Advanced Restructuring Compilers

AdvancedRestructuring

Compilers

Advanced Topics Spring 2008

Prof. Robert van Engelen

HPC II Spring 2008 22/19/08

Overview

Data dependence classification

Data dependences in loops

Dependence direction and distance Loop-independent dependences

K-level loop-carried dependences

Testing dependence Dependence equations

Dependence solvers

Two example loop transformations Fission

Interchange


Data Dependence

Definition

There is a data dependence from statement S1 tostatement S2 iff

1) both statements S1 and S2 access the same memorylocation and at least one of them stores into it, and

2) there is a feasible run-time execution path fromstatement S1 to S2


Data Dependence Classification

Definition

A dependence relation δ is

A true dependence (or flowdependence), denoted S1 δ S2

An anti dependence, denotedS1 δ-1 S2

An output dependence,denoted S1 δo S2

S1 X = …S2 … = X

S1 … = XS2 X = …

S1 X = …S2 X = …

S1 δ S2

S1 δ-1 S2

S1 δo S2


Compiling for Parallelism

Theorem

Any instruction statement reordering transformationthat preserves every dependence in a programpreserves the meaning of that program

Bernstein’s conditions for loop parallelism: Iteration I1 does not write into a location that is read by iteration

I2

Iteration I2 does not write into a location that is read by iterationI1

Iteration I1 does not write into a location that is written into byiteration I2


Dependence in Loops

The statement instances S1[i]for iterations i = 1,…,Nrepresent the loop execution

We have the following flowdependences:

S1[1] δ S1[2]S1[2] δ S1[3]…S1[N-1] δ S1[N]

DO I = 1, NS1 A(I+1) = A(I) + B(I) ENDDO

S1[1] A(2) = A(1) + B(1)S1[2] A(3) = A(2) + B(2)S1[3] A(4) = A(3) + B(3)S1[4] A(5) = A(4) + B(4) …

Statement instances with flowdependences


Representing Dependenceswith Data Dependence Graphs

It is generally infeasible torepresent all datadependences that arise in aprogram

Usually only static datadependences are recorded S1 δ(=) S2 means S1[i] δ S2[i]

for all i = 1, …, 10000 S2 δ(<) S2 means S1[i] δ S2[j]

for all i,j = 1, …, 10000 with i < j

A data dependence graphcompactly represent datadependences in a loop nest

DO I = 1, 10000S1 A(I) = B(I) * 5S2 C(I+1) = C(I) + A(I) ENDDO

Static data dependences foraccesses to A and C:S1 δ(=) S2 and S2 δ(<) S2

Datadependencegraph

S1

S2

(=)

(<)


Iteration Vector

Definition

Given a nest of n loops, the iteration vector i of aparticular iteration of the innermost loop is a vector ofintegers

i = (i1, i2, …, in)

where ik represents the iteration number for the loop atnesting level k

The set of all possible iteration vectors spans an iterationspace over loop statement instances Sj[i]


Iteration Vector Example

The iteration space of thestatement at S1 is the set ofiteration vectors {(1,1), (2,1),(2,2), (3,1), (3,2), (3,3)}

Example: at iteration i = (2,1)statement instance S1[i]assigns the value of A(1,2) toA(2,1)

DO I = 1, 3 DO J = 1, IS1 A(I,J) = A(J,I) ENDDO ENDDO

(1,1)

(2,1) (2,2)

(3,2) (3,3)(3,1)

I

J

HPC II Spring 2008 102/19/08

Iteration Vector Ordering

The iteration vectors are naturally ordered according to alexicographical order For example, iteration (1,2) precedes (2,1) and (2,2)

Definition

Iteration i precedes iteration j, denoted i < j, iff

1) i[1:n-1] < j[1:n-1], or

2) i[1:n-1] = j[1:n-1] and in < jn

HPC II Spring 2008 112/19/08

Cross-Iteration Dependence

Definition

There exist a dependence from S1 to S2 in a loop nest iffthere exist two iteration vectors i and j such that

