Algebraic Techniques To Enhance Common Algebraic Techniques To Enhance Common Sub-expression Extraction for Polynomial Sub-expression Extraction for Polynomial

System SynthesisSystem Synthesis

Sivaram Gopalakrishnan Synopsys Inc., Hillsboro, OR – 97124

Priyank KallaDepartment of Electrical and Computer Engineering,

University of Utah, Salt Lake City, UT- 84112

OutlineOutline Problem context: Polynomial datapath synthesisProblem context: Polynomial datapath synthesis

• Our Focus: Integrating CSE and Algebraic methodsOur Focus: Integrating CSE and Algebraic methods

• Applications: DSP for audio, video, multimedia….Applications: DSP for audio, video, multimedia….

MotivationMotivation Previous Work and LimitationsPrevious Work and Limitations Integrated Approach Integrated Approach

• Square-free factorizationSquare-free factorization

• Common Coefficient ExtractionCommon Coefficient Extraction

• Common Cube ExtractionCommon Cube Extraction

• Algebraic DivisionAlgebraic Division

Results: Area OptimizationResults: Area Optimization Conclusions & Future WorkConclusions & Future Work

The Synthesis FlowThe Synthesis Flow

Polynomial representation?Polynomial representation?

Quadratic filter design for polynomial signal processingQuadratic filter design for polynomial signal processing

y = ay = a0 0 . x. x1122 + a + a1 1 . x. x11 + b + b0 0 . x. x00

22 + b + b1 1 . x. x00 + c . x + c . x0 0 . x. x11

MotivationMotivation

PP11 = x = x22 + 6xy + 9y + 6xy + 9y22

PP22 = 4xy = 4xy22 + 12y + 12y3 3

PP33 = 2zx = 2zx22 + 6xyz + 6xyz

PP11 = x(x+ 6y) + 9y = x(x+ 6y) + 9y22

PP22 = 4xy = 4xy22 + 12y + 12y3 3

PP33 = x(2zx + 6yz) = x(2zx + 6yz)

PP11 = x(x+ 6y) + 9y = x(x+ 6y) + 9y22

PP22 = y = y22(4x+ 12y) (4x+ 12y) PP33 = xz(2x + 6y) = xz(2x + 6y)

Direct Implementation17 Mults & 4 Adds

Horner form15 Mults & 4 Adds

Factorization + CSE12 Mults & 4 Adds

MotivationMotivation

dd11 = x + 3y = x + 3y PP11 = d = d11

22

PP22 = 4d = 4d11yy22

PP33 = 2xzd = 2xzd11

dd11 is a good building block is a good building block

How to identify such building blocks across How to identify such building blocks across multiple polynomial datapaths?multiple polynomial datapaths?

Need an methodology to expose many common Need an methodology to expose many common expressions!!!expressions!!!

Our Approach8 Mults & 1 Add

Conventional MethodsConventional Methods

Extracting control-dataflow graphs (CDFGs) from RTL Extracting control-dataflow graphs (CDFGs) from RTL

• SchedulingScheduling

• Resource sharing Resource sharing

• RetimingRetiming

• Control synthesisControl synthesis

Algebraic Transforms for arithmetic designsAlgebraic Transforms for arithmetic designs

• Factorization [Factorization [Hosangadi et alHosangadi et al, ICCAD 04], ICCAD 04]

• Common Sub-expression Elimination [Common Sub-expression Elimination [Hosangadi et alHosangadi et al, VLSI 05], VLSI 05]

• Term-rewriting [Term-rewriting [Arvind et alArvind et al, IEEE. Micro 98], IEEE. Micro 98]

• Tree-Height Reduction [Tree-Height Reduction [De Micheli De Micheli 94]94]

Lack of symbolic computer algebra manipulationLack of symbolic computer algebra manipulation

Conventional Methods…Conventional Methods… Kernel/Co-kernel Extraction (Factorization + CSE)Kernel/Co-kernel Extraction (Factorization + CSE)

Integrates CSE with cube/coefficient extractionIntegrates CSE with cube/coefficient extraction

