Unified Quadratic Programming Unified Quadratic Programming Approach Approach for Mixed Mode Placementfor Mixed Mode Placement

Bo Yao, Hongyu Chen, Chung-Kuan Cheng, Bo Yao, Hongyu Chen, Chung-Kuan Cheng, Nan-Chi Chou*, Lung-Tien Liu*, Peter Suaris*Nan-Chi Chou*, Lung-Tien Liu*, Peter Suaris*

CSE DepartmentCSE DepartmentUniversity of California, San DiegoUniversity of California, San Diego

*Mentor Graphics Corporation*Mentor Graphics Corporation

OutlineOutlineIntroduction to the mixed mode placementIntroduction to the mixed mode placementUnified cost functionUnified cost functionDCT based cell density costDCT based cell density costExperimental resultsExperimental resultsConclusions Conclusions

Mixed Mode PlacementMixed Mode Placement

Common design needsCommon design needs Mixed signal designs Mixed signal designs

(analog and RF parts are (analog and RF parts are macros)macros)

Hierarchical design styleHierarchical design style IP blocksIP blocks Memory blocksMemory blocks

Challenges for placementChallenges for placement Huge amount of Huge amount of

componentscomponents Heterogeneous module Heterogeneous module

sizes/shapessizes/shapes

Memory

IPAnalog

Previous Works on Mixed Mode Previous Works on Mixed Mode PlacementPlacement

Combined floorplanning and std. cell placementCombined floorplanning and std. cell placement CapoCapo (Markov, ISPD 02, ICCAD 2003)(Markov, ISPD 02, ICCAD 2003)

Multi-level annealing placement Multi-level annealing placement mPG-MS mPG-MS (Cong, ASPDAC 2003)(Cong, ASPDAC 2003)

Partitioning based approachesPartitioning based approaches Feng Shui Feng Shui (Madden, ISPD 04)(Madden, ISPD 04)

Force-directed / analytical approachesForce-directed / analytical approaches Kraftwork Kraftwork (Eisenmann and Johannas, DAC 98 )(Eisenmann and Johannas, DAC 98 ) FastPlaceFastPlace (Chu, ISPD 04)(Chu, ISPD 04) APlace (Kahng, ISPD 04, ICCAD 04)APlace (Kahng, ISPD 04, ICCAD 04)

UPlace: Optimization FlowUPlace: Optimization Flow

Analytical Placement

Discrete Optimization

DetailedPlacement

Unified Cost FunctionUnified Cost FunctionCombined object function for global placementCombined object function for global placement

DP: Penalties for un-even cell densitiesDP: Penalties for un-even cell densities WL: Wire length cost functionWL: Wire length cost function

Quadratic analytical placementQuadratic analytical placement

WL = WL = 1/2x1/2xTTQx+px +1/2yQx+px +1/2yTTQy+pyQy+py Bounding box wire length for discrete optimizationBounding box wire length for discrete optimization

DPghtDensityWeiWLWLWeight **

Cell Density Cell Density Common strategyCommon strategy Partition the placement area into Partition the placement area into NN

by by NN rooms roomsCell density matrix Cell density matrix D D = = {{ddijij} } ddijij is the total cell area in room (i,j) is the total cell area in room (i,j)

A

00000.250.50.2500.510.500.250.50.250

D

DCT: Cell Density in Frequency DCT: Cell Density in Frequency DomainDomain

2-D Discrete Cosine Transform (DCT)2-D Discrete Cosine Transform (DCT)Cell density matrixCell density matrix D D => Frequency matrix => Frequency matrix FF = { = {ffijij}}where where ffij ij is the weight of density pattern (i,j)is the weight of density pattern (i,j)

Njv

NiudjCiC

Nf

N

u

N

vuvij 2

)12(cos2)12(cos)()(2 1

0

1

0

Otherwise

iiC

1

02/1)(

Properties of Frequency MatrixProperties of Frequency Matrix

Each Each ffuvuv is the weight of is the weight of frequency (frequency (u,vu,v))

