dependence testing

Dependence TestingOptimizing Compilers for Modern

Architectures, Chapter 3Allen and Kennedy

Presented by Rachel Tzoref and Rotem Oshman

Part II

Roadmap Single Subscript Tests:

SIV tests MIV tests

GCD Test Banerjee Inequality Trapezoidal Banerjee Inequality

Testing in coupled groups: Delta Test Empirical results

MIV SubscriptsDO i1 = L1, U1

DO i2 = L2, U2...DO in = Ln, Un

S1 A(f(i1, ..., in)) = ...S2 ... = A(g(i1, ..., in))

ENDDO...ENDDO

ENDDO

MIV Subscripts We will assume that

MIV Subscripts Rearranging, we get

?

GCD Test Theorem: the equation

has a solution iff

No solution ) no dependence!

divides

GCD Test: drawbacks Doesn’t take into account loop bounds The most common GCD is… 1

The Banerjee Inequality Input:

A pair of array references A(f(i1, ..., in)), A(g(i1, ..., in))

Loop boundsL1, U1, …, Ln, Un

A direction vector D = (D1, …, Dn) Output: whether or not a dependency exists

in direction D

The Banerjee Inequality (cont’d)

Define

Let R be the region defined by

The Banerjee Inequality (cont’d)

Does h(x1, …, xn, y1, … yn) = 0 have a solution in the region R?

iff

The Banerjee Inequality (cont’d)

Recall:

Define:

The Banerjee Inequality (cont’d)

The problem is reduced to:

Finding is a plane:

Finding Define

Finding

What about the direction vector?

?

Lemma

Lemma: proof

Finding

The Banerjee Inequality We have:

The Banerjee Inequality Similarly, we obtain:

The Banerjee Inequality:Wrap-up

Dependency exists in direction D

Example Example:DO i = 1, 100DO j = 1, i - 1

A(j) = A(i + j - 1) + CENDDO

ENDDO

D = ( = , = )

Example:DO i = 1, 100DO j = 1, i - 1

A(j) = A(i + j - 1) + CENDDO

ENDDO

Example (cont’d)

D = ( = , = )

Dependence!

Upon closer examination…DO i = 1, 100DO j = 1, i - 1

A(j) = A(i + j - 1) + CENDDO

ENDDO

Example (cont’d)

No dependence!

Trapezoidal Banerjee Inequality Assume linear bounds:

Trapezoidal Banerjee Inequality Recall:

Dependence Old strategy:

Minimize / maximize each separately:

Calculate for each independently.

Trapezoidal Banerjee Inequality (cont’d)

New strategy:

We would like to do:

.

.

.

.

.

.

min

max

Trapezoidal Banerjee Inequality (cont’d)

The problem: ‘s bounds depend on outer indices:

We can use H- and H+, but we will get

Trapezoidal Banerjee Inequality (cont’d)

Step 1:

Obtain:

Rinse and repeat

Trapezoidal Banerjee Inequality (cont’d)

Algorithm:

Trapezoidal Banerjee Inequality (cont’d)

A closer look:

Suppose Di = “*”. We want to use:

xj or yj ?

source sink

Trapezoidal Banerjee Inequality (cont’d)

To prove the bound for Di = “*“, we use

Lemma:

x’ s AND y’ s !x’s or y’s?

Trapezoidal Banerjee Inequality (cont’d)

To prove the bound for Di = “=“, we use

Lemma:

source sink

Trapezoidal Banerjee Inequality (cont’d)

What happens if we choose wrong? Bounds are still correct

… we gave the loop index “more freedom” than it has in reality

Bounds may not be tight) the test is not precise.

Using the Banerjee Inequality

Banerjee Inequality

2 array referencesLoop boundsDirection vector

Dependenceexists(yes / no)

?2 array referencesLoop bounds

Set ofdirectionvectors

The desired test:

Testing For All Direction Vectors

Naïve approach (brute force):

Testing For All Direction Vectors

Burke & Cytron’s algorithm:

(*, *, *) ?(<, *, *) (=, *, *) (>, *, *)? ?(=, <, *) (=, =, *) (=, >, *)

…

Testing For All Direction Vectors

Burke & Cytron’s algorithm: D: current directionvector

k: currentcoordinate

dvlist: set ofvectors foundin prev. calls

Roadmap Single Subscript Tests:

ZIV SIV MIV

Testing in coupled groups: Delta Test Empirical results

Reminder Coupled group:

DO i = 1, 100DO j = 1, 100 A(i + 10, j) = A(i, i - 5)

ENDDOi + 10 ij i - 5

, ,< > < >

coupled

Motivation

Delta

TestDirectionvectors

CoupledSubscriptgroup

Motivation What happens if coupled subscripts are tested

separately? Result = “independent”

really independent Result = “dependent”

not necessarily dependent Goal: use information provided by other

subscripts where the index appears

DO i = 1, 5 A(i + 10, i) = A(i, i)ENDDO

DO i = 1, 100A(i + 1, i) = A(i, i + 1) + C

ENDDO

Subscript 1: < Subscript 2: > No dependence.