Uses coefficients and variables to identify cubes (co-kernels)Uses coefficients and variables to identify cubes (co-kernels)

to obtain kernelsto obtain kernels

Subsequently uses CSE for further optimizationSubsequently uses CSE for further optimization

P = 5P = 5xx22 + 10y + 10y3 3 + 15pq+ 15pq;;

Uses {5, 10, 15, x, y, p, q} for kernel/co-kernel extractionUses {5, 10, 15, x, y, p, q} for kernel/co-kernel extraction

Does not perform algebraic divisionDoes not perform algebraic division

Cannot determine decomposition Cannot determine decomposition 5(x5(x22 + 2y + 2y3 3 + 3pq)+ 3pq)

P = xP = x22 + 2xy + y + 2xy + y22; -> (x+y); -> (x+y)22

Cannot determine the above decomposition Cannot determine the above decomposition

Symbolic algebra techniquesSymbolic algebra techniques Polynomial models for complex computational blocksPolynomial models for complex computational blocks

Guiding Synthesis engines using Gröbner’s basis Guiding Synthesis engines using Gröbner’s basis

[[Peymandoust and De MicheliPeymandoust and De Micheli, TCAD 02], TCAD 02]

• Given polynomial F and Library elements <IGiven polynomial F and Library elements <I11, …, I, …, Inn>>

• F = hF = h11 I I11 + …… + h + …… + hnn I In n

• Restricted to library elementsRestricted to library elements

Datapath optimization using word-length information

[Gopalakrishnan et al, ICCAD 07]• Restricted to fixed-size datapathsRestricted to fixed-size datapaths

• Cannot address systems of polynomialsCannot address systems of polynomials

Optimization techniquesOptimization techniques

• Canonical Form repreCanonical Form representationsentation

∑∑cckkYYk k

• cck k : : Coefficient in the range (0 ≤ Coefficient in the range (0 ≤ cckk ≤ ≤ b bkk))

• YYk k : Falling factorial : Falling factorial

• F = F = 3x3x22yy2 2 - 3x- 3x22yy - 3xy- 3xy2 2 + 3xy = 3x(x-1)y(y-1)+ 3xy = 3x(x-1)y(y-1)

ff11 = 5x = 5x33yy22 - 5x - 5x33yy - 15x- 15x22yy2 2 + 15x+ 15x22yy + 10xy+ 10xy2 2 - 10xy + 3z- 10xy + 3z22

ff22 = 3x = 3x22yy2 2 - 3x- 3x22yy - 3xy- 3xy2 2 + 3xy + z + 1+ 3xy + z + 1

dd11 = x(x-1)y(y-1) = x(x-1)y(y-1)

ff11 = 5d = 5d11(x-2) + 3z(x-2) + 3z22

ff22 = 3d = 3d11 + z + 1 + z + 1

Optimization techniquesOptimization techniques Square-free factorizationSquare-free factorization

Let F be an integral domain ZLet F be an integral domain Z

A polynomial u in F[x] is square-free if there is no A polynomial u in F[x] is square-free if there is no polynomialpolynomial v in F[x] v in F[x]

with deg(v, x) > 0, such that vwith deg(v, x) > 0, such that v22 | u. | u.

uu11 = x = x22 + 3x + 2; u + 3x + 2; u11 = (x+1)(x+2) is square-free = (x+1)(x+2) is square-free

uu22 = x = x44 + 7x + 7x3 3 + 18x+ 18x22 + 20x + 8; + 20x + 8;

uu22 = (x+1)(x+2) = (x+1)(x+2)22 is not square-free!!! is not square-free!!!

