High level abstractions for irregular computations...Programming irregular computations for...

Highlevelabstractionsforirregularcomputations

DavidPaduaDepartmentofComputerScience

UniversityofIllinoisatUrbana-Champaign

1.INTRODUCTION:THEPROBLEM

2December4,2017

Programmingforperformance

• Islaborious– Particularlysowhenthetargetmachineisparallel.

• Mustdecidewhattoexecuteinparallel• Inwhatordertotraversethedata• Howtopartitionthedata…

• Today,tuningmustbedonemanually.• Maintenanceisdifficult

– andtheresultisnotportable.

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Programmingforperformance

• Thereisstillmuchroomforprogressinhighperformanceprogrammingmethodology.

• Ourgoalistofacilitate(automate)tuningandenablemachineindependentprogramming.

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2.IRREGULARCOMPUTATIONS

5December4,2017

Whatareirregularcomputations?

• Irregularcomputationsaredifficulttodefine.• Onepossibledefinitionis:

– Computationson“irregular”datastructures.– Irregulardatastructures=notdensearrays.

• Pingali etal.Thetao ofparallelisminalgorithms.PLDI’11

– Andsubscriptexpressionsnon-linear.

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Whatareirregularcomputations?

• Examplesinclude– Subscriptedsubscripts:

• A[C[i]]– Linkedliststraversal:

• leftCell = leftCell->ColNext;• result+=leftCell->Value;

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Programmingirregularcomputationsforperformance

• Today,programmingofirregularcomputationsistypicallyatalowlevelofabstraction.

8G.HagerandG.Wellein.IntroductiontoHighPerformanceComputingforScientistsandEngineers.CRC Press

do diag=1,Nj, 2 ! two-way unroll amd Jamdiaglen=min( ( jd_ptr(diag+1)-jd_ptr(diag) ) , \

( jd_ptr(diag+2)-jd+ptr(diag+1) ) )offset1 = jd_ptr(diag) - 1offset2 = jd_ptr(diag+1) - 1do i=1, diagLen

C(i) = C(i) + val(offset1+i)*B(col_idx(offsetr1+i))C(i) = C(i) + val(offset2+i)*B(col_idx(offset2+i))

end do! peeled off iterationsoffset1 = jd_ptr(diag)do i=(diagLen+1), (jd_ptr(diag+1)-jd_ptr(diag))

C(i) = C(i)+val(offset1+i)*B(col_idx(offswt1+i))end do

end do

December4,2017

Programmingirregularcomputationsforperformance

• Programmingandoptimizingirregularcomputationsmorechallengingthanforlinear-subscriptscomputations.

• Wanttohavecompilersupportforprogrammingata lowlevelofabstractiontoenable– Automaticallymappingcomputationontodiversemachineclasses– Localityenhancement– Othercompileroptimizations(e.g.commonsubexpression eliminations)

• Inaddition,oftenusinghigherlevelsofabstractionsandapplyingoptimizationsatthatlevelisabettersolution.

• Bothapproacheswillbediscussedinthispresentation

9December4,2017

Whencomputationsarenotirregular

• Compilercandoagoodjobattransformingprograms:– Loopinterchange– Loopdistribution– Tiling– …

• Andwehavethethetechnologytogenerateefficientcodeforavarietyofplatformsevenwhentheinitialcodeissequential– Vectorextensions– Multicores– Distributedmemorymachines– GPUs

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ExampleAutomatictransformationofaregularcomputation

• Considertheloop

for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

a[1][1]=a[1][0]+a[0][1]

a[1][2]=a[1][1]+a[0][2]

a[1][3]=a[1][2]+a[0][3]

a[1][4]=a[1][3]+a[0][4]

12

j=1

j=2

j=3

j=4

a[2][1]=a[2][0]+a[1][1]

a[2][2]=a[2][1]+a[1][2]

a[2][3]=a[2][2]+a[1][3]

a[2][4]=a[2][3]+a[1][4]

i=1 i=2

• Computedependences(part1)

ExampleAutomatictransformationofaregularcomputation

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

a[1][1]=a[1][0]+a[0][1]

a[1][2]=a[1][1]+a[0][2]

a[1][3]=a[1][2]+a[0][3]

a[1][4]=a[1][3]+a[0][4]

13

j=1

j=2

j=3

j=4

a[2][1]=a[2][0]+a[1][1]

a[2][2]=a[2][1]+a[1][2]

a[2][3]=a[2][2]+a[1][3]

a[2][4]=a[2][3]+a[1][4]

i=1 i=2

• Computedependences(part2)

