
Once
decide
are
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exact
to eliminate
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bounds
on T give.:
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the coefficients
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ables
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to be upper
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involving
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constraints
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Performing
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variables.
remain.
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Motzkin
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in practice.
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integers;
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no unbounded
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time
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scheme
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ideas
coefficients.
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to
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milliseconds
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4.5
are possible:
to the
nightmares
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designed
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process,
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expect
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uations
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designed
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coefficients
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To demonstrate
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with
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to
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nightmares
variables
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the variable
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and lower
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eliminate.
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simplifying
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Problems
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analysis
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use
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2, the Omega
in Section
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x,
of
is a solution
allow
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add
elimination.
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problems.
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constraint
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integer.
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negation
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coefficients
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based
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All
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constraints,
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Three
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problem
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protected
simplification.
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One
problem
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When
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simplify
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(not stopping
when an integer solution
is first veriif simplifying
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fied). This could be expensive
many
inexact
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tion
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always)
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ables.
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In order
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protected
to eliminate
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2.2)
variable
sible
constraint.
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For
we report
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explore
those.
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the
current
the
gcd
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i
for
non-protected
problem
protected
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k
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variables
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6
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and
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on the
MXM
treatment
decomposition,
and
NAS
Since
arrays,
them.
vectors
distributed
as several
arrays.
for index
to include
files
6 of
direction
NAS
algorithms,
as well
2 and
bounds
of direc-
this
NASA
tool
sets
applied
the
decomposition,
of wavefront
use of index
treatment
We
7 of
tiny
in loop
exact
compressed
tiny).
tiny
examples),
misleading
have
the
by
in Wolfe’s
compute
to the
5 and
test
max expressions
and
Cholesky
Programs
extensive
and
(aa opposed
(which
cial
min
Omega
constants,
generated
programs
suite
the
on
on
scan
as induction
one common
test
in the
code,
ple”
implemented
results
FORTRAN
feel
Performance
have
Omega
are
avoiding
were
dependence
at least
cases
both
dependence
between
based
is substantially
by
[A191].
We
input
#7:vPENTA
loop
Ths
substitution
CHOLSKY
#5:
to determine
forward
or
to
such
compute
Program
#1:
required
not
programs
could
time
those
(thus
and
the
below
(such
regions
the
array)
workstation
Bounds
be used
here;
subscripts
optimizations
share
timed
analyze
to convert
an
MIPS
#4:
Loop
reported
the
is a dependence
that
#3:
Determining
figures
and
We did
of
be merged.
5.3
Performance
the
recognition
We
affected
regions
those
150
Test
scanning
time
location
test
of another.
merge
be used
approximate
100
(~sec)
3: Omega
Standard
dependence
when
Omega
analysis
by hand.
solu-
be changed.
the
in
while
bounds).
to
have
is a subset
Omega
HK90])
the
by
limited
will
work,
region
Omega
regions
used
more
use the
is not
actually
detected
an
affected
included
not
so
variables
gives
array
Figure
represented.
be
With
to
can
are accurately
check
however,
affected
that
and
the
problem
time
W_
I
con-
constants,
summary
75
copy
vari-
appropriate
of the
The
loop
expressions,
symbolic
50
complex
I
problem
all
protecting
simplified
can
how
as described
subscript
“-’‘1
~~,
t
statement.
and
adding
while
locations’
intersect.
is unclear
and
/’
100
of an array
programming
index
locations
The
locations
I
por-
Omega
assignment
array
the
of the
as strides
regions
can
bounds,
statement.
such
of the
an integer
each
problem,
polyhedron.
can
a single
this
indices
the
call
loop
summary
to characterize
test
constants,
accurate
References
the
symbolic
for
to analyze.
a procedure
up
250
seen
by
by
time
(psecs)
and
example
use
affected
the
(<,=,*,=)
difficult
We
ables
for
Ai
be affected
by setting
Simplifying
of
may
for
straints
sigu
IJT91].
variables
on.
