Multilevel Generalized Force-directed Multilevel Generalized Force-directed Method for Circuit PlacementMethod for Circuit Placement

Tony ChanTony Chan11, Jason Cong, Jason Cong22,, Kenton Sze Kenton Sze11

11UCLA Mathematics DepartmentUCLA Mathematics Department22UCLA Computer Science DepartmentUCLA Computer Science Department

This work is partially supported by SRC, NSF, and ONR.

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OutlineOutline A Brief History of mPL

Recent Progress in Analytical PlacementRecent Progress in Analytical Placement

Our new contributions and enhancements [mPL5]Our new contributions and enhancements [mPL5] Generalization of force-directed method (GFD)Generalization of force-directed method (GFD)

More accurate approximation of half-perimeter wirelengthMore accurate approximation of half-perimeter wirelength

More accurate computation of cell spreading forcesMore accurate computation of cell spreading forces

Systematic scaling of the cell spreading forcesSystematic scaling of the cell spreading forces

Multilevel implementation of GFDMultilevel implementation of GFD Overview of mPL multilevel frameworkOverview of mPL multilevel framework

mPL5 frameworkmPL5 framework

ConclusionsConclusions

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Relative Wirelength

mPL 1.0 [ICCAD00]• Recursive ESC clustering• NLP at coarsest level• Goto discrete relaxation• Slot Assignment legalization• Domino detailed placement

year2000 2001 2002 2003 2004

A Brief History of mPL

mPL 1.1• FC-Clustering• added partitioning to legalization

mPL 2.0 • RDFL relaxation• primal-dual netlist pruning

mPL 3.0 [ICCAD 03]• QRS relaxation• AMG interpolation• multiple V-cycles• cell-area fragmentation

UNIFORM CELL SIZE

NON-UNIFORM CELL SIZE

mPL 4.0• improved DP• better coarsening • backtracking V-cycle

mPL 5.0• Multilevel Force-Directed

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Recent Progress on Analytical PlacementRecent Progress on Analytical Placement Force-directed method [Eisenmann and Johannes 98]Force-directed method [Eisenmann and Johannes 98]

Efficient spreading force computation using a fast Poisson solverEfficient spreading force computation using a fast Poisson solver

Interleave with quadratic placementInterleave with quadratic placement

Limitations:Limitations:• Inaccurate objective functionInaccurate objective function• Require ad hoc tuning of forces for good convergence Require ad hoc tuning of forces for good convergence

Aplace [Kahng and Wang 04] Aplace [Kahng and Wang 04] More accurate approximation to half-perimeter wirelengthMore accurate approximation to half-perimeter wirelength

• Log-sum-exp [Naylor. et al 01]Log-sum-exp [Naylor. et al 01] Solving the non-linear optimization problem in a multilevel frameworkSolving the non-linear optimization problem in a multilevel framework

Limitations:Limitations:• Local smoothing of density functionsLocal smoothing of density functions• Penalty formulation lumps all constraints togetherPenalty formulation lumps all constraints together

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Basic Formulation of Our ApproachBasic Formulation of Our Approach

Minimize the half-perimeter wirelength subject to even Minimize the half-perimeter wirelength subject to even

density constraint:density constraint:

)( min xW

,)( .. cxdts area. coreby divded area cells total where c

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Choices of Wirelength Objective FunctionsChoices of Wirelength Objective Functions

HPWL

Log-Sum-Exp

Quadratic

Lp-norm

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Bin based Density FormulationBin based Density Formulation

Average bin densityAverage bin density

Equality constraintEquality constraint

Average bin density = Average bin density = utilization ratioutilization ratio

However, density function is However, density function is

highly non-smoothhighly non-smooth

1

1

3

2

432

m

n

v6

v5

v4

v3

v2

v1v7

= a13(v7) = fractional area of cell v7 in bin B13

7

1

area)bin /()(k

kijij vaD

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Smoothing Density FunctionSmoothing Density FunctionSmoothing operator:Smoothing operator:

Larger epsilonLarger epsilon

More local smoothingMore local smoothing

Slow convergenceSlow convergence

Smaller epsilonSmaller epsilon

More global smoothingMore global smoothing

Faster convergenceFaster convergence

dI )(

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Smoothed Constrained WL Minimization ProblemSmoothed Constrained WL Minimization Problem

MMinimize smooth objective wirelength subject to smooth inimize smooth objective wirelength subject to smooth

density function:density function:

,)( .. xts)( min xW

.),()( where11cxdx

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Solving Density Constrained WL MinimizationSolving Density Constrained WL Minimization

Using the Uzawa algorithm, we iteratively solveUsing the Uzawa algorithm, we iteratively solve

can be viewed as “generalized force”can be viewed as “generalized force”

Advantages:Advantages: Individual scaling factor at each binIndividual scaling factor at each bin

Systematic updates of these scaling factorsSystematic updates of these scaling factors

No Hessian inversion is requiredNo Hessian inversion is required

)()(1 kkk

W xλx

)()()(1

ijijkijkλλ

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Summary of Generalized Force-directed (GFD) AlgorithmSummary of Generalized Force-directed (GFD) Algorithm

If initial solution not given:If initial solution not given: Use unconstrained quadratic minimizerUse unconstrained quadratic minimizer

Set stopping criterionSet stopping criterion

Iteratively solve:Iteratively solve: Poisson equation to get forcesPoisson equation to get forces

Updating the scaling factor (Lagrange multiplier) for forces based Updating the scaling factor (Lagrange multiplier) for forces based on the smoothed densityon the smoothed density

The nonlinear equation by stabilized fixed point iterationThe nonlinear equation by stabilized fixed point iteration

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Important Ingredients of GFDImportant Ingredients of GFD

Use of accurate objective functionsUse of accurate objective functions

Optimization-based bin-density constraint formulationOptimization-based bin-density constraint formulation

Global smoothing of density functionGlobal smoothing of density function

Use of Uzawa algorithm enables:Use of Uzawa algorithm enables: Systematic bin-level adjustment of force-scaling factors Systematic bin-level adjustment of force-scaling factors

Convergence to a well defined solution via fixed-point iterationConvergence to a well defined solution via fixed-point iteration

Applying multilevel optimization can lead to better runtime Applying multilevel optimization can lead to better runtime

and wirelengthand wirelength

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Overview of mPL multilevel frameworkOverview of mPL multilevel framework

Coarsening:Coarsening: build a hierarchy of problem approximations by build a hierarchy of problem approximations by

First ChoiceFirst Choice clustering clustering

Relaxation:Relaxation: improve the placement at each level by iterative improve the placement at each level by iterative

optimizationoptimization

Interpolation:Interpolation: transfer coarse-level solution to adjacent, finer transfer coarse-level solution to adjacent, finer

level (AMG declustering)level (AMG declustering)

Multilevel Flow:Multilevel Flow: multiple traversals over multiple hierarchies multiple traversals over multiple hierarchies

(V-cycle variations)(V-cycle variations)

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mPL5 FrameworkmPL5 Framework

Level at which GFD is applied

Level 3

Level 2

Level 1

C

C

I

I

C+I

C+I

I

I

C Coasening

I Interpolation

Keep coarsening until # cells less than 500

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Improvement by Our Multilevel FrameworkImprovement by Our Multilevel Framework

Improvement by multilevel GFD over flat GFDImprovement by multilevel GFD over flat GFD

CircuitCircuit % WL improved% WL improved % runtime reduced% runtime reduced

Ibm01Ibm01 13%13% 42%42%

Ibm05Ibm05 37%37% 43%43%

Ibm10Ibm10 23%23% 59%59%

Ibm15Ibm15 20%20% 67%67%

Ibm18Ibm18 31%31% 66%66%

AverageAverage 24.8%24.8% 55.4%55.4%

Experiments carried out on ISPD2004 FastPlace IBM benchmarks.

