SPIRAL: Tuning DSP Transforms to

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12/5/2002 IBM-Thomas T. J. Watson Res. Center

SPIRAL: SPIRAL: Tuning DSP Transforms to Tuning DSP Transforms to Computing PlatformsComputing Platforms

• Jeremy Johnson (Drexel)• Robert Johnson (MathStar Inc.)• David Padua (UIUC)• Viktor Prasanna (USC)• Markus Püschel (CMU)• Manuela Veloso (CMU)

• Franz Franchetti (TU Vienna)• Gavin Haentjens (CMU)• Pinit Kumhom (Drexel)• Neungsoo Park (USC)• David Sepiashvili (CMU)• Bryan Singer (CMU)• Yevgen Voronenko (Drexel)• Jianxin Xiong (UIUC)

Faculty Students

http://www.ece.cmu.edu/~spiral

José M. F. Moura

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SponsorSponsor

Work supported by DARPA (DSO), Applied & Computational

Mathematics Program, OPAL, through grant managed by

research grant DABT63-98-1-0004 administered by the Army

Directorate of Contracting.

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Moore’s Moore’s Law and Law and High(estHigh(est)) Performance Performance Scientific ComputingScientific Computing

arithmetic cost model not accurate for predicting runtime (one cache miss = 10 floating point ops)better performance models hard to getbest code is machine dependent (registers/caches size, structure)hand-tuned code becomes obsolete as fast as it is writtencompiler limitationsfull performance requires (in part) assembly coding

Moore’s Law: processor-memory bottleneckshort life cycles of computersvery complex architectures

• vendor specific• special instructions (MMX, SSE, FMA, …)• undocumented features

(single processor, off-the-shelf)

Effects on software/algorithms:

Portable performance requires automation

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AutomaticAutomatic Performance Tuning: ResearchPerformance Tuning: Research

Linear Algebra:ATLAS (J. Dongarra et al.) LAPACKPhiPACK (J. Demmel et al.)

Signal Processing: FFTW (M. Frigo and S. Johnson)SPIRAL

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SPIRALSPIRALAutomates

cuts development costscode less error-prone

takes advantage of architecture specific featuresporting without loss of performance

systematic exploration of alternatives both at algorithmic and code level

are performance critical

Implementation

Platform-Adaptation

Optimization

of DSP algorithms

A library generator for highly optimized signal processing algorithms

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SPIRALSPIRAL ApproachApproach

DSP Transform(DFT, DCT, Wavelets etc.)

Computing Platform

given

given

PossibleImplementations

PerformanceEvaluation

Inte

llig

ent

Sea

rchPossible

Algorithms

SP

IRA

L S

earc

h S

pac

e

(Pentium III, Pentium 4, Athlon, SUN, PowerPC, Alpha, … )

adaptedimplementation

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OrganizationOrganization

Mathematical Framework

Formula Generator

SPL and SPL Compiler

Search Engine

SPIRAL system

Conclusions

Transforms, Rules, and Formulas

Transform → Algorithm

Algorithm → Implementation

How to find the best implementation

Everything taken together

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DSPDSP Algorithms: Example 4Algorithms: Example 4--point DFTpoint DFT

Cooley/Tukey FFT (size 4):

product of structured sparse matricesmathematical notation

−

−

−−

=

−−−−−−

1000

0010

0100

0001

1100

1100

0011

0011

000

0100

0010

0001

1010

0101

1010

0101

11

1111

11

1111

iii

ii

4222

42224 )()( LDFTITIDFTDFT ⋅⊗⋅⋅⊗=

Fourier transform

Identity Permutation

Diagonal matrix (twiddles)

Kronecker product

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DSPDSP Algorithms: TerminologyAlgorithms: Terminology