psecs
we need
involving
and
the
=),
most
Array
analysis,
of an array
[Tri85,
for
of (<,>,*,
is the
“A.”-” -
anrdysis
possibilities
example
In interprocedural
tions
the
vectors
This
in our
500
(Redundant)
vectors
excluding
and
the
loop
the
and
the
bounds
dependence
be-
the array
tween
Across
ing
break-down
about
1/2
7
pairs
a range
of
for
the
test
how
time
programs,
time
was
was
spent
we found
spent
the
by the
dealing
follow-
Omega
test:
wit h inequality
con-
lyzed
to remove
effective
each
and
any
redundant
normally).
array
pair
constraints
Based
on
was classified
simple
Any case that
dist antes.
the
(this
is not
simplified
section,
The
time
constraints,
as follows:
apply
the
takes
time.
coupled
dependence
time,
A
case
coefficient
10; Ai
the
but
(for
inequality
at least
<
10}
constraints
one constraint
example,
+ 2Aj
is the
{O ~ AJ
– the
define
Normalizing
last
a
constraint
makes
(the
because
<
Motzkin
th~
pass
time
the
we
might
we
a substitution
O(rnrLp)
At
the
in the
before
worst-case
required
unbound.
to eliminate
least
one variable
last.
and
checking
constraints
for
requires
is only
directly
O(mn)
expected,
not
exworst-
is used).
subproblems
variable
size of the
takes
except
bound
that
performing
constraints
hashing
Producing
+ Aj
become
the
fact
time,
value
substitutions
of passes
or redundant
time
case,
and
worst-case
absolute
the
variables
in each
contradictory
has a non-unit
~ 10; O ~ Ai
that
is eliminated
largest
from
number
of constraints
2.2 to eliminate
log ICI)
log [Cl
constraint,
unbound
p
where
all the variables
where
region
arises
perform
for
of variables.
in Section
the
of the
bounds
are accurate.
number
number
O(mn
with
cost
can eliminate
O(nm)
the
the
is
on parts
time
algorithms
by the methods
the
Eliminating
does not involve
time
denote
constraint
This
to
pected
convex
taken
constraint.
cost-
the
bounds
polynomial
polynomial
C is coefficient
have
time
describe
we use m to denote
equality
where
regular
A case where dependence
dist antes are coupled,
but afl inequality
constraints
have unit coefficients
(for
example,
{Ai > O;Ai + Aj > O}).
convex
other
bounds
generaf
then
n to denote
one
results,
the set of constraints
describdistances for each array pair were ana-
some
and
where
In this
elimination.
To analyze
our
ing the dependence
test,
cases
time
describe
Omega
straints,
about 1/4 of the time was spent on dealing with
equality
constraints,
and I/4 of the time was spent examining simplified
constraints
to construct
direction
vectors.
None of our test cases required
inexact
Fourier-Motzkin
variable
Polynomial
We first
resulting
elimination
subproblems
takes
from
time
Fourier-
proportional
to the
produced.
non-regular).
complex
A csse
non-convex
the
region.
one shown
the
where
We only
below
lower
inequality
and
bound
encounted
another
of the
constraints
define
two
such
one identical
i loop
7.1
a
During
cases,
except
Special
lOdo
= a(j)
end for
flow/anti
above
are
4; –7
{Ai=
test
speed
improvement.
loop
array
dependence
building
<
Aj
[MHL91]
found
in the
The
<
except
the
produce
produce
a hash
2-4 times
adding
or
that
this
cases
a dependence
for
the
the
larger
problem
dependence
the
have
performance
the
pairs,
faster
overall
time
not
for
this
it
at
be
was
cases
percentage
applied
resolve
in
just
exact
They
in
Residue”
over
exactly
by
elimi-
eliminations
a process
found
1/4
algorithm
[Sho81]
each constraint
In a set
x,.
this
variable
property.