Optimization techniquesOptimization techniques Common Coefficient ExtractionCommon Coefficient Extraction

P = 8x + 16y + 24z;P = 8x + 16y + 24z;

PP11 = 2(4x + 8y + 12z); = 2(4x + 8y + 12z);

PP22 = 4(2x + 4y + 6z); = 4(2x + 4y + 6z);

PP33 = 8(x + 2y + 3z); best transformation = 8(x + 2y + 3z); best transformation

Use GCD computationUse GCD computation

Get the coefficients (aGet the coefficients (aisis))

Compute GCD of every pair (aCompute GCD of every pair (aii, a, ajj))

Retain GCDs > atleast (aRetain GCDs > atleast (aii, a, ajj))

Arrange GCDs in decreasing order, perform extractionArrange GCDs in decreasing order, perform extraction

Update GCD list and continue…Update GCD list and continue…

Optimization techniquesOptimization techniques Common Coefficient Extraction (Example)Common Coefficient Extraction (Example)

P = 8x + 16y + 24z + 15a + 30b;P = 8x + 16y + 24z + 15a + 30b; Coefficients {8, 16, 24, 15, 30}Coefficients {8, 16, 24, 15, 30}

GCD list {8, 8, 1, 2, 8, 1, 2, 1, 6, 15}GCD list {8, 8, 1, 2, 8, 1, 2, 1, 6, 15}

Reduced GCD list {8, 15} -> decreasing order {15, 8}Reduced GCD list {8, 15} -> decreasing order {15, 8}

Extracting 15 results in Extracting 15 results in

P = 8x + 16y + 24z + 15(a + 2b);P = 8x + 16y + 24z + 15(a + 2b);

Similarly, extracting 8 results in Similarly, extracting 8 results in

P = 8(x + 2y + 3z) + 15(a + 2b);P = 8(x + 2y + 3z) + 15(a + 2b);

Optimization techniquesOptimization techniques Common Cube ExtractionCommon Cube Extraction

Similar to kernel/co-kernel extraction (for variables…)Similar to kernel/co-kernel extraction (for variables…)

PP11 = x = x22y + xyz;y + xyz;

PP22 = ab = ab22cc33 + b + b22cc22x;x;

PP33 = axz + x = axz + x22zz22b; b;

kernel/co-kernel extraction results inkernel/co-kernel extraction results in PP11 = xy(x + z); = xy(x + z);

PP22 = b = b22cc22(ac + x);(ac + x);

PP33 = xz(a + xzb); = xz(a + xzb);

Optimization techniquesOptimization techniques Polynomial long divisionPolynomial long division

Given two polynomials a(x) and b(x), algebraic division determines Given two polynomials a(x) and b(x), algebraic division determines

q(x) and r(x) such thatq(x) and r(x) such that

a(x) = b(x) q(x) + r(x)a(x) = b(x) q(x) + r(x)

a(x) = xa(x) = x44 - 2x - 2x3 3 + 5; + 5;

b(x) = xb(x) = x22 + 3x + 3x - 2; - 2;

a(x) = b(x) (xa(x) = b(x) (x22 – 5x – 5x + 17) – 61x + 39+ 17) – 61x + 39

q(x) r(x)q(x) r(x)

Optimization techniquesOptimization techniques Common Sub-Expression EliminationCommon Sub-Expression Elimination

Identify isomorphic patterns in an arithmetic expression tree and Identify isomorphic patterns in an arithmetic expression tree and

merge them!!!merge them!!!

k = x + y;k = x + y;

m = x + y + z;m = x + y + z;

n = xy + x + y;n = xy + x + y;

k = x + y;k = x + y;

m = k + z;m = k + z;

n = xy + k;n = xy + k;

Integrated approachIntegrated approach

Input: The polynomial system PInput: The polynomial system Porigorig (list of arrays) (list of arrays)

Perform Canonization, Square-free factorizationPerform Canonization, Square-free factorization

Get best initial cost: CGet best initial cost: Cinitialinitial

Perform Coefficient extraction: PPerform Coefficient extraction: Pcce cce

Perform cube extraction: PPerform cube extraction: Pcce_cubecce_cube, get linear blocks, get linear blocks

Get the lists representing the systemGet the lists representing the system

For every linear block, for each list perform algebraic divisionFor every linear block, for each list perform algebraic division

Pick the best costPick the best cost

IllustrationIllustration

Integrated approach (Example)Integrated approach (Example)

PP11 = 13x = 13x22 + 26xy + 13y + 26xy + 13y22 + 7x - 7y + 11; + 7x - 7y + 11;

PP22 = 15x = 15x22 - 30xy + 15y - 30xy + 15y22 + 11x + 11y + 9; P + 11x + 11y + 9; Porigorig

Square-free factorization does not work!!!Square-free factorization does not work!!!