Transform

Rule

Formula

( ) ( ) PDFTIDIDFTDFT mnmnnm ⋅⊗⋅⋅⊗→

( ) ( )( ) PFIIDIFDFT ⋅⊗⊗⋅⋅⊗= 222428

parameterized matrix

• a breakdown strategy • product of sparse matrices

• recursive application of rules• uniquely defines an algorithm• efficient representation• easy manipulation

nDFT

Ruletree 8DFT

2DFT 4DFT

2DFT 2DFT

• few constructs and primitives• uniquely defines an algorithm• can be translated into code

DFT is symmetric ⇒ transpose the rule:

CT rule transposed

Commuting tensor product factorsA and B square size m and n

Commutation property ⇒ further variations

Different patterns for access, storage and flow of data

( )mn mn

n mB A L A B L⊗ = ⊗

( ) ( )( ) ( )

RS RS RS RS

N S S R R S R S R

RS RS

N R S S S S R

F L I F L T I F L

F F I T L F I

= ⊗ ⊗

= ⊗ ⊗

( ) ( )RS RS

RS S R S S R SF L I F T F I= ⊗ ⊗

( ) ( )0 0

1 12 2 2 2

2 2

3 3

1 0 1 0 1 1 0 00 1 0 1 1 1 0 01 0 1 0 0 0 1 10 1 0 1 0 0 1 1

x xx xxF I I Fx xx x

−−

− −

= =⊗ ⊗

MoreMore CooleyCooley--Tukey RulesTukey Rules

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DSPDSP TransformsTransforms

( )[ ]nkliDFT n /2exp π=

222 DFTDFTWHT k ⊗⊗=

( )( )[ ]nlkDCT nII /2/1cos)( π+=

( )( )[ ]nlkDCT nIV /)2/1(2/1cos)( π++=

( )[ ]nklDST nI /sin)( π=

( )[ ]nlnkMDCT nn /)2/1)(2/)1((cos2 π+++=×

Others: filtering, Haar, Hartley, …

discrete Fourier transform

Walsh-Hadamard transform

discrete cosine and sineTransforms (16 types)

modified discrete cosine transform

TT ⊗two-dimensional transform

( ) ( )H

Wl

n

n 22

2

2222 1-n1-1-n1-nn ILIDTWTDTWT −⊗⊕=discrete wavelet transform

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RulesRules = Breakdown Strategies= Breakdown Strategies

( ) ( )Qn

IVn

IIn

IIn FIDCTDCTPDCT 22/

)(2/

)(2/

)( ⊗⋅⊕⋅→

DDCTSDCT IIn

IVn ⋅⋅→ )()(

( ) 2)(

2 2/1,1 FdiagDCT II ⋅→

rIV

n MMDCT 1)( →

1 1 12 2 2 21

( )n n n n n ni i i t

n

i

WHT I WHT I+ + + +− +

=→ ⊗ ⊗∏ … …

nnn SinDFTjCosDFTDFT ⋅+→

)(4/2/

IInnn DCTCosDFTCosDFT →

)(4/2/

IInnn DCTSinDFTSinDFT →

( ) ( ) PDFTIDIDFTDFT mnmnnm ⋅⊗⋅⋅⊗→

CDSTDCTBDFT In

Inn ⋅⊕⋅→ )( )(

2/)(2/

PDCTSMDCT IVnnn ⋅⋅→×

)(2

base case

recursive

translation

iterative

recursive

recursive

recursive

recursive

recursive

translation

iterative/recursive

( ) ( )H

Wl

n

n 22

2

2222 1-n1-1-n1-nn ILIDTWTDTWT −⊗⊕=

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Partition TreesPartition Trees

4

1 1 1 14

1 3

1 11

4

1 3

1 2

11iterative

mixed

recursive

Each WHT algorithm can be represented by a tree,

where a node labeled by n corresponds to nWHT2

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AlgorithmsAlgorithms = Ruletrees = Formulas= Ruletrees = Formulas)(

