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ScalarOptimization

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Organization

��Whatkindofoptimizationsareuseful�

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analysisfordeterminingopportunitiesforoptimization�

data�owanalysis�

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latticealgebra

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solvingequationsonlattices

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applicationstodata�owanalysis

��Speedingupdata�owanalysis�

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exploitationofstructure

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sparserepresentations�controldependence�SSAform�sparse

data�owevaluatorgraphs

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Optimizationsperformedbymostcompilers

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Constantpropagation�replaceconstant�valuedvariableswith

constants

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Commonsub�expressionelimination�avoidre�computingvalue

ifvaluehasbeencomputedearlierinprogram

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Loopinvariantremoval�movecomputationsintolessfrequently

executedportionsofprogram

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Strengthreduction�replaceexpensiveoperations�like

multiplication�withsimpleroperations�likeaddition�

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Deadcoderemoval�eliminateunreachablecodeandcodethat

isirrelevanttooutputofprogram

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Optimizationexample�

DO

J

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DO

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BaseAddress�A

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OnlythetermI�

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restofcomputationis

invariantintheinnerloopandcanbehoistedoutofit�

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Furtherhoistingofinvariantcodeispossiblesinceonlythesubterm

J�

�dependsonJ�

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SinceIandJ

areincrementedby�eachtimethroughtheloop�

expressionslikeI�

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replacingthemwithadditions�

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BaseAddress�A�

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Terminology

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Ade�nitionofavariableisastatementthatmayassignto

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Auseofavariableisastatementthatmayreadthe

valueofthatvariable�

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Aliasing�occursinaprogramwhentwoormorenamesrefertothe

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Ourposition�

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Thisimplieswecandeterminesyntacticallywhereallde�nitions

andusesofavariableare�

Morere�nedapproach�performaliasanalysis�

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Forthenextfewslides�wewillfocusonconstantpropagationto

illustratethegeneralapproachtodata�owanalysis�

Examples�

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ifx�ztheny����fi�

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Constantpropagationmaysimplifycontrolflowaswell�

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Whydoopportunitiesforconstantpropagationariseinprograms�

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constantdeclarationsformodularity

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macros

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procedureinlining�smallmethodsinOOlanguages

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machine�speci�cvalues

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Overviewofalgorithm�

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makes�owofcontrolinprogram

explicit

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simplifyexpressionsandcontrol�ow�

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Step��buildthecontrol�owgraph�CFG��

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if�yxtheny����fi�

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11

13

1

35

51

x := 1

STAR

T

y:= x + 2

y > x

y := 5

merge

...y....

control flow graph (C

FG)

state vectorson CFG

edges

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AlgorithmforbuildingCFG�easy�onlycomplicationbeingbreak�s

andGOTO�s�needtoidentifyjumptargets�

BasicBlock�straight�linecodewithoutanybranchesormergingof

control�ow

NodesofCFG�statements�orbasicblocks�switchesmerges

EdgesofCFG�representpossiblecontrol�owsequence

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SymbolicevaluationofCFGforconstantpropagation

Propagatevaluesfromfollowinglattice�

false true .... -1 0 1 ...

definitely not a constant

may or m

ay not be constant

join(T,0) = Tjoin(0,-1) = T

meet(0,-1) =

meet(T,1) = 1

Twooperators�

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join�a�b��lowestvalueabovebothaandb�alsowrittenas

a�

b�

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meet�a�b��highestvaluebelowbothaandb�alsowrittenas

a�

b�

Symbolicinterpretationofexpressions�EVAL�e�Vin��ifany

argumentofeis�

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respectively��

otherwise�evaluateenormallyandreturnthatvalue

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��AssociateonestatevectorwitheachedgeoftheCFG�

initializingallentriesto��Initializework�listtoempty�

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placethisedgeontheworklist�

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Getedgefromworklist�

LetstatevectoronedgebeVin�

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if�targetnodeisassignmentstatementx��e�

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elseif�targetnodeisswitch�p��

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elseifEVAL�p�Vin�istrue�PropagateVintotruesideof

elsePropagateVintofalsesideofswitch�

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else��targetnodeismerge

Propagatejoinofstatevectorsonallinputstooutput�

Ifthischangestheoutputstatevector�enqueueoutputedge

onworklist�

od�

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Runningexample�

11

13

1

35

51

x := 1

STAR

T

y:= x + 2

y > x

y := 5

merge

...y....

control flow graph (C

FG)

state vectorson CFG

edges

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Algorithm

canquitesubtle�x := 1

STAR

T

...........

p

x := x +1

merge

..x....