ExampleAutomatictransformationofaregularcomputation

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

1 2 3 4 …

1

2

3

4

j

i

14

1,1

or

ExampleAutomatictransformationofaregularcomputation

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

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• Findparallelism

ExampleAutomatictransformationofaregularcomputation

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

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• Findparallelism

ExampleAutomatictransformationofaregularcomputation

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for (i=1; i<n; i++) {for (j=1; j<n; j++) {

a[i][j]=a[i][j-1]+a[i-1][j];}}

17

• Transformthecode

for (k=4; k<2*n; k++) forall (i=max(2,k-n):min(n,k-2)) a[i][k-i]=...

ExampleAutomatictransformationofaregularcomputation

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Analyzingirregularcomputations

• Analysisofthefollowingcodeisdifficultandsometimesimpossible

• Dependencesareafunctionofthevaluesofc andd• Intheabsenceofadditionalinformation,thecompilermustassumetheworst.

• Andthisprecludesautomatictransformations(parallelization,vectorization,datadistribution,…)

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for (i=1; i<n; i++) {a[c[i]]=a[d[i]]+1;

}

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Irregularityisabeastprogramminglanguageresearchershavebeen

fightingforalongtime

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Ithasdefeatedussometimes

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IgnoringtheimportanceofirregularcomputationswasoneoftheReasonsforthefailureofHighPerformanceFortran

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Tamingtheirregularbeast

• CompilersandRuntime(Section3)

• Betternotations(Section4)

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3.COMPILERANDRUNTIMESOLUTIONSFORLOWLEVELIRREGULARPROGRAMMING

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Twoapproachestodealingwithirregularcomputations

• Analyzingirregularcodesdirectly– Statically– Dynamically

• Convertingintohigherlevelofabstraction(densearraycomputations)

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Staticanalysis

• Insomecases,itispossibletoanalyzeirregularalgorithmswritteninconventionalnotation.

• Moreeffortthanforregular/linearsubscriptcomputations

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Staticanalysis

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Dependenceanalysis

do k = 1, nq = 0do i = 1, p

if ( x(i) > 0 ) thenq = q + 1ind(q) = i

end ifend dodo j = 1, q

x(ind(j)) = x(ind(j))* y(ind(j))end do

end do

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Staticanalysis

26

do k = 1, nq = 0do i = 1, p

if ( x(i) > 0 ) thenq = q + 1ind(q) = i

end ifend dodo j = 1, q

x(ind(j)) = x(ind(j))* y(ind(j))end do

end do

Dependenceanalysis

Needinformationaboutind(j)

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Staticanalysis

27

do k = 1, nq = 0do i = 1, p

if ( x(i) > 0 ) thenq = q + 1ind(q) = i

end ifend dodo j = 1, q

x(ind(j)) = x(ind(j))* y(ind(j))end do

end do

Dependenceanalysis

Needinformationaboutind(j)Canbefoundbyanalyzingtheprogram

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Staticanalysis

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do i=1, ndo j=2, iblen(i)

do k=1, j-1x(pptr(i)+k-1) = ...

end doend dodo j=1, iblen(i)-1

do k=1, j... = x(iblen(i)+pptr(i)+k-j-1)

end doend do

end do

Arrayprivatization

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Staticanalysis

29

do i=1, ndo j=2, iblen(i)

do k=1, j-1x(pptr(i)+k-1) = ...

end doend dodo j=1, iblen(i)-1

do k=1, j... = x(iblen(i)+pptr(i)+k-j-1)

end doend do

end do

Arrayprivatization

Todecideifx canbeprivatized

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Staticanalysis

30

do i=1, ndo j=2, iblen(i)

do k=1, j-1x(pptr(i)+k-1) = ...

end doend dodo j=1, iblen(i)-1

do k=1, j... = x(iblen(i)+pptr(i)+k-j-1)

end doend do

end do

Arrayprivatization

Todecideifx canbeprivatizedIssufficienttoshowthatawriteprecedeseachread

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Staticanalysis

• Apossibleapproachistoquerythecontrolflowgraph,boundingthesearch.

31

YuanLinandDavidPadua.Compileranalysisofirregularmemoryaccesses.(PLDI'00).

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Dynamicanalysis

• Embeddeddependenceanalysis• Inspectorexecutor• Speculation

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Embeddeddependenceanalysis

• ThisisaclevermechanismduetoZhuandYew.