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Comparison on Standard Cell DesignsComparison on Standard Cell Designs

Capo9.01.09, 2.29

Dragon3.011.03, 12.38

FastPlace1.01.08, 0.18

Fengshui5.01.06, 2.03mPL5

1,1mPL5-fast1.07, 0.30

123456789

10111213

0.98 1 1.02 1.04 1.06 1.08 1.1

Scaled w irelength

Sca

led

ru

nti

me

Experiments carried out on ISPD2004 FastPlace IBM benchmarks.

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Scalability ComparisonScalability Comparison

y = 0.0001x1.2409

(mPL5-fast)

y = 5E-06x1.4995

(FastPlace1.0)

0

200

400

600

800

0 50000 100000 150000 200000

#Cells

Run

tim

e

FastPlace1.0 mPL5-fast

mPL5-fast is slightly more scalable than FastPlace1.0

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Comparison on Mixed-Size Placement BenchmarksComparison on Mixed-Size Placement Benchmarks

mPL51, 1

Capo9.01.23, 0.97

Fengshui5.01.10, 0.62

0

0.2

0.4

0.6

0.8

1

1.2

0.95 1 1.05 1.1 1.15 1.2 1.25

Scaled Wirelength

Sca

led

Run

tim

e

mPL5 has 18% shorter wirelength than Capo 9.0

mPL5 has 9 % shorter wirelength than Fengshui 5.0

Experiments carried out on ICCAD2004 Mixed-size benchmarks.

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Placement Plot of Placers on ICCAD2004 Mixed-size Placement Plot of Placers on ICCAD2004 Mixed-size IBM02IBM02

mPL5

Rel. WL = 1.00

Fengshui 5.0

Rel. WL = 1.11

Capo 9.0

Rel. WL = 1.17

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Placement Plot of Placers on ICCAD2004 Mixed-size Placement Plot of Placers on ICCAD2004 Mixed-size IBM10IBM10

mPL5

Rel. WL = 1.00

Fengshui 5.0

Rel. WL = 1.15

Capo 9.0

Rel. WL = 1.28

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Results on PEKO BenchmarksResults on PEKO Benchmarks

1.001.201.401.601.802.002.202.402.602.80

12506 27220 45639 68685 83709 182980

#Cells

Qualit

y r

atio

Capo9.0 Dragon3.01 Fengshui5.0

FastPlace1.0 mPL5 mPL5-fast

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mPL5 placement on ICCAD2004 Mixed-size IBM02mPL5 placement on ICCAD2004 Mixed-size IBM02

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Conclusions Conclusions

mPL5 is a highly scalable multilevel placer based on bin-mPL5 is a highly scalable multilevel placer based on bin-

density constrained optimization formulationdensity constrained optimization formulation

Provides a mathematically sound foundation for force-Provides a mathematically sound foundation for force-

directed methodsdirected methods

mPL5 produces the best wirelength with competitive mPL5 produces the best wirelength with competitive

runtime on both standard cell and mixed-size designs.runtime on both standard cell and mixed-size designs. 3% to 9% shorter WL on standard cell designs3% to 9% shorter WL on standard cell designs

9% to 18% shorter WL on mixed size designs9% to 18% shorter WL on mixed size designs

compared the best-known academic placerscompared the best-known academic placers

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AcknowledgementAcknowledgement

Financially supported by SRC, NSF, and ONR.Financially supported by SRC, NSF, and ONR.

Thank Min Xie for implementation of detailed placementThank Min Xie for implementation of detailed placement

Thank Joseph Shinnerl and Min Xie for valuable Thank Joseph Shinnerl and Min Xie for valuable

discussionsdiscussions

Thank Chris Chu and Natarajan Viswanathan for providing Thank Chris Chu and Natarajan Viswanathan for providing

ISPD04 FastPlace IBM benchmarks.ISPD04 FastPlace IBM benchmarks.

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End of the PresentationEnd of the Presentation

Thank you!Thank you!