Firsttimethroughloop�useofx

inloopisdeterminedtobe

constant��Nexttimethroughloop�itreaches�nalvalue��

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Complexityofalgorithm�

Heightoflattice����

eachstatevectorcanchangevalue��

V

times�

Sowhileloopinalgorithmisexecutedatmost��

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V

times�

Costofeachiteration�O�V��

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Questions�

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Canweusesamework�listbasedalgorithm

withdi�erent

latticestosolveotheranalysisproblems�

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Canweimprovethee�ciencythealgorithm�

Needtoseparatewhatisbeingcomputedfrom

howitisbeing

computed��

usealgebrasonceagain��

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Latticealgebraicapproachtodata�owanalysis

Abstractly�ourwork�listalgorithm

canbeviewedasonesolution

procedureforsolvingasetoflatticealgebraicequations�

Data�owlattices�

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partiallyordersetof�niteheight

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meetandjoinoperationswithappropriatealgebraicproperties

arede�nedforallpairsofvaluesfrom

po�set�

Thesepropertiesimplythatthelatticehasaleastandagreatest

element�

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Examples�

false true .... -1 0 1 ...

{x,y,z}

{x,y}{x,z}

{y,z}

{y}{x}

{z}{}

lattice for constant propagationpow

er set of variables {x,y,z}��

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Monotonicfunction�IfD

isapartiallyorderedsetandf�D

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D�

fissaidtobemonotonicifx�

y��f�x��

f�y��

Intuitively�iftheinputofamonotonicfunctionisincreased�the

outputeitherstaysthesameorincreasesaswell�

ExamplesofmonotonicfunctionsonCPlattice�

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identity�f�x��x

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bottomfunction�f�x���

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constantfunction�f�x���

ExamplesofnonmonotonicfunctionsonCPlattice�

f�x��if�x����thenelse�

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Theorem�LetD

bealatticeof�niteheightandf�D�

D

be

monotonic�Then�theequationsx�f�x�hasaleastandagreatest

solutiongivenbythelimitsofthechains��f����f���������and

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Proof�

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f����definitionof��

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f�����monotonicityoff�

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Sincethelatticehas�niteheight�thischainhassomelargest

elementl�andf�l��l�Solsolvestheequation�Itisalsoeasyto

showthatlistheleastsolutiontotheequations�

Asimilarargumentshowsthatagreatestsolutionexists�

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��greatest solution

least solution

x .x

Examples�

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f�x���

limit���f������������

limit���f������f������������

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limit���f�������������

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Corollary�

Iff�g�hetcaremonotonicfunctions�thesystemofequations

x�f�x�y�z����

y�g�x�y�z����

z�h�x�y�z��������

hasleastandgreatestsolutionsgivenbythelimitsoftheobvious

chains�eg�leastsolutionisobtainedbystartingwith�

forall

variables�substitutingintorighthandsidestogetnewvaluesfor

allvariables�anditeratingtillconvergenceoccurs��

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Connectionbetweenconstantpropagationandlatticeequations�

iterativeprocedureisjustamethodtosolvelatticeequations�

S0 = [T,T]S1 = S0{1 / x}S2 = S1{2 / y }S3 = if (S2[y] < S2[x]) or (S2[y]=T) or (S2[x] = T) then S2 else S3 .....S6 = S4 U

S5......

x := 1

STAR

T

y:= x + 2

y > x

y := 5

merge

...y.... S0S1S2

S3

S4

S5

S6

(2 variables)S0, S1, S2,..... : C

xC

Lattice: Ctrue false ... -1 0 1 ....