33

Zhu,C.Q.,&Yew,P.C.(1987).SCHEMETOENFORCEDATADEPENDENCEONLARGEMULTIPROCESSORSYSTEMS.IEEETransactionsonSoftwareEngineering,SE-13(6),726-739

for i=1; i<n; i++a(k(i)) = c(i) + 1

a_tag(k(:)) = 0parallel for i=1;i<n;i++

critical a(k(i)) if a_tag(k(i)) < i

a(k(i)) = c(i) + 1a_tag(k(i)) = i

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Inspectorexecutor

• Parallelscheduleandcommunicationaggregationcomputedatexecutiontime– Analyzingthememoryaccesspatternofacodesegment(typicallyaloop)

• Overheadreducedwhenthecodesegment(loop)isexecutedmultipletimeswiththesameaccesspattern

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SeeEncyclopediaofParallelComputingRunTimeParallelization(Saltz andDas)

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Speculation

• Thereisanextensivebodyofliteratureonspeculation.• Acodesegmentisexecutedinparallelinthehopethatthereisnoproblem.

• Ifthereis,executionbacktracksandthecodesegmentisre-executedserially

See LawrenceRauchwerger,DavidA.Padua:TheLRPD Test:SpeculativeRun-TimeParallelizationofLoopswithPrivatizationandReductionParallelization.PLDI 1995:218-232

EncyclopediaofParallelComputingThreadlevelspeculation(Torrellas)TransactionalMemories(Herlihy)

35December4,2017

Convertingintodensearraycomputations

• Theobjectiveistoconvertsparsecomputationsintoequivalentdenseform.

• Thisincreasesthenumberofoperationsinvolvingzeroes(oridentity)– Sometimesmakingthecomputationimpossible

• Butitallowstheapplicationofcompile-timetransformations.

• Sparsity isrecoveredbyapplyingafinaltransformation(seebelow)

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Convertingintodensearraycomputations

37

for(col=0; col<cols; col++) {for(row=0; row<dimensions; row++) {

leftCell = left.Rows[row];while( leftCell != NULL ) {

result[row][col] +=leftCell->Value * right[leftCell->ColIndex][col];

leftCell = leftCell->ColNext;

for( col = 0; col < cols; col++ ) for( row = 0; row < dimensions; row++ )

result[row][col] += A_Valuep[row][:] * right[:][col];

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Convertingintodensearraycomputations

38

Harmen L.A.vanderSpek,HarryA.G.Wijshoff:Sublimation:ExpandingDataStructurestoEnableDataInstanceSpecificOptimizations.LCPC2010:106-120

Anand Venkat,MaryHall,andMichelleStrout.2015.Loopanddatatransformationsforsparsematrixcode.In Proceedingsofthe36thACMSIGPLANConferenceonProgrammingLanguageDesignandImplementation(PLDI'15).

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Wheredowestand?

• Thereisanextensivebodyofliteratureoncompilationandruntimetofacilitatetheparallelprogrammingofirregularcomputations.

• Althoughmoreideasareimportant,whatweneedmosturgentlyisanevaluationoftheireffectiveness.– Ameansforrefinementandprogress.

• Allcompilertechniquessufferfromlackofunderstandingoftheireffectiveness.

39December4,2017

Needtoolstoevaluateprogress

• Measuringadvancesincompilertechnologyiscrucialforprogress.

• Anidea:apubliclyavailablerepository– Containingextensiveandrepresentativecodecollections.

– Keeptrackofresultsforcompilers/techniquesalongtime.

• Ongoingworkononesuchrepository:– LORE:ALoopRepositoryfortheEvaluationofCompilersbyZ.i Chen,Z.Gong,J.

JosefSzaday,D.Won,D.Padua,A.Nicolau ,A.Veidenbaum ,N.Watkinson,Z.Sura,S.Maleki,J.Torrellas,G.DeJong.IISWC-2017

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4. NOTATIONS

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4. NOTATIONS4.1ABSTRACTION

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WhatareProgrammingabstractions?

§ Anabstractionisformedbyreducinginformation.

§ Ineverydaylife,itformstheworldofideaswherewecanreason.No(naturallanguage)statementcanbemadewithoutabstractions

43December4,2017

WhatareProgrammingabstractions?

§ Abstractmachinelanguage/lowerlevelimplementation§ Scalarcomputations:

§ a=b*c+d^3.4§ Noloads,stores,registerallocation.

§ Scalarloops:§ for (i= …..) {…}§ Noifstatements,branch/goto instructions.§ Givesstructuretotheprogram.