8IIDCT

)(4

IIDCT)(

4IVDCT

R1)()( 2/2

)(2/

)(2/

)(n

IVn

IIn

IIn IFDCTDCTPDCT ⊗⋅⊕⋅→

2FR3 R6

2FR4

R3

R1

R6

2F

2FR4

2)(

22

1FDCT II ⋅→

)(2

IIDCT

)(2

IIDST

)(2

IVDCT

)(2

IIDST

)(2

IIDCT

R1R6 SDCTPDCT II

nIV

n ⋅⋅→ )()(

)(4

IIDCT)(2

IVDCT

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Mathematical Mathematical Framework for Filters Framework for Filters and Discreteand Discrete--Time Wavelet TransformsTime Wavelet Transforms

Overlapped direct sum of matrices A (m x n) and B (r x s)

Overlapped tensor product

column overlap

row overlap

A

Bv=⊕ BA v

( ) ( )Svn IIBA ⊕⊕=

AB

v

=⊕ BA v ( )( )BAII ⊕⊕= rv

m

foldk

vvvvk

−

⊕⊕⊕=⊗ BBBBI

foldk

vvvvk

−

⊕⊕⊕=⊗ BBBBI

column overlap

row overlap

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PropertiesProperties of Overlapped Direct Sums of Overlapped Direct Sums and Tensor Productsand Tensor Products

( ) ( ) x x x 1 1

I B B I I B I Ikk

k v m n m n v n k m n k v nl l= =

⊗ = ⊕ ⊕ = ⊗ ⊗ i i

Column and row overlap direct sum

Column and row overlapped tensor product

AB

v

=⊕ BA v

T

=AT

BTv ( )A B

TT Tv= ⊕

( )Tvn s n v s⊕ = ⊕I I I I

( ) ( )

( ) ( )

x x x 1 11

x

I B I B I I B

I I I B

T Tk k kT Tvk m n v m m n k v m m n

l ll

vk m k m n

= ==

⊗ = ⊕ ⊕ = ⊗ ⊕

= ⊗ ⊗

i i

i

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FiltersFiltersLinear convolution in matrix form

Block convolutionsOverlap-save rule

Overlap-add rule

Circular convolution rule for circulant K

( ) [ ]T110

1I −−⊗= l

lnn hhhhC

( ) ( ) ( )( )hChC bbnlbl

n/bn ⊗⊗= −+−

/11 III

( ) ( )( )( ) 1,111T

/ III −−−+−⊗⊗= lllblm/bbbmn EhChC m=n+l-1

( ) ( ) nnn hhK DFTDFTdiagDFT 1 ⋅⋅⋅= −h-first column of K

n-length of h

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DiscreteDiscrete--TimeTime Wavelet TransformWavelet Transform

DTWT scaling and wavelet expansion coefficients

Recursive algorithm (Mallat)

( )( ) 1,...,02

2

1,

,0

−=−=

−=

−−∑

∑Jjnkhxd

nkhxc

jJjJ

nnkj

JJ

nnk

scaling coefficients

wavelet coefficients

h(-n)

h1(-n)

2

2

h(-n)

h1(-n)

2

2cJ,k

cJ-1,k

dJ-1,k

cJ-2,k

dJ-2,k

VJ-1

VJ-2

VJ WJ-1

WJ-2

1

, 2 1, , 2 1,j k m k j m j k m k j mm m

c h c d h c− + − += =∑ ∑

Haar wavelets = square waves

First stage: V2 ⇒V1⊕W1

The process is repeated for the upper half of the output

11 1

2 2[ 1, 1], [ 1, 1]h h= = −

( )4 4

2 2 2 21

21

1 1 0 0 1 1 0 0

0 0 1 1 1 1 0 0

1 1 0 0 0 0 1 1

0 0 1 1 0 0 1 1

H L L I Fc

H cd

−−

− −

= = ⊗

= =

( ) ( )1 1 1 1

2

2 2 22 2 2 22,

n

n n n nn

H

HTHT I L I F HT F− − − −= ⊕ ⊗ =

HaarHaar Wavelets Wavelets –– ExampleExample

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DiscreteDiscrete--TimeTime Wavelet TransformWavelet TransformDiscrete-Time Wavelet Transform (DTWT) rule