1) i < j and there is a path from S1 to S2

2) S1 accesses memory location M on iteration i and S2

accesses memory location M on iteration j

3) one of these accesses is a write

HPC II Spring 2008 122/19/08

Dependence Example

Show that the loop has nocross-iteration dependence

Answer: there are no iterationvectors i and j in {(1,1), (2,1),(2,2), (3,1), (3,2), (3,3)} suchthat i < j and S1 in i writes to the same

element of A that is read at S1

in iteration j

or S1 in iteration i reads anelement A that is written at S1

in iteration j

DO I = 1, 3 DO J = 1, IS1 A(I,J) = A(J,I) ENDDO ENDDO

(1,1)

(2,1) (2,2)

(3,2) (3,3)(3,1)

I

J

HPC II Spring 2008 132/19/08

Dependence Distance andDirection Vectors

Definition

Given a dependence from S1 on iteration i to S2 oniteration j, the dependence distance vector d(i, j) isdefined as d(i, j) = j - i

Given a dependence from S1 on iteration i and S2 oniteration j, the dependence direction vector D(i, j) isdefined for the kth component as

D(i, j)k = “<” if d(i, j)k > 0“=” if d(i, j)k = 0“>” if d(i, j)k < 0

HPC II Spring 2008 142/19/08

Example 1

DO I = 1, 3 DO J = 1, IS1 A(I+1,J) = A(I,J) ENDDO ENDDO

Flow dependence between S1 and itself on:i = (1,1) and j = (2,1): d(i, j) = (1,0), D(i, j) = (<, =)i = (2,1) and j = (3,1): d(i, j) = (1,0), D(i, j) = (<, =)i = (2,2) and j = (3,2): d(i, j) = (1,0), D(i, j) = (<, =)

S1(<,=)

1 2 31

2

3

J

I

HPC II Spring 2008 152/19/08

Example 2

DO I = 1, 4 DO J = 1, 4S1 A(I,J+1) = A(I-1,J) ENDDO ENDDO

J

I1 2 3

1

2

3

4

4

Distance vector is (1,1)S1 δ(<,<) S1

S1(<,<)

HPC II Spring 2008 162/19/08

Example 2

I1 2 3

1

2

3

4

4

Distance vector is (1,-1) S1 δ(<,>) S1

DO I = 1, 4 DO J = 1, 5-IS1 A(I+1,J+I-1) = A(I,J+I-1) ENDDO ENDDO

J

S1(<,>)

HPC II Spring 2008 172/19/08

Example 3

I1 2 3

1

2

3

4

4

Distance vectors are (0,1) and (1,0)S1 δ(=,<) S1

S2 δ(<,=) S1

S2 δ(<,=) S2

DO I = 1, 4 DO J = 1, 4S1 B(I) = B(I) + A(I,J)S2 A(I+1,J) = A(I,J) ENDDO ENDDO

J

S1(=,<)

S2

(<,=)

(<,=)

HPC II Spring 2008 182/19/08

Loop-Independent Dependences

Definition

Statement S1 has a loop-independent dependence on S2

iff there exist two iteration vectors i and j such that

1) S1 refers to memory location M on iteration i, S2 refersto M on iteration j, and i = j

2) there is a control flow path from S1 to S2 within theiteration

HPC II Spring 2008 192/19/08

K-Level Loop-CarriedDependences

DefinitionStatement S1 has a loop-carried dependence on S2 iff1) there exist two iteration vectors i and j such that S1refers to memory location M on iteration i and S2 refers toM on iteration j2) d(i, j) > 0, that is, D(i, j) contains a “<” as its leftmostnon-“=” component

A loop-carried dependence from S1 to S2 is1) lexically forward if S2 appears after S1 in the loop body2) lexically backward if S2 appears before S1 (or if S1=S2)

The level of a loop-carried dependence is the index ofthe leftmost non-“=” of D(i, j) for the dependence

HPC II Spring 2008 202/19/08

Example

DO I = 1, 10 DO J = 1, 10 DO K = 1, 10S1 A(I,J,K+1) = A(I,J,K) ENDDO ENDDO ENDDO

S1 δ(=,=,<) S1

All loop-carried dependencesare of level 3, D(i, j) = (=, =, <)