On
n2 + n constraints
of
[M HL91]
of
the
unique
Residue
algorithm.
found
that
in
of the
1/4
Thus,
Perfect
the
Club
Banerjee’s
using a dependence
analysis
than the Ome~a
test would
alnot
Loop
Hennessy
encountered
and
could
Generalized
cases,
Residue
cases
of
with
form
test
can
constraints
will
take
and
be
O(n3
) time
by the
Loop
Lam
could
their
can
eliminating
be solved
in
is exact
there
after
algorithm
encounted
in
x, >
elimination
Hennessy
Residue
unique
be applied
[MHL91]
be
applied
study
of the
benchmark.
Maydan,
for
Maydan,
can
is of the form
n variables,
this
that
the
Con-
unique
of constraints,
Omega
imagine
be
time.
applied
preserves
most
that
speed
could
Per
which
(a higher
can
number
Fourier-Motzkin
they
suggests
the
the
builds
of the
performing
) worst-case
property,
the
This
test
“Loop
twice
problem.
increase
and
a set of constraints
analyz-
1/3
can
pairs.
was
in
test
to resolve
that
to
variables
O(mn2
Variable
com-
separately).
redudant
effort
less than
Omega
has
additional
applied,
Benchmark
typically
much
code
not
any
has
problem
be
[MHL91]
the
pairs
the
‘Single
for many
spent
is difficult
in much
and
problem
spent
of the
However,
problem
was
Thus,
Test”
that
know
can
considered
variables.
constraint
perform
the
Club
iust those cases where
zj+c,
x,>c,
ore>
no overall
subscripts
problem.
than
We
array
iff
[MHL91]
to see if any
other
encountered.
The
to
savings
little
do
to
test
were
checks
with
we
to be applicable
unbound
takes
the
caching
significant
of scanning
be added
of computing
be about
cost
problems.
to copy
cases
that
need
Perfect
test
contradictory
applies
cases
that
Omega
constraints
for
not
“Acyclic
those
for
memorization
could
therefore
may
problem.
to significant
cost
would
and
of building
the majority
of array
gorithm
significantly
lead
the
hit
cost
a dependence
required
use
Memorization
not
or even
to improve
9}
do
This
if duplicate
example
{ –4
<
[MHL91]
and
build
the cost
resulting
and
found
and
the
copying
as large
the
trying
time
to
the
pairs
nearly
ing
cases
that
bounds
2-4 times
Lam
However,
would
simple
found
satisfy
the
the
in
a contradiction),
(SVPC)
nating
a problem,
typical,
We
< 10; Ai
a cache
for
Omega
that
Az + Aj
and
test.
verifying
cost
for
distances
performance.
Omega
and
distances
=0}.
better
copying
the
the
Hennessy
to obtain
key
all
9;Aj
Maydan,
to the
dependence
AZ – Aj,
~
moduced
straint”
or
involved
checking
putation.
a(i-j)
The
(and
solu~ions
forj=Oto4do
endf
are
If not
is 2.
not
fori=ito
normalization,
variables
that
cases
they
tests
GCD
found
could
Lam
found
that
be determined
that
be appIied
tests.
their
91%
of the
by constant
Of
the
SVPC,
in 86%
remaining
Acyclic
of the
cases
tests
and
9?Z0 of
or
Loop
unique
cases.
for
depen-
improvements.
The
11
Delta
test
[GKT91]
works
by
searching
dence
distances
propagating
that
that
possible
to easily
cisely.
In
the
mial
test
also
into
treats
the
pendence
analysis
without
problem
methods
tests)
can be solved
In their
study
the
of the
can
could
RiCEPS,
be
tight
any
Since
the
Omega
combination
Acyclic
and
test
time
of the
test,
the
that
Perfect,
SPEC
Loop
solve
Delta
test.
benchmarks
97%
requiring
do
equality
Variable
Residue
be able
than
any
effectively
use
one
Per
test
and
in
be solved
Constraint
and
the
to solve
more
of them
alone.