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Question�sinceequationshavemanysolutionsingeneral�which

oneshouldwecompute�

ForCP�leastsolutiongivesmoreaccurateinformationthanother

solutions�

x := 1

STAR

T

merge

y := ...

p(y)

...x... [1,T]

[T,T]

[T,T]

[T,T]

[T,T]

[T,T]

[T,T]

[1,T]

[1,T]

[1,T]

[1,T]

[1,T]

[.....]: least solution

[.....]: greatest solution

Ingeneral�ifcon�uenceoperatorisjoin�computeleastsolution�

otherwisecomputegreatestsolution�

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Generalspeci�cationofdata�owproblem�

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Lattice��niteheight

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Rulesforwritingdownequationsfrom

CFG

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Con�uenceoperator

Nospecialargumentsaboutterminationorcomplexityareneeded�

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Constantpropagationisexampleof

FORWARD�FLOW�ALL�PATHSproblem�

Intuitively�dataispropagatedforwardinCFG�andvalueis

constantatapointponlyifitisthesameconstantforallpaths

fromstarttop�

Generalclassi�cationofdata�owproblems�reaching definitions

very busy expressionslive variables

constant propagation

available expressions

BA

CK

WA

RD

FOR

WA

RD

ALL PA

THS

AN

Y PA

TH

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Availableexpressions�FORWARDFLOW�ALLPATHS

De�nition�Anexpression�xopy�isavailableatapointpifevery

pathfrom

STARTtopcontainsanevaluationofpafterwhich

therearenoassignmentstoxory�

Lattice�powersetofallexpressionsinprogram

orderedby

containment

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y := x + 1z := x + 1

merge

...x+1...

STAR

T

<---- x+1 is "available" here

x := y op z E0E1

E1 = {y op z} U (E0 - Ex)

(where Ex is all expressions involving x)

STAR

TE0 = { }

EQU

ATIO

NS:

Lattice: powerset of all expressions in procedure

confluence operator: meet (intersection)

compute greatest solution

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Reachingde�nitions�FORWARDFLOW�ANYPATH

Ade�nitiondofavariablevissaidtoreachapointpifthereisa

pathfrom

STARTtopwhichcontainsd�andwhichdoesnot

containanyde�nitionsofvafterd�

a := REA

D()

b := REA

D()

d := b - a

d := b + d

d > 0

d := a + b

merge

print (d)

STAR

T

merge

d > b

END

d0d1d2

{}{d0}

{d0,d1}

{d0,d1,d2}

{d0,d1,d2}

{d0,d1,d4}

d3

d4

{d0,d1,d2,d4}

{d0,d1,d3}

{d0,d1,d2,d3}

{d0,d1,d2,d3}

Lattice: powerset of definitions in procedure

Equations:STAR

T{}

x := e Din

Dout

Dout =

d{d} U

(Din - dx)

(dx is set of all definitions of x)

Com

pute least solution

Confluence operator: join (union)

Com

plexity: D*E*D

(D is num

ber of definitions)

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Manyintermediaterepresentationsrecordreachingde�nitions

informationingraphicalform�

def�usechain�edgewhosesourceisade�nitionofvariablev�and

whosedestinationisauseofvreachedbythatde�nition

use�defchain�reverseofdef�usechain

a := REA

D()

b := REA

D()

d := b - a

d := b + d

d > 0

d := a + b

merge

print (d)

STAR

T

merge

d > b

END

d0d1d2

{}{d0}

{d0,d1}

{d0,d1,d2}

{d0,d1,d2}

{d0,d1,d4}

d3

d4

{d0,d1,d2,d4}

{d0,d1,d3}

{d0,d1,d2,d3}

{d0,d1,d2,d3}

def-use chains

��

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Livevariableanalysis�BACKWARDFLOW�ANYPATH

Avariablexissaidtobeliveatapointpifxisusedbeforebeing

assignedonsomepathfromptoEND�usedinregisterallocation��

Lattice: powerset of variables ordered by containm

ent

Equations:

END

{}

x := y op z

E0 E1 = {y,z} U (E0 - {x})

Confluence operator: join (union)

Com

pute least solution

��

��

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Verybusyexpressions�FORWARDFLOW�ALLPATHS

Anexpressione��yopz�issaidtobeverybusyatapointpifit

isevaluatedoneverypathfromptoENDbeforeanassignmentto

yorz�

Lattice: powerset of expressions ordered by containm

ent

Equations:

END

{}

x := y op z

E0 E1 = {y op z} U (E0 - Ex)

Com

pute greatest solution

(Ex is set of expressions containing x)

Confluence operator: m

eet (intersection)

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Pragmaticsofdata�owanalysis�

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Computeandstoreinformationatbasicblocklevel�

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Usebitvectorstorepresentsets�

Question�canwespeedupdata�owanalysis�

Twoapproaches�

�

exploitstructureincontrol�owgraph

�

exploitsparsity

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OptimizingData�owAnalaysis

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ConstantpropagationonCFG�O�EV��

Reachingde�nitionsonCFG�O�EN��

AvailableexpressionsonCFG�O�EA��

Twoapproachestospeedingupdata�owanalysis�

�

exploitstructureintheprogram

�

exploitsparsityinthedata�owequations�usually�adata�ow

equationinvolvesonlyasmallnumberofdata�owvariables

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Exploitingprogram

structure

�

Work�listalgorithm

didnotenforceanyparticularorderfor

processingequations

�

Shouldexploitprogram

structuretoavoidrevisitingequations

unnecessarily

p

y = ....x....

x = 2x = 3

e1e2

e3

- we should schedule e3 after w

e have processed e1 and e2; otherw

ise e3 will have to be done tw

ice

- if this is within a loop nest, can be a big w

in

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Generalapproachtoexploitingstructure�elimination

�

IdentifyregionsofCFGthatcanbepreprocessedbycollapsing

regionintoasinglenodewiththesameinput�outputbehavior

asregion

�

Solvedata�owequationsiterativelyonthecollapsedgraph�

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Interpolatedata�owsolutionintocollapsedregions�

Whatshouldbearegion�

�

basic�blocks

�

basic�blocks�if�then�else�loops

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intervals

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Structuredprograms�limitinwhichnoiterationisrequired

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Example�reachingde�nitionsinstructuredlanguage

Tosummarizethee�ectofaregion�computegenandkillfor

region�

Data�owequationforregioncanbewrittenusinggenandkill�

inout

regionR

gen[R]: set of definitions in R

from w

hich there is a path to exit free of other definitions of the sam

e variable

kill[R]: set of definitions in program

that do not reach

out = gen[R] U

(in - kill[R])

exit of R even if they reach the beginning of R

��

��

��a = b + c

d

R

p

R1

R2

R1

R2

p R1

R

R

R

gen[R] = {d}

kill [R] = D

a(all definitions of a)

out[R] = gen[R

] U (in[R

] - kill[R])

gen[R] = gen[R

2] U (gen[R

1] - kill[S2])kill[R

] = kill[R2] U

kill[R2]

gen[R] = gen[R

1] U gen[R

2]kill[R

] = kill[R1]

Ukill[R2]

gen[R] = gen[R

1]kill[R

] = kill[R1]

in[R2] = gen[R

1] U (in[R

] - kill[R1])

in[R1] = in[R

]

in[R1] = in[R

2] = in[R]

in[R1] = in[R

] U gen[R

]

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Observations�

�

Forstructuredprograms�wecansolvedata�owproblemslike

reachingde�nitionspurelybyelimination�withoutany

iteration��complexity�O�EV���

�

Forstructuredprograms�wecanevensolvethedata�ow

problemdirectlyontheabstractsyntaxtree�noneedtobuild

thecontrol�owgraph��

�

Forlessstructuredprograms�likereducibleprograms��we

mustbuildthecontrol�owgraphtoidentifyregionslike

intervals�butthereisstillnoneedtoiterate�

��

��

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Exploitingsparsitytospeedupdata�owanalysis

Example�constantpropagation

�

CFGalgorithmforconstantpropagationusedcontrol�ow

graphtopropagatestatevectors�

�

Propagatinginformationforallvariablesinlock�stepforcesa

lotofuselesscopyinginformationfromonevectortoanother

�consideravariablethatisde�nedattopofprocedureand

usedonlyatbottom��

Solution�

�

doconstantpropagationforeachvariableseparately

�

propagateinformationdirectlyfromde�nitionstouses

skippingoverirrelevantportionsofcontrol�owgraph

Subtlepoint�inwhatordershouldweprocessvariables

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Constantpropagationusingdef�usechains

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Associatecellwitheachlhsandrhsoccurenceofallvariables�

initializeto��

�

Propagate�

alongeachdef�useedgeoutofSTART�and

enqueuetargetstatementsofdef�useedgesontoworklist�

�

Enqueueallde�nitionswithconstantRHSontoworklist�

�

while�worklistisnotempty�do

dequeuedefinitiondfromwork�list�

evaluateRHSofdusingcellvaluesforRHSvariables

andupdateLHScell�

ifthischangesLHScellvalue�

propagatenewvaluealongdef�usechainstoeachuse

�takejoinofcellvalueatuseandLHScellvalue��

ifcellvalueatusechangesandtargetstatementisadefinit

enqueuetargetstatementontoworklist�

od�

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Example�

x := 1

STAR

T

y:= x + 2

y > x

merge

...y....