§ Abstractalgorithmimplementation§ min(A(1:n))§ it = find (myvec.begin(), myvec.end(), 30);§ Hidealgorithm,datarepresentation

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Whatarethebenefitofusingabstractions?

• Programmerproductivity(expressiveness)– Codesareeasiertowrite– Tounderstand,maintain

• Portability• Optimization(manualandautomatic)

– Programsareeasiertoanalyze– Easiertotransform– Deepertransformationsareenabled

45December4,2017

Abstractionsandirregularcomputations

• Whatabstractionsshouldweuseforprogrammingirregularcomputationsatahigherlevel?

• Bestanswerseemstobetorepresentirregularalgorithmsintermsof:

Densearrayoperations

46December4,2017

4. NOTATIONS4.2.ARRAYNOTATIONANDIRREGULARCOMPUTATIONS

47December4,2017

Whyarraynotation

• Couldaccessarraysinscalarmode• Butoperationsonaggregatesare(often)easiertoread.

for (i=0; i<n; i++) for (j=0;j<n;j++)

a[i][j]=b[i][j]+c[i][j]

a=b+c

• Theyarewellunderstoodabstractions.• Powerful.

48December4,2017

Arrayoperationsandperformance

• Butusefulnessofarraynotationgobeyondexpressiveness.

• Thereareatleastthreereasonstousearrayabstractionsforperformance– Torepresentparallelism– Toreducetheoverheadofinterpretation.– Toenableautomaticoptimizations.

49December4,2017

Representationofparallelism• Popularintheearlydays:

• Usedtodaybutnotwidely

• Tensorflow• Futhark

– T.Henriksen,N.Serup,M.Elsman,F.Henglein,andC.Oancea.Futhark:PurelyFunctionalGPU-ProgrammingwithNestedParallelismandIn-PlaceArrayUpdates PLDI 2017

50

do 10 i = 1, 100, 2

M (i) = .true.M (i+l) = .false.

10 continueA(*) = B(*) + A(*)

C(M(*)) = B(M(*)) + A(M(*))

Illiac IVFortran

C[i][ j] = __sec_reduce_add( a[ i][:] * b[:][j]);

Cilk plus

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Array operations and optimizationsAnexample

• Applyingarithmeticrulessuchasassociativityofmatrix-matrixmultiplicationtoreducethenumberofoperations.– Whichoneisbetter?

• (((A*B)*C)*D)*E,• (A*((B*C)*D))*E,• (A*B)*((C*D)*E),• …

– Findbestusingdynamicprogramming– YoichiMuraokaandDavidJ.Kuck.1973.Onthetimerequiredforasequenceofmatrixproducts.Commun.ACM16,1(January1973),22-26.

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Reduceoverheadofinterpretation

• SeeHaichuan Wang,DavidA.Padua,PengWu:Vectorization ofapplytoreduceinterpretationoverheadofR.OOPSLA 2015:400-415

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Arrayoperationsandirregularcomputations

• Arrayoperationsareanidealnotationtomanipulatesparsearrays.

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SparsearraysinMATLABload west0479.matA = west0479;S = A * A' + speye(size(A));pct = 100 / numel(A);

figurespy(S)title('A Sparse Symmetric Matrix')nz = nnz(S);xlabel(sprintf('nonzeros = %d (%.3f%%)',nz,nz*pct));

• Arraysappearasdense,butrepresentedinternallyasspaarse.• Theinterpreterandlibrarieshandlethesparseness

54

S = sparse(m,n)

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Sparsearrayscanbeabstractedasdensearraysincompiledlanguages

• BikandWijshoff*developedastrategytotranslatedensearrayprogramsontosparseequivalents.

55

do i=1,nacc=acc+a(i,:)*<1:n>

do i=1,ndo j=1,n

if (i,j) ∈ E a thenacc=acc+a’(σa(i,j))*j

do i=1,ndo ad∈PADa

j=π2σa-1(ad)acc=acc+a’(ad)*j

do i=1,ndo ad=alow(i),ahigh(j)

j=aind(ad)acc=acc+aval(ad)*j

*Aart J.C.Bik,HarryA.G.Wijshoff:CompilationTechniquesforSparseMatrixComputations.InternationalConferenceonSupercomputing1993:416-424December4,2017

Arrayoperationsandirregularcomputations

• Notonlycanarraynotationbeusedtorepresentdensecomputations,itcanbeusedforanumberofotherdomains.

• Awiderangeofdomains

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Howwide?