Related
work
on
Exact
algorithm
test
modified
the
test,
we
Delta
Constraint-Matrix
how
test
efficiently
Lu
and
algorithm
it
works
Chen
for
appears
can
that
to
use of the
terminate
la~],
sim-
not
dependence
an
analysis.
expensive
integer
their
[Tri85]
resenting
affected
Triolet
(22
used
found
array
than
array
have
The
been
Power
combines
the
tightening,
in interprocedural
using
test
Banerjee’s
and
take
no special
tion
except
action
Fourier-Motzkln
for
Motzkin
when
the
is
If
the
sample
use of branch
the
existence
need
to
[WT90]
that
due
ination,
where
to
Maydan,
the
simpler
they
Ancourt
tests
and
Irigoin
of
have
Both
and
should
be
used
describe
If so, then
or
P’
= ~ and
to
P’
of P’,
so that
solutions.
and
they
is
These
has
appropriate
for
to use for
In
we
is exact,.
with
respect
p.sezdo-linear
solutions
con-
iff
P
constraints
determining
the
P ~ P’,
redundant
pseudo-linear
are difficult
from
reduction.
elimination
integer
problem.
are redundant
If so, since
add
~ is
results.
differs
is an exact
they
Since
a separate
constraints
not
la~a~ I –
better
in P that
then
P’
a:
~
P
difference
P“.
gives
~
the
P
the
to be at least
~ P.
t~at
in
produce
a problem
use force
P“
P’
if P’
they
– ak + 1 for
using
generate
constraint
loop
bounds.
determining
has
appear
the
How-
existence
solutions.
they
recent
report
dependence
The
methods
is not
(based
used
clear
that
the
whole
the
work
fully
described,
described
out
integer
in
is not
but
many
programming
high
ana-
[A191]).
the
basic
in Section
of
simply
is fast
a very
to
in
constraints
that
and
5.1,
[A191]
cases,
efficient.
techniques
cost
a production
in their
implementation
a noticeable
amount
mentions
is used
described
elimination
parallelization
project
to that
inccurs
PIPS
PIPS
pseudo-linear
point
using
analysis
sinces
the
elimination
the
not
variable
also state that
does not take
suggest
are
They
state
on
similar
how
handled.
on
variable
appears
ceptable
Hen-
[IJT91]
Fourier-Motzkin
dependence
(that
for
is
system).
ac-
They
dependence
testing
of time compared
with
process.
or disprove
not
found
and
the
Tseng
suggest
variable
instead
in
the
use of
9
Source
A
C language
available
elim-
rectory
situations
to be accurate.
[A191]
to verify
consider
they
– Iajl
check
constraints
They
elimina-
[MHL91]
Fourier-Motzkin
a
it is
solutions.
also
see if those
P.
Fourier-Motzkin
They
methods
they
a sample
so-
Wolfe
Lam
that
the
are
constraint
Maydan,
to verify
(they
far).
Hennessy
expense
are known
thus
by
integral,
methods
solutions
this
by
are
over
but
attempting
the constraints
they
lyze
It
[WT90]
conservative.
of the other
to determine
is not
bound
of integer
implement
and
solution
and
test,
being
used
There
iterate
inequalities,
bounds
P“,
actually
to
framework
and-
Tseng
an inexact
as possibly
that
elimination,
except
laka~l
of P’sand
some
that
rep-
variable
elimination.
performing
result
elimination
the
GCD
variable
nessy and Lam [M HL91]
if none
use They
use back substitution
lution.
for
use in dependence
and
when
lower
than
check
words,
A
expensive
methods
by Wolfe
Generalized
Fourier-
to flag
be
Fourier-Motzkin
described
described
their
introducing
problem.
loops
They
P“
and
to
do not
of integer
regions).
of
on
analysis.
to
simpler
data
bv
the
of integer
P’.
to our
respect
ever,
rep-
our
upper
they
useful
comfor
producing
an inexact
to
conservative
integer
method
techniques
techniques
implementations
elimination
ysis.
longer
affected
SeveraJ
regions
Fourier-Motzkln
to 28 times
resenting
Fourier-Motzkin
haneasily
test.
by a set of linear
performing
straints
piler.