y := 45y := 5

control flow graph (C

FG)

def-use edges

cell for value at definition/use

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Complexity�O�sizeofdef�

usechains�

ThiscanbeaslargeO�N�V�whereN

issizeofsetofCFGnodes�

However�withSSAform�canbereducedtoO�EV��

Problemwithalgorithm�lossofaccuracy�

Propagationalongdef�usechainscannotdeterminedirectlythat

y��

��isdeadcode�solastuseofyisnotmarkedconstant�

Onepossibility�repeatedcyclesofreachingde�nitioncomputation�

constantpropagationanddeadcodeelimination�

Isthereabetterway

Keyidea�

�

�ndunreachablestatementsduringconstantpropagation

�

donotpropagatevaluesoutofunreachablede�nitions

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��

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Oneapproach�usecontroldependenceanddef�usechains

Intuitiveideaofcontroldependence�Noden

iscontroldependent

onpredicatep

ifp

determineswhethern

isexecuted�

Convention�assumeSTARTisapredicatesounconditionally

executedstatementsarecontroldependentonSTART�

CDG�Controldependencegraph

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��

x := 1

STAR

T

y:= x + 2

y > x

merge

...y....

y := 45y := 5

control flow graph (C

FG)

control dependence edge

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Algorithm�Propagate�liveness�alongcontroldependenceedges

whilepropagatingconstantsalongdef�usechains�

x := 1

STAR

T

y:= x + 2

y > x

merge

...y....

y := xy := 5

control dependence edges

def-use chains

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Constantpropagation

�

Associatecellwitheachlhsandrhsoccurenceofallvariables�

andwitheachstatement�initializedto�

�

Propagate�

alongeachdef�useedgeandcontroldependence

edgeoutofSTART�Ifvalueinanytargetcellchanges�enqueue

targetstatementontoworklist�

�

while�worklistisnotempty�do

dequeuestatementdfromwork�list�

ifcontroldependencecellofstatementis��top�

switch�typeofd��

case�definition��

EvaluateRHSofdusingcellvaluesforRHSvariables

andupdateLHScell�

IfthischangesLHScellvalue

propagatenewvaluealongdef�usechainstoeachuse

�takejoinofcellvalueatuseandLHScellvalue��

��

��

��

Ifcellvalueatusechanges�enqueuetargetstatementonto

�case�switch��

�Evaluatepredicateandpropagatealongappropriatecontrold

edgesoutofpredicate�

Ifcellvalueattargetchanges�

enqueuetargetstatementontoworklist�

�

fi�od�

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Observations�

�

Wedonotpropagateinformationoutofdead�unreachable�

statements�

�

However�precisionofinformationisstillnotasgoodasCFG

algorithm�westillpropagateinformationoutofstatements

thatareexecutedbutareirrelevanttooutput�othersortof

deadstatements�asinSlide����

�

Needanalgorithmtocomputecontroldependenceingeneral

graphs�

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SizeofCDG�O�EN��canbereduced�

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��

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Solutions�

�

Requirethatavariableassignedononesideofaconditionalbe

assignedonbothsidesofconditional�byinsertingdummy

assignmentsofformx

��

x��Programmersdon�twanttodo

this�

�

Makecompilerinsertdummyassignments�Hardto�gureout

inpresenceofunstructuredcontrol�ow�

�

UseSSAform�ensurethateveryuseisreachedbyexactlyone

de�nitionbyinserting�functionsatmergestocombine

multiplereachingde�nitions��

�

��

��

SSAalgorithm

forConstantPropagation

x := 1

STAR

T

y:= x + 2

y > x

y := 5

merge

...y....

Φ

- phi-function is like pseudo-assignment

One possibility: one phi-function for every variable

at every merge in C

FG.

- phi-function combines different reaching definitions

at a merge into a single one at output of m

erge

- control dependence at merge: com

pute for each side of the m

erge separately

Constant propagation:

Where should phi-functions be placed?

similar to previous algorithm

, but at merge,

propagate join of inputs only from live sides of m

erge

Com

puting minim

al SS

A form

: O(|E

|) per variable (P

ingali and Bilardi P

LDI 96)

(eg. variable x in example)

Minim

al SS

A form

: permit def-use chains to bypass

a merge if sam

e definition reaches all sides of merge

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