Inverse DCT recovers the Inverse DCT recovers the cell densitycell density

Nvj

NuifvCuC

Nd

N

u

N

vuvij 2

)12(cos2)12(cos)()(2 1

0

1

0

…

…

…

(0,0) (1,0) (3,0)

(0,1) (1,1)

(0,3) (3,3)

…

0001000000000000

D

0.070.140.180.140.140.250.330.250.180.330.430.33

0.140.250.330.25

F

…

…

…

(0,0) (1,0) (3,0)

(0,1) (1,1)

(0,3) (3,3)

…

Frequency Matrix: An ExampleFrequency Matrix: An ExampleDensity matrix D and frequency matrix FDensity matrix D and frequency matrix F

Properties of DCTProperties of DCTCell density energy Cell density energy ddijij

22 = = ffijij22

Cell perturbation and frequency matrixCell perturbation and frequency matrix

Uniform density Uniform density ffij ij = 0= 0

0001-000100000000

D

0.250.460.600.460.27-0.5-0.65-0.5-0.100.190.250.19

0000

)DCT(D

1111111111111111

D

0000000000000004

DCT(D)

Density Cost of a PlacementDensity Cost of a PlacementWeighted sum of fWeighted sum of fijij

22

Higher weight for lower frequencyHigher weight for lower frequency

otherwise j)1/(i0ji 0

w ),f(w DP ij2

ijij

Approximate the density cost with a quadratic functionApproximate the density cost with a quadratic function

DP = ½aDP = ½aiixxii22+ b+ biixxii+c+cii

Make DP convexMake DP convex aai i >= 0 to ensure>= 0 to ensure

Matrix formMatrix form

DP =DP = ½x½xTTAx+Bx Ax+Bx

A = diag(aA = diag(a11, a, a22, …, a, …, ann), ),

B=(bB=(b11, b, b22, …, b, …, bnn))TT

Approximation of the Density CostApproximation of the Density Cost

xi

DP

x- x x+ DP

xi

x- x x+

ai > 0

ai = 0

UPlace: Minimize Combined UPlace: Minimize Combined Objective FunctionObjective Function

Combine quadratic objectivesCombine quadratic objectives WL + WL + DPDP WL = WL = ½½ xxTTQx+pxQx+px DP =DP = ½x½xTTAx+BxAx+Bx

Solve linear equation for each minimizationSolve linear equation for each minimization (Q + (Q + A)x = -p - A)x = -p - BB

Use Lagrange relaxation to reduce cell Use Lagrange relaxation to reduce cell congestioncongestion (k+1)(k+1) = = (k)(k) + + (k)(k)* DP* DP 00 = 0, = 0, 00 = Const = Const (k+1)(k+1) = = (k)(k) * * , 0< , 0< 1 1

Discrete OptimizationDiscrete Optimization Try Try -distance moves in four directions. Pick the -distance moves in four directions. Pick the

best position.best position.

Sweep all the cells in each iterationSweep all the cells in each iteration

A

A

A

A

A

Legalization/Detailed Placement – Legalization/Detailed Placement – Zone RefinementZone Refinement

One cell a time, ceiling -> floorOne cell a time, ceiling -> floor

Two directional alternationTwo directional alternation

A

Experimental Results: Wire lengthExperimental Results: Wire length

00.20.40.60.8

11.21.41.6

1 2 3 4 5 6 7 8 9 10

Capo mPG-MS Feng Shui APlace UPlace

No r

mal

i ze d

Wi re

Len

gth

Experimental results: CPU timeExperimental results: CPU time

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10

Capo mPG-MS Feng Shui APlace UPlace

CP

U (M

in)

UPlace: Placement ResultsUPlace: Placement Results

IBM-02IBM-02

ConclusionsConclusions

We propose a unified cost function for global We propose a unified cost function for global optimization, which provides good trade-offs optimization, which provides good trade-offs between wire length minimization and cell between wire length minimization and cell spreading.spreading.We introduce a DCT based cell density We introduce a DCT based cell density calculation method, and a quadratic calculation method, and a quadratic approximation.approximation.The unified placement approach The unified placement approach generates promising results on mixed generates promising results on mixed mode designs.mode designs.

Thank You !