Triolet
for
could
performance
into
existence
as opposed
If
programming
However,
they
elimination
to use them
the
the
known
clear
for use in a production
any
inexact
how
equaJ
other
The
and
describe
described
is similar
to
[LC90]
Their
key differences
techniques
GCD
space
constraints
programming.
special
Generalized
for
more
exactly
in practice.
describe
prohibitively
fail
an iter-
is as follows:
although
useful
They
makes
integer
not
any
useful
between
Dependence
[Wa188]
for
The
constraints,
handle
problem
any
test,
problems
describe
constraints
with
Constraint-Matrix
plex
ours.
work
over
inequalities.
of
Analysis
The
their
in Section
for iterating
of linear
with
pseudo-linear
Rather,
8
do
disprove
(effecby
not
unclear
percent
the
They
Thev
to as MIV
Omega
that
that
a set
an inte-
described
algorithm.
or approx-
by the
problem
Single
it should
efficiently
can
any
and
or project,
concept
bounds
by
overlaps
work
dling
When
polynomial
expect
our
They
de-
tests.
tive)
loop
described
use Banerjee’s
achieves
by the
time
without
the
as a sin-
refer
found
in-
Since
Therefore,
they
between
in polyno-
algorithms
they
solved
has significant
the
to simplify,
(the
constraints,
be solved
to what
EISPACK,
work
problem
test.
in polynomial
and
cases
MIV
that
resorting
tests,
it automatically
Delta
to exponential
(i.e.,
LINPACK
of
of the
resorting
imate
and
problem,
effects
space
converting
analysis
4) so as to determine
pre-
ation
constraints.
dependence
programming
propagation
and
equality
(and
equality
eliminate
problem
determine
MIV
it efficiently
variable
ger programming
it
distances
of
Motzkin
then
of making
can
use
of solving
constraints
constraints
test
gle integer
test
algorithm
determine
inequality
equality
the
dependence
the
and
intent
their
without
will
determined,
the
other
by a combination
tightening
Omega
easily
with
where
distance
time)
be
determine
cases
a dependence
Omega
can
information
Fourier-
12
for
code
availability
implementation
anonymous
pub/omega.
of
f tp
from
the
Omega
f tp.
test
is freely
cs. umd. edu
in
di-
10
Conclusions
Conservative
cious
dependence
for
forming
the
SIMD
Also,
lems
that
analysis
loop
have
that
only
adequate
FORTRAN
program
used
to determine
efficiently
direction
and
distance
easier
transformation
made
bounds
Omega
prob-
problems
and
build
For example,
loop
program
it can
in
work
manner
other
uses.
and
that
for deter-
[A191],
and
considers
Thanks
to
everyone
who
Z. Li and P. Yew.
Some results
on exact data deIn D. Gelernter,
A. Nicolau,
and
pendence
anrdysis.
D. Padua, editors,
Languages
and Compders
for Parallel Computing.
The MIT Press, 1990.
[LYZ89]
Z. Li, P. Yew, and C. Zhu. Data dependence
analysis
on multi-dimensional
array references.
In Proceedings
International
Conference
on Superoj the 1989
ACM
computing,
June 1989.
[MHL91]
D. E. Maydan,
J. L. Hennessy,
and M. S. Lam.
Efficient and exact data dependence
analysis.
In A Ckf
Conference
on PTogTamming
Language
SIGPLAN’91
Design and Implementation,
1991.
[Mur71]
Y. Muraoka.
Parallelism
EzposuTe
and Ezplottatton
in Programs.
PhD thesis, Dept. of Computer
Science,
University
of Illinois
at Urbana-Champaign,
February
1971.
[Pug91]
William
Pugh.
Uniform
methods
for loop optimization.
In 1991 International
Conjer-ence
on Supercomputing,
Cologne,
Germany,
June 1991.
[Sh081]
R. Shostak.
Deciding
linear inequalities
loop residues.
Journal
oj the A CM,
October
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[Tri85]
R. Triolet.
structuring
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Champaign,
[wa188]
D. Wallace.
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of multi-dimensional
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In Proceedings
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[Pug91].
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