Practical Dependence Test

Gina Goff, Ken Kennedy, Chau-Wen TsengPLDI ’91

presented byChong Liang Ooi

Contribution

Efficient and precise dependence test is essential

General tests (Banerjee and GCD) are unnecessary

Most array refs in scientific Fortran program are simple

Based on these simple cases, this paper proposedPartition-Based Algorithm

Classification of Subscripts

1. ComplexityNo of unique array indices a subscript hasi. ZIV– Zero Index Variableii. SIV – Single Index Variableiii. MIV – Multiple Index Variable

DO 10 i

DO 10 j

DO 10 k

10 A(5, i+1, j) = A(N, i, k) + c

Classification of Subscripts

2. Separability

Separable if indices do not occur in other subscriptOtherwise, coupled

A(i, j, j) = A(i, j, k) + c

Partition-Based Algorithm

Partition subscripts separable & minimal coupled group

Label each subscript as ZIV, SIV or MIV

Apply Single Subscript Test based on complexity

Apply Multiple Subscript Test to coupled group

If any test yields independence, no dependence exist

Otherwise, merge all direction vectors into single set

Single Subscript Test – ZIV

ZIV takes 2 loop invariant expressions

Proves 2 expressions cannot be equal

Can be extended for symbolic expressions

If differences is non-zero constant independence

SS Test – Strong SIV

Ref pair of form: <ai+c1, ai’+c2> for a Є [1,10]

Dependence Distance, d = i’-i = (c1-c2)/a

Dependence exist, if |d| <= U - LDependence Direction =

< if d > 0= if d = 0> if d < 0

Exact & efficientExtendable to Symbolic Expr by eval d symbolically

SS Test – Weak-Zero SIV

Ref pair of form: <a1i+c1, a2i’+c2> for a1 != a2

a1i+c1=a2i’+c2 is a line in 2D space of i vs i’

Check whether line intersect with any integer points

Weak-zero SIVFor a1=0 or a2=0

Let a2=0 i=(c2-c1 )/a1

Check i Є I and |i| < U - L

SS Test – Weak-Zero SIV

Usually, i=0 or last iterationLoop peeling transformation can help

DO 10 i=1, N10 Y(i, N)=Y(1,N)+Y(N,N)

Y(1,N)=Y(1,N)+Y(N,N)DO 10 i=2, N-1

10 Y(i, N)=Y(1,N)+Y(N,N)Y(N,N)=Y(1,N)+Y(N,N)

SS Test – Weak-Crossing SIV

Weak-Crossing SIVFor a2 = -a1

Let i=i’ i=(c2-c1 )/2a1

Check |i| < U – L and i Є I or 1/2

Typically in Choleskey decomposition

SS Test – Weak-Crossing SIV

Loop splitting transformation can help

DO 10 i=1, N10 A(i)=A(N-i+1)+C

DO 10 i=1, (N+1)/210 A(i)=A(N-i+1)+C

DO 20 i=(N+1)/2+1, N20 A(i)=A(N-i+1)+C

SS Test – Restricted Double Index Var

Ref pair of form: <a1i+c1, a2j+c2>

SIV Tests can be used with 2 loop bounds for i & j

Coupled Subscripts – Delta TestSubscript-by-subscript test may yield false dep.Delta Test Algorithm

Delta Test ConstraintsAssertions on indices derived from subscripts<a1i+c1, a2i’+c2> a1i - a2i’= c2 - c1

Constraint vector, C=(del1, del2, …)

one constraint for each index in the coupled group

Del can bedependence line: <ax+by=c>dependence distance: <d>dependence point: <x,y>

Intersecting Delta Test ConstraintsIf the intersect of all constraints is empty set

no dependence

The End

Questions?