Scaling (lowpass) and wavelet (highpass) filter coefficients

DTWT - convolution rule

( ) ( )H

Wl

n

n 22

2

2222 1-n1-1-n1-nn ILIDTWTDTWT −⊗⊕=

[ ]( ) ( ) ( )( )([ ]( ) ( ) ( )( ) )

⊗

⋅⋅⊕⋅⊗

⊕⋅⊕⋅⊗=

nn

mmm

nmmm

heChoC

leCloCH

I1

1LI11

LI11

2T

2/T

2/2/

2T

2/T

2/2/

′′′

=−

−

110

110

l

l

hhh

hhhW

lo-lowpass odd coeffs., le-lowpass even coeffs.

ho-highpass odd coeffs., he-highpass even coeffs.

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DTWTDTWT RuletreeRuletree –– ExampleExample

DTWTn

DTWTn/2 C(lo) Cm/2(le) C(ho) C(he)

DTWT-convolution rule

DTWTn/4filters

Overlap-add rule

Cb(le)

Circulant rule

K(le0)

Circular convolution rule

1DFT−s sDFT

m=n+l-1

le0=le padded with b-1 zeroes

s=l/2+b-1

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FormulaFormula for a DCT, size 16for a DCT, size 16

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HelpfulHelpful ConceptConcept

DSP Transforms Formal Languages

DSP transform (of size)

Rule

Formula/Algorithm

Non-terminal symbol (with attribute)

Rule (production)

Element in Language (only terminals)

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MathematicalMathematical Framework: SummaryFramework: Summary

fast algorithms represented as ruletrees (easy generation/manipulation)and as formulas (can be translated into code)

formulas built from few constructs and primitives

many different algorithms/formulas generated from few rules (combinatorial explosion)

these algorithms are (essentially) equal in arithmetic cost, but differ in data flow

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Formula Generator


Search Engine

SPIRAL system

Conclusions






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FormulaFormula GenerationGeneration

recursive application

Formula Generator

data base (extensible!)

data type

rules control

transforms formulasruletrees

translation

search engine formula translation

(spl compiler)export

runtime

written in GAP/AREP (computer algebra system)

all computation/manipulation is symbolic

exact arithmetic

easy extensible rule and transform data base

verification of rules and formulas

cut here for otheroptimization problems

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Formula Generator: SummaryFormula Generator: Summary

Concept:

Implementation:

• easy extensible• formulas as ruletrees (efficient representation, easy manipulation)• interface to spl• allows implementation of search methods

• efficient implementation moves bottleneck to spl compiler• fully symbolic derivation/manipulation of formulas• verification• tools to analyze structure, arithmetic cost of formulas• interfaces with search engine

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Formula Generator


Search Engine

SPIRAL system

Conclusions






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FormulasFormulas in SPLin SPL

( compose( diagonal ( 2*cos(1/16*pi) 2*cos(3/16*pi) 2*cos(5/16*pi) 2*cos(7/16*pi) ) )( permutation ( 1 3 4 2 ) )( tensor

( I 2 )( F 2 )

)( permutation ( 1 4 2 3 ) )( direct_sum

( compose( F 2 )( diagonal ( 1 sqrt(1/2) ) )

)( compose

( matrix( 1 1 0 )( 0 (-1) 1 )

)( diagonal ( cos(13/8*pi)-sin(13/8*pi) sin(13/8*pi) cos(13/8*pi)+sin(13/8*pi) ) )( matrix( 1 0 )( 1 1 )( 0 1 )

)( permutation ( 2 1 ) )

• • • •

• • • •

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SPLSPL Syntax (Subset)Syntax (Subset)matrix operations:

(compose formula formula ...)(tensor formula formula ...)(direct_sum formula formula ...)

direct matrix description:(matrix (a11 a12 ...) (a21 a22 ...) ...)(diagonal (d1 d2 ...))(permutation (p1 p2 ...))