• Atthetime(ca.1959),Illinois,withthefirstofits‘ILLIAC’supercomputers,wasagreatcentre ofcomputing,andGolubshowedhisaffectionforhisAlmaMaterbyendowingachairthere50yearslater.Rumour hasitthatthefundsforthegiftcamefromGooglestockacquiredinexchangeforsomeadviceonlinearalgebra.Google’sPageRanksearchtechnologystartsfromamatrixcomputation— aneigenvalueproblemwithdimensionsinthebillions.Hardlysurprising,Golub wouldhavesaid:everythingislinearalgebra.Nature2007

57December4,2017

New operatorsArraysforgraphsalgorithms

• AsimplealgorithmforSSSP(Bellman-Ford)canberepresentedasfollows:

58

for(i=1; i<=n; i++)d = d ⨁ A ⨂ d

JeremyKepner andJohnGilbert.2011.GraphAlgorithmsintheLanguageofLinearAlgebra.SIAMPressT.Mattson,D.Bader,J.Berry,A.Buluc,J.Dongarra,C.Faloutsos,J.Feo,J.Gilbert,J.Gonzalez,B.Hendrickson,J.Kepner,C.Leiserson,A.Lumsdaine,D.Padua,S.Poole,SteveReinhardt,M.Stonebraker,S.Wallach,A.Yoo.StandardsforGraphAlgorithmPrimitivesDecember4,2017

4. NOTATIONS4.3.EXTENSIONSTOARRAYNOTATIONFORPARALLELISM

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repmat(h, [1, 3])

circshift( h, [0, -1] )

transpose(h)

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Communicationasarrayoperations

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A00 A01

A10

A20

A11

A21 A22

A12

A02 B01

B10

B20

B11

B21 B22

B12

B02B00

A00B00

A01B11

A02B22

A12B21

A11B10

A10B02

A22B20

A20B01

A21B12

A00B00

A01B11

A02B22

A12B21

A11B10

A10B02

A22B20

A20B01

A21B12

initial skew

shift-multiply-add

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Cannon’salgorithm

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for k=1:nc = c + a × b;a = circshift(a,[0 -1]); b = circshift(b,[-1 0]);

end

Cannnon’s Algorithm

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JamesC.Brodman,G.CarlEvans,MuratManguoglu,AhmedH.Sameh,María Jesús Garzarán,DavidA.Padua:AParallelNumericalSolverUsingHierarchicallyTiledArrays.LCPC 2010:46-61

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Barrierelimination

63

X(3)=Y(3)*2X(1)=Y(1)*2 X(2)=Y(2)*2 X(4)=Y(4)*2

Z(1)=X(1) Z(2)=X(3)

Z(3)=X(2) Z(4)=X(4)

X(1:4)=Y(1:4)*2;Z(1:2)=X([1,3]);Z(3:4)=X([2,4]);

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Barrierelimination

64December4,2017

Asynchronoussemantics

• Insomecases,itcouldbedesirabletodelayupdatesacrossthewholemachine.Atsomepoint,aglobalupdateismade.

• Thisisaparticularexampleofcommunicationinasynchronousalgorithmswheredataassignmentsufferdelays.

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TiledLinearAlgebra– SSSPExample

// the distance vectord = inf; d(1) = 0;// the mask vectorL = 0; L(1) = 1;while (L.nnz())for i = 1:LOC_ITERr <- d + (trans(A)*(d*.L);L <- (r != d);d <- r;

r += d;L <- (r != d);d <- r;

Partialmultiplication

Localcom

putatio

n

Globalreduction

66

Saeed Maleki,G.CarlEvans,DavidA.Padua:TiledLinearAlgebraaSystemforParallelGraphAlgorithms.LCPC2014:116-130December4,2017

4. NOTATIONS4.3.WHEREDOWESTAND?

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Arraynotation

• Arraynotationhasalonghistory• Isapowerfulnotationthatcanbeusedforawiderangeofapplications.

• Forparallelism,itisanunfulfilledpromise• Thehighlymathematicalnotationisachallenge• Weseeincreaseuseofthenotationandexpectittobecomepopularalsoforirregularalgorithms.

• Challenges:– developthenotation– Designcompileroptimizations

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5.EPILOG

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• In1976notmuchwasknownaboutparallelprogrammingofirregularcomputations.

• 40yearslaterifisimpressivehowmuchhavebeenlearnedoncompilationandnotations.

• Butthedevelopmentofwidelyusedtoolsbasedontheseideasremainachallenge.

70December4,2017