Example 1

Some information can begained by intersectingdirection vectors.

Example 2

DO i = 1, 100A(i + 1, i + 2) = A(i, i) + C

ENDDO

Subscript 1: < Subscript 2: < Dependence with direction vector D = ( “<“ ) … except not.

Example 2 (Take Two)

DO i = 1, 100A(i + 1, i + 2) = A(i, i) + C

ENDDO

Subscript 1: d = 1 Subscript 2: d = 2 No dependence.

More complete informationcan be gained byintersecting distancevectors.

The Delta Test Main idea:

Test SIV subscripts (easiest), and get: Independence, or Constraint

E.g., d = 1 Apply constraints, hoping to:

Prove independenceE.g., (d = 1) (d = 2) = or

Reduce MIV subscripts to SIV or ZIV

DO i = 1, 100DO j = 1, 100

A(i + 1, i + j) = A(i, i + j - 1)ENDDO

ENDDO

Example: Reducing Subscript Complexity

MIV Subscript!

What can be?

Constraints Maintain a constraint vector

constraints for the ith indexDO i1 = 1, 100DO i2 = 1, 100

…

Constraint Types Initially: <none>

Eventually (we hope): ; Independence

Constraint Types What constraints can we infer from a SIV

test?

Strong SIV

d

Weak SIV

Constraint Types

Constraint Propagation What can we do with constraints? Example 1:

MIV

SIV

Constraint Propagation What can we do with constraints? Example 2:

MIV

SIV

Constraint Propagation Can we always eliminate an index? Example 3:

MIV

Constraint Propagation Can we always eliminate an index?

Answer: no. May still gain information.

Constraint Propagation

DO iDO j

A(i, 2*j + i) = A(i, 2*j – i + 5)ENDDO

ENDDO

Dependence

Constraint Propagation

DO iDO j

A(i, 2*j + i) = A(i, 2*j – i + 5)ENDDO

ENDDO

No dependence

The Delta Test: Summary

Delta Test: Example

DO iDO j

DO kA(j–i, i+1, j+k) = A(j–i, i,

j+k)ENDDO

ENDDOENDDO

Test untested SIV subscripts:

Delta Test: Example

DO iDO j

DO kA(j–i, i+1, j+k) = A(j–i, i,

j+k)ENDDO

ENDDOENDDO

Propagate constraints:

j j - 1

Delta Test: Example

DO iDO j

DO kA(j, i+1, j+k) = A(j–1, i, j+k)

ENDDOENDDO

ENDDO

Test untested SIV subscripts:

Delta Test: Example

DO iDO j

DO kA(j, i+1, j+k) = A(j–1, i, j+k)

ENDDOENDDO

ENDDO

Propagate constraints:

k k+1

Delta Test: Example

DO iDO j

DO kA(j, i+1, k) = A(j–1, i, k+1)

ENDDOENDDO

ENDDO

Test untested SIV subscripts:

Delta Test

Is the Delta Test Exact?

Single

Subscript

Tests

Constraint

Propagation

Constraint

Intersection

??Imprecise whenan index can’tbe eliminated

Intersection of lines and points

Is the Delta Test Exact? Additional sources of imprecision:

Triangular or trapezoidal bounds Coupled MIV subscripts may remain:

DO iDO j

A(i + j, i – j) = A(2i + 3j, j – i)ENDDO

ENDDO

Roadmap Single Subscript Tests:

ZIV SIV MIV

Testing in coupled groups: Delta Test Empirical results

Empirical Results

SIV

ZIV Strong Weak-zero

Weak-crossing

Exact MIV Delta

Of all tests 45% 34% 7% 0.7% 0.1% 5.1% 8.4%

Of successful tests

31% 52% 8% 0.6% 0.2% 3% 6%

Of proved independencies

85% 5% 1.5% 0.1% 0% 2.7% 5.3%

Non-linear subscripts: 6%

DO i DO j DO k A(j, i+1, k) = A(j–1, i, k+1) ENDDO ENDDOENDDO

Strong SIV

dDO i = 1, 100

DO j = 1, 100 A(i + 10, j) = A(i, i

- 5)ENDDO

DO i = 1, 5 A(i + 10, i) = A(i, i)ENDDO

Weak SIV

DO i DO j A(i + j, i – j) = A(2i + 3j, j – i) ENDDOENDDO

DO i DO j A(i + j, i – j) = A(2i + 3j, j – i) ENDDOENDDO

THEEND

THEEND

DO j = 1, 100 A(j - 1, j) = A(j, i)ENDDO