parameterized matrices:(I n)(F n)

scalars:1.5, 2/7, cos(..), w(3), pi, 1.2e-04

definition of new symbols:(define name formula)(template formula (i-code-list)

directives for code generation#codetype real/complex#unroll on/off

allows extension of SPL

controls loop unrolling

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SPLSPL Compiler, 4Compiler, 4--point FFTpoint FFT

(compose (tensor (F 2) (I 2)) (T 4 2)(tensor (I 2) (F 2)) (L 4 2))

f0 = x(1) + x(3)f1 = x(1) - x(3)f2 = x(2) + x(4)f3 = x(2) - x(4)f4 = (0.00d0,-1.00d0)*f(3)y(1) = f0 + f2y(2) = f0 - f2y(3) = f1 + f4y(4) = f1 - f4

r0 = x(1) + x(5)r1 = x(1) - x(5)r2 = x(2) + x(6)r3 = x(2) - x(6)r4 = x(3) + x(7)r5 = x(3) - x(7)r6 = x(4) + x(8)r7 = x(4) - x(8)y(1) = r0 + r4y(2) = r1 + r5y(3) = r0 - r4y(4) = r1 - r5y(5) = r2 + r7y(6) = r3 - r6y(7) = r2 - r7y(8) = r3 + r6

fast algorithmas

formulaas

SPL program#codetype

complex real

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SPLSPL Compiler: SummaryCompiler: Summary

Parsing

Intermediate Code Generation

Intermediate Code Restructuring

Target Code Generation

Symbol Table Abstract Syntax Tree

I-Code

I-Code

C, FORTRAN function

Template Table

SPL Formula Template DefinitionSymbol Definition

OptimizationI-Code

SPL Program

Built-in optimizations:

• single static assignment code• no reuse of temporary vars• only scalar temporary vars• constants precomputed

Extensible through templates

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TemplatesTemplates

(template(F n)[ n >= 1 ]( do i=0,n-1

y(i)=0do j=0,n-1y(i)=y(i)+W(n,i*j)*x(j)

endend ))

Pattern

I-code

Condition

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Code Generation and Template MatchingCode Generation and Template Matching

(F 2) matches pattern (F n) and assigns 2 to n.Because n=2 satisfies the condition n>=1,the following i-code is generated from the template:

do i = 0,1y(i) = 0do j = 0,1

y(i) = y(i)+W(2,i*j)*x(j)end

end

Y(0)=x(0)+x(1)y(1)=x(0)-x(1)

Unrolling & Optimization

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SIMDSIMD Short Vector ExtensionsShort Vector Extensions

+ x

vector length = 4(4-way)

Extension to instruction set architectureAvailable on most current architectures (SSE on Pentium, AltiVec on Motorola G4)Originally for multimedia (like MMX for integers)Requires fine grain parallelismLarge potential speed-up

SIMD instructions are architecture specificNo common API (usually assembly hand coding)Performance very sensitive to memory accessAutomatic vectorization very limited

Problems:

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Vector Vector Code Generation from SPL FormulasCode Generation from SPL Formulas

Naturally vectorizable construct

A

x y

4IA ⊗vector length

iiii

k

ii QEIADP )(

1υ⊗∏

=

Pi, Qi permutationsDi, Ei diagonalsAi arbitrary formulasν SIMD vector length

(Current) generic construct completely vectorizable:

Vectorization in two steps:

1. Formula manipulation using manipulation rules2. Code generation (vector code + C code)

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SPL SPL Compiler: Vector Code GenerationCompiler: Vector Code Generation

( ) ( )( )⋅′⋅⊗⋅⊗=16444

84416 TIDFTLIDFT

( ) ( ) 16444

1644416 LDFTITIDFTDFT ⋅⊗⋅⋅⊗=

( )( )( ) ( ) ( )( )82444

8442

164

824 LIIDFTLIILLI ⊗⋅⊗⋅⊗⊗⊗

Symbolic vectorization(automatic formula manipulation)