Initial cost: 16 M and 10 AInitial cost: 16 M and 10 A

After common coefficient extraction (PAfter common coefficient extraction (Pccecce))

PP11 = 13(x = 13(x22 + 2xy + y + 2xy + y2)2) + 7(x – y) + 11; + 7(x – y) + 11;

PP22 = 15(x = 15(x22 - 2xy + y - 2xy + y2)2) + 11(x + y) + 9; + 11(x + y) + 9;

Linear blocks: (x – y), (x + y)Linear blocks: (x – y), (x + y)

Integrated approach (Example…)Integrated approach (Example…)

After common cube extraction (PAfter common cube extraction (Pcce_cubecce_cube))

PP11 = 13(x(x + 2y) + y = 13(x(x + 2y) + y2)2) + 7(x – y) + 11; + 7(x – y) + 11;

PP22 = 15(x(x- 2y) + y = 15(x(x- 2y) + y2)2) + 11(x + y) + 9; + 11(x + y) + 9;

Linear blocks: (x – y), (x + y), (x + 2y), (x – 2y)Linear blocks: (x – y), (x + y), (x + 2y), (x – 2y)

Perform algebraic division using the linear blocksPerform algebraic division using the linear blocks

PPccecce is the best cost implementation with (x+y) (x-y) is the best cost implementation with (x+y) (x-y)

dd11 = x + y; d = x + y; d22 = x - y; = x - y;

PP11 = 13d = 13d1122 + 7d + 7d22 + 11; + 11;

PP22 = 15d = 15d2222 + 11d + 11d11 + 9; + 9;

Cost: 6 M and 6 ACost: 6 M and 6 A

ResultsResults

Average area improvement: 42%

Benchmark Var/Deg/m Factor/CSE Proposed ↑Area % ↑Delay %

SG3X2 2/2/16 204805 102386 50 21.3

SG4X2 2/2/16 449063 197599 55.9 -24.1

SG4X3 2/3/16 690208 557252 19.2 -16.3

SG5X2 2/2/16 570384 271729 52.3 -13.9

SG5X3 2/3/16 1365774 614955 54.9 -20.7

Quad 2/2/16 36405 30556 16 -9.5

Mibench 3/2/8 20359 8433 58.6 -3.7

MVCS 2/3/16 31040 22214 28.4 -32

ResultsResults

Average area improvement: 42%

Benchmark Var/Deg/m Factor/CSE Proposed ↑Area % ↑Delay %

SG3X2 2/2/16 204805 102386 50 21.3

SG4X2 2/2/16 449063 197599 55.9 -24.1

SG4X3 2/3/16 690208 557252 19.2 -16.3

SG5X2 2/2/16 570384 271729 52.3 -13.9

SG5X3 2/3/16 1365774 614955 54.9 -20.7

Quad 2/2/16 36405 30556 16 -9.5

Mibench 3/2/8 20359 8433 58.6 -3.7

MVCS 2/3/16 31040 22214 28.4 -32

Conclusions & Future WorkConclusions & Future Work

Polynomial decomposition approach for arithmetic datapaths Polynomial decomposition approach for arithmetic datapaths Arithmetic datapaths modeled as polynomial systemsArithmetic datapaths modeled as polynomial systems

Integrating CSE with algebraic manipulationIntegrating CSE with algebraic manipulation

Performing algebraic decomposition to enhance the power of CSEPerforming algebraic decomposition to enhance the power of CSE

Impressive area savingsImpressive area savings But delay penalty!!!But delay penalty!!!

Future Work: Future Work: • Address the concerns in delay!!!Address the concerns in delay!!!

• Retarget the approach towards power savings??? Retarget the approach towards power savings???

Questions???