Level-k dependences aresometimes denoted by Sx δk Sy

S1 δ3 S1Alternative notation for alevel-3 dependence

HPC II Spring 2008 212/19/08

Combining Direction Vectors

A loop nest can have multipledifferent directions at the sameloop level k

We abbreviate a level-kdirection vector componentwith “*” to denote any direction

DO I = 1, 10 DO J = 1, 10S1 A(J) = A(J)+1 ENDDO ENDDO

DO I = 1, 10S1 S = S + A(I) ENDDO

DO I = 1, 9S1 A(I) = …S2 … = A(10-I) ENDDO

S1 δ(*) S2≡

S1 δ(<) S2

S1 δ(=) S2

S1 δ(>) S2

S1 δ(*) S1

S1 δ(*,=) S1

HPC II Spring 2008 222/19/08

How to DetermineDependences?

A system of Diophantine dependence equations is set upand solved to test direction of dependence (flow/anti) Proving there is (no) solution to the equations means there is (no)

dependence Most dependence solvers require affine array index expressions

of the forma1 i1 + a2 i2 + … + an in + a0

where ij are loop index variables and ak are integer

Non-linear subscripts are problematic Symbolic terms Function values Indirect indexing etc

HPC II Spring 2008 232/19/08

Dependence Equation:Testing Flow Dependence

DO I = 1, NS1 A(I+1) = A(I) ENDDO

DO I = 1, NS1 A(f(I)) = A(g(I)) ENDDO

DO I = 1, NS1 A(2*I+1) = A(2*I) ENDDO

2α+1 = 2β has no solution

α+1 = β has solution α=β-1

write read

To determine flow dependence:prove the dependence equation

f(α) = g(β)has a solution for iteration

vectors α and β, such that α < β

HPC II Spring 2008 242/19/08

Dependence Equation:Testing Anti Dependence

DO I = 1, NS1 A(I+1) = A(I) ENDDO

DO I = 1, NS1 A(f(I)) = A(g(I)) ENDDO

DO I = 1, NS1 A(2*I) = A(2*I+2) ENDDO

2α = 2β+2 has solution α=β+1

α+1 = β has no solution

To determine anti dependence:prove the dependence equation

f(α) = g(β)has a solution for iteration

vectors α and β, such that β < αwrite read

HPC II Spring 2008 252/19/08

Dependence EquationExamples

DO I = 1, NS1 A(5) = A(I) ENDDO

DO I = 1, NS1 A(5) = A(6) ENDDO

To determine flow dependence:prove there are iteration vectors α < β such that f(α) = g(β)

5 = 6 has no solution

5 = β is solution if 5 < N

DO I = 1, N DO J = 1, NS1 A(I-1) = A(J) ENDDO ENDDO

α = (αI, αJ) and β = (βI, βJ)αI-1 = βJ has solution

HPC II Spring 2008 262/19/08

Dependence System:Loop Normalization

DO I = 2, 100 DO J = 1, I-1S1 A(I,J) = A(J,I) ENDDO ENDDO

Consider normalized loopiteration spaces Set lower bound to zero by

adjusting limits andoccurrence of loop variable

A dependence system consistsof a dependence equationalong with a set of constraints: Solution must lie within loop

bounds

Solution must be integer

Need dependence distance ordirection vector (flow/anti)

DO I = 0, 98 DO J = 0, I+2-2S1 A(I+2,J+1) = A(J+1,I+2) ENDDO ENDDO

HPC II Spring 2008 272/19/08

Dependence System:Formulating Flow Dependence



A dependence systemconsists of a dependenceequation along with a set ofconstraints: Solution must lie within loop

bounds



DO I = 0, 98 DO J = 0, I+2-2S1 A(I+2,J+1) = A(J+1,I+2) ENDDO ENDDO

αI+2 = βJ+1αJ+1 = βI+2

0 < αI , βI < 980 < αJ , βJ < αI0 < αJ , βJ < βI

αI < βI

Dependenceequations

Loopconstraints

Constraint for (<,*) dep. direction

α < β such that f(α) = g(β)where α = (αI, αJ) and β = (βI, βJ)