Mapping to C code + vector API

LOAD_VECT(xl0, x + 0);LOAD_VECT(xl4, x + 16);f0 = SIMD_SUB(xl0, xl4);LOAD_VECT(xl1, x + 4);LOAD_VECT(xl5, x + 20);f1 = SIMD_SUB(xl1, xl5);...yl7 = SIMD_SUB(f1, f4);STORE_L_8_4(yl6, yl7, y + 24);yl2 = SIMD_SUB(f0, f5);yl3 = SIMD_ADD(f1, f4);STORE_L_8_4(yl2, yl3, y + 8);

SSESSE2AltiVec…

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Formula Generator


Search Engine

SPIRAL system

Conclusions






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WhyWhy Search?Search?Toy problem

DCT IV- size 24

~31000 formulas

Search in algorithm space in SPIRAL:Exhaustive & Random Search, DP, Hill climbing, Genetic AlgorithmsBeyond search: design of optimal tree

DCT IV- size 24

~31000 formulas

scheduled

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WhyWhy Search?Search?

DCT, type IV, size 16

• maaaany different formulas• large spread in runtimes, even for modest size• precisely equal arithmetic cost• best formula is platform-dependent

~31000 formulas

Toy problem:scheduled

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NumberNumber of Formulas/Algorithmsof Formulas/Algorithms

k

123456789

# DFT, size 2^k

16

40296

27744162570361280~1.01 • 10^27~2.31 • 10^61

~2.86 • 10^133

# DCT-IV, size 2^k

110

12631242

19244433627343815121631354242

~1.07 • 10^38~2.30 • 10^76

~1.06 • 10^153

differ in data flow not in arithmetic cost exponential search space

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SearchSearch Methods Available in SPIRALMethods Available in SPIRAL

Exhaustive SearchDynamic Programming (DP)Random SearchHill ClimbingSTEER (similar to a genetic algorithm)

Good100s-1000sAllHill Climbing

(very) good100s-1000sAllSTEER

fair/goodUser decidedAllRandom

(very) good10s-100sAll DP

BestAllVery smallExhaust

ResultsTimedSizesFormulasPossible

Search over • algorithm space and • implementation options (degree of unrolling)

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STEERSTEERPopulation n:

Population n+1:

……

……

Mutation

Cross-Breeding expanddifferently

swapexpansions

Survival of Fittest

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LearningLearning to Generate Fast Algorithmsto Generate Fast Algorithms

• Learns from given dataset (formulas+runtimes) how to design a fast algorithm (breakdown strategy)• Learns from a transform of one size, generates the best algorithm for many sizes• Tested for DFT and WHT

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Some Experimental ResultsSome Experimental Results

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DCTDCT Type IV Size 16Type IV Size 16

Fastest Found Formulas Number of Formulas Timed

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Experimental Experimental ResultsResults

high performance code(compared with FFTW)

different transforms

search methods(applicable to all transforms)

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Some Experimental ResultsSome Experimental Results

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GeneratedGenerated DFT Code: Pentium 4, SSEDFT Code: Pentium 4, SSE(P

seu

do

) g

flo

ps

DFT 2n single precision, Pentium 4, 2.53 GHz, using Intel C compiler 6.0

n

speedups (to C code) up to factor of 3.1

0

1

2

3

4

5

6

7

4 5 6 7 8 9 10 11 12 13

Spiral SSEIntel MKL interl.FFTW 2.1.3Spiral CSpiral C vectSIMD-FFT

hand-tuned vendor assembly code

compiler vectorization

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GeneratedGenerated DFT Code: Pentium 4, SSE2DFT Code: Pentium 4, SSE2g

flo

ps

DFT 2n double precision, Pentium 4, 2.53 GHz, using Intel C compiler 6.0

n


0

0.5

1

1.5

2

2.5

3

3.5

4 5 6 7 8 9 10 11 12

Intel MKL splitSpiral SSE2FFTW 2.1.3Spiral CSpiral C vectSpiral SSE2 exh.