HPC II Spring 2008 282/19/08

Dependence System:Matrix Notation



A dependence system consistsof a dependence equationalong with a set of constraints: Solution must lie within loop

bounds



1

-1

βJ

αJ

βI

αI

0

-1

1

0

-1

0

0

1=

00-11

10-10

-1000

010-1

0-100

0010

00-10

0

0

0

0

0

0

1

-1

βJ

αJ

βI

αI

<

-1

0

0

0

0

98

0

98

0

HPC II Spring 2008 292/19/08

Fourier-Motzkin Projection

-1-2

-11

11

10

x2

x1

-7

5

9

6

<

x1

x2

Projections on x1 and x2

System of linearinequalities

Ax < b

x2 < 6

x1 + x

2 < 9

x 1 -

x 2 <

5-2x

1 - x2 < -7

HPC II Spring 2008 302/19/08

Fourier-Motzkin VariableElimination (FMVE)

FMVE procedure:

1. Select an unknown xj

2. L = {i | aij < 0}

3. U = {i | aij > 0}4. if L=∅ or U=∅ then xj is

unconstrained (delete it)5. for i ∈ L ∪ U

A[i] := A[i] / |aij| bi := bi / |aij|

6. for i ∈ L for k ∈ U add new inequality A[i] + A[k] < bi + bk

7. Delete old rows L and U

-1-2

-11

11

10

x2

x1

-7

5

9

6

<

0-1

0-2

02

01

x2

x1

2

-1

14

11

<

Select x2: L = {3,4}, U = {1,2} new system:

max(1/2,-2) < x1 < min(11,7)

HPC II Spring 2008 312/19/08

GCD Test

Requires that f(α) and g(β)are affine:f(α)= a0+a1 α1+…+an αn

g(β)= b0+b1 β1+…+bn βn

Reordering gives linearDiophantine equation:a1α1-b1β1+…+anαn-bnβn = b0-a0

which has a solution ifgcd(a1,…,an,b1,…,bn) dividesb0-a0

DO I = 1, NS1 A(2*I+1) = A(2*I) ENDDO

DO I = 1, NS1 A(4*I+1) = A(2*I+3) ENDDO

DO I = 1, 10 DO J = 1, 10S1 A(1+2*I+20*J) = A(2+20*I+2*J) ENDDO ENDDO

Which of these loopshave dependences?

HPC II Spring 2008 322/19/08

Loop Fission

Compute the acyclic condensationof the dependence graph to find alegal order of the loops

S1 DO I = 1, 10S2 A(I) = A(I) + B(I-1)S3 B(I) = C(I-1)*X + ZS4 C(I) = 1/B(I)S5 D(I) = sqrt(C(I))S6 ENDDO

S2

S3

S4

S5

0

01

1

Dependence graph

S2 S5

S3 S4

Acyclic condensation

S1 DO I = 1, 10S3 B(I) = C(I-1)*X + ZS4 C(I) = 1/B(I)Sx ENDDOSy DO I = 1, 10S2 A(I) = A(I) + B(I-1)Sz ENDDOSu DO I = 1, 10S5 D(I) = sqrt(C(I))Sv ENDDO

S3 δ(<) S2

S4 δ(<) S3

S3 δ(=) S4

S4 δ(=) S5

HPC II Spring 2008 332/19/08

Loop Interchange

Compute the direction matrix andfind which columns can bepermuted without violatingdependence relations in originalloop nest

S1 DO I = 1, NS2 DO J = 1, MS3 DO K = 1, LS4 A(I+1,J+1,K) = A(I,J,K) + A(I,J+1,K+1)S5 ENDDOS6 ENDDOS7 ENDDO

S4 δ(<,<,=) S4

S4 δ(<,=,>) S4

< < =< = >

Direction matrix

< = <= > <

< < =< = >

Invalid

< < == < >

< < =< = >

Valid

HPC II Spring 2008 342/19/08

Further Reading

[High] Chapters 5, 7-9 (parts)

Advanced Restructuring Compilers

Documents

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