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GeneratedGenerated DFT Code: Pentium III, SSEDFT Code: Pentium III, SSEg

flo

ps

DFT 2n single precision, Pentium III, 1 GHz, using Intel C compiler 6.0n


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

4 5 6 7 8 9 10 11 12 13

Intel MKLSpiral SSESpiral C vectSpiral CFFTW 2.1.3

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Other Other transformstransforms

0

0.5

1

1.5

2

2.5

3

3.5

4 5 6 7 8 9 10 11 12 13 14

Spiral floatSpiral C vectSpiral C SSE

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

4x4 8x8 16x16 32x32 64x64 128x128

Spiral SSESpiral C vectSpiral C float

gfl

op

s

transform size transform size

2-dim DCT 2n x 2n

Pentium 4, 2.53 GHz, SSEWHT 2n

Pentium 4, 2.53 GHz, SSE

speedups (to C code) up to factor of 3

WHT has only additionsvery simple transform

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Best Best DFT Trees, size DFT Trees, size 210 = 1024= 1024

scalar

C vect

SIMD

Pentium 4float

Pentium 4double

Pentium IIIfloat

AthlonXPfloat

10

6

4

4

10

87

5

10

8

6

4

2

2

2

2 2

2 2

2 2

2

21

2

2 3

10

8

5

2

2

2 3

10

97

5

12

22 3

10

6

2

4

2 2

2 2

4

10

4

2

6

2 4

2 2

2

10

5

2

5

2 3 3

10

6

3

4

2 2 3

10

8

5

2

2

2 3

10

6

4

4

2 2

2 2

2

10

5

2

5

2 3 3

trees platform/datatype dependent

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CrosstimingCrosstiming of best trees on Pentium 4of best trees on Pentium 4S

low

dow

n f

acto

r w

.r.t

. bes

t

DFT 2n single precision, runtime of best found of other platforms

n

software adaptation is necessary

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

4 5 6 7 8 9 10 11 12 13

Pentium 4 SSEPentium 4 SSE2AthlonXP SSEPentiumIII SSEPentium 4 float

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Formula Generator


Search Engine

SPIRAL system

Conclusions






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SPIRAL SPIRAL SystemSystem

DSP transformspecifies

user

goes for a coffee

Formula Generator

SPL Compiler Sea

rch

En

gin

e

runtime (performance)

controlsimplementation options

controlsalgorithm generation

fast algorithmas SPL formula

C/Fortran/SIMDcodeS P

I R

A L

(or an

espresso

for sm

all transfo

rms)

platform-adaptedimplementation

comes back

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SPIRALSPIRAL SystemSystem

Available for download (v3.1): www.ece.cmu.edu/~spiral

Easy installation (Unix: configure/make; Windows: install shield)

Unix/Linux and Windows 98/ME/NT/2000/XP

Current transforms: DFT, DHT, WHT, RHT, DCT/DST type I – IV,

MDCT, Filters, Wavelets, Toeplitz, Circulants

Extensible

New version (4.0) in preparation

~ 30 Publications in the areas of Signal Processing, High

Performance Computing, Compilers, Machine Learning,

Mathematics

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SPIRALSPIRAL System: SummarySystem: Summary

Available for download: www.ece.cmu.edu/~spiral

Easy installation (Unix: configure/make; Windows: install shield)

Unix/Linux and Windows 98/ME/NT/2000/XP

Current transforms: DFT, DHT, WHT, DCT/DST type I – IV,

MDCT, Filters, Wavelets, Toeplitz, Circulants

Extensible

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Formula Generator


Search Engine

SPIRAL system

Conclusions






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ConclusionsConclusions

Mathematical computer representation of algorithmsAutomatic translation of algorithms into code

SPIRAL closes the gap between math domain (algorithms) andimplementation domain (programs)

High level: Mathematical manipulation of algorithmsLow level: Coding degrees of freedom

SPIRAL does automatic optimization by intelligent search/learning in the space of alternatives

a new paradigm of software development

Documents

SPIRAL: Tuning DSP Transforms to