Highly Efficient Gradient Computation for Highly Efficient Gradient - - PowerPoint PPT Presentation

Highly Efficient Gradient Computation for Highly Efficient Gradient Computation for Density Density-

• Constrained Analytical Placement Methods

Constrained Analytical Placement Methods

Jason Cong and Guojie Luo Jason Cong and Guojie Luo UCLA Computer Science Department { cong, gluo } @ cs.ucla.edu

This work is partially supported by NSF CCF-0430077 and CCF-0528583

Outline Outline

• Related Work, Motivation & Contributions

Related Work, Motivation & Contributions

• NLP Formulation and Gradient Computation

NLP Formulation and Gradient Computation

• Nonlinear programming formulation

Nonlinear programming formulation

• Iterative methods

Iterative methods

• Density and smoothing techniques

Density and smoothing techniques

• Efficient Gradient computation

Efficient Gradient computation

• Experimental Results

Experimental Results

• Summary and Future Work

Summary and Future Work

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Related Work Related Work

• Top performing placers in ISPD

Top performing placers in ISPD’ ’06 placement contest 06 placement contest

• Kraftwerk [Eisenmann & Johannes DAC

Kraftwerk [Eisenmann & Johannes DAC’ ’ 98] 98]

• mPL6 [Chan, Cong, et al., ISPD

mPL6 [Chan, Cong, et al., ISPD’ ’06] 06]

• NTUplace [Chen, Chang, et al. ICCAD

NTUplace [Chen, Chang, et al. ICCAD’ ’06] 06]

• Iterative, density

Iterative, density-

• driven overlap removal

driven overlap removal

• Guided by

Guided by “ “force force” ”, ,

• r by descend direction of density penalty
• r by descend direction of density penalty

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An Example of Analytical Placement An Example of Analytical Placement

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• Blue: standard cells

Blue: standard cells

• Green: movable macros

Green: movable macros

• Yellow: fixed macros

Yellow: fixed macros

Motivation & Contributions Motivation & Contributions

• Common strategies in analytical placers

Common strategies in analytical placers

• Density smoothing techniques

Density smoothing techniques

• Some smooth operators without closed

Some smooth operators without closed-

• form formula involved

form formula involved

• Iterative unconstrained optimizations

Iterative unconstrained optimizations

• Frequent gradient computation is necessary

Frequent gradient computation is necessary

• Our contributions

Our contributions

• Unify a wide range of smoothing techniques

Unify a wide range of smoothing techniques

• Derive a highly efficient gradient computation method

Derive a highly efficient gradient computation method

• Show good quality results for mixed

Show good quality results for mixed-

• size placements

size placements

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Nonlinear Programming Formulation for Placement Nonlinear Programming Formulation for Placement

• Wirelength

Wirelength-

• driven placement with density constraints

driven placement with density constraints

• NLP formulation with inequality constraints

NLP formulation with inequality constraints

• minimize: total wirelength

minimize: total wirelength

• subject to:

subject to: cell_density cell_density ≤ ≤ capacity capacity

• NLP formulation with equality constraints

NLP formulation with equality constraints

• minimize: total wirelength

minimize: total wirelength

• subject to:

subject to: cell_density cell_density + + filler_density filler_density = = capacity capacity

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Density Functions (1/3) Density Functions (1/3)

• Original Density

Original Density

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Density Functions (2/3) Density Functions (2/3)

• Bell

Bell-

• shaped Density (Naylor, APlace, NTUplace)

shaped Density (Naylor, APlace, NTUplace)

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Density Functions (3/3) Density Functions (3/3)

• Poisson/Helmholtz

Poisson/Helmholtz-

• smoothed Density (Kraftwerk, mPL)

smoothed Density (Kraftwerk, mPL)

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Iterative Methods Iterative Methods

• Problem formulation

Problem formulation

• minimize: wirelength

minimize: wirelength

• subject to: cell_density = capacity

subject to: cell_density = capacity

• Unconstrained problems

Unconstrained problems

• Penalized objective function

Penalized objective function

• OBJ

OBJμ

μ = wirelength +

= wirelength + μ μ •

• Penalty

Penalty

• Penalty function

Penalty function

• e.g. squared deviation: Penalty =

e.g. squared deviation: Penalty = 2

2 +

+ 2

2

• Iterative method

Iterative method

• loop { minimize OBJ

loop { minimize OBJμ

μ; if (converge) break; else increase

; if (converge) break; else increase μ μ; } ; }

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Density Penalty Functions (1/3) Density Penalty Functions (1/3)

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cell i

• ther cells

Density Function Density Function (high dimensional) Density Penalty Function Density Penalty Function (projected in xi)

• Bell

Bell-

• shaped density penalty function

shaped density penalty function

Density Penalty Functions (2/3) Density Penalty Functions (2/3)

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Density Penalty Function (projection in xi) cell i

• ther cells
• Helmholtz

Helmholtz-

• smoothed density penalty function

smoothed density penalty function

Bell Bell-

• shaped

shaped Helmholtz Helmholtz-

• smoothed

smoothed

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cell i

• ther cells

Density Penalty Functions (3/3) Density Penalty Functions (3/3)

Density Smoothing Techniques Density Smoothing Techniques

• Local smoothing

Local smoothing

• Locally smooth the non

Locally smooth the non-

• differentiable edges

differentiable edges

• Global smoothing

Global smoothing

• Globally distribute the original density but maintain consistenc

Globally distribute the original density but maintain consistency y

• We unified these global smoothing techniques

We unified these global smoothing techniques

• An integral transformation

An integral transformation

• Including but not limited to

Including but not limited to

• Poisson smoothing (Kraftwerk)

Poisson smoothing (Kraftwerk)

• Helmholtz smoothing (mPL)

Helmholtz smoothing (mPL)

• Gaussian smoothing (NTUplace)

Gaussian smoothing (NTUplace)

• Keywords: Elliptic PDE, Green

Keywords: Elliptic PDE, Green’ ’s function s function

• Our gradient computation works with such smoothing techniques

Our gradient computation works with such smoothing techniques

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1 1

( , ) ( , ) ( , , , ) ψ u v D u v G u v u v du dv ′ ′ ′ ′ ′ ′ = ∫ ∫

• Na

Naï ïve computation by finite difference scheme ve computation by finite difference scheme

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Gradient Computation (1/2) Gradient Computation (1/2)

cell i

• ther cells

Δ xi

Penalty

i i

x x ∂ − ≈ ∂ Δ

Penalty Function Penalty Function

Density Function Density Function

Gradient Computation (2/2) Gradient Computation (2/2)

• Our efficient method

Our efficient method

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cell i

• ther cells

Penalty

i i

x w ∂ − = ∂

Smoothed Density Smoothed Density Twice Smoothed Density Twice Smoothed Density

Comparison of the Gradient Computation Methods Comparison of the Gradient Computation Methods

• Na

Naï ïve computation ve computation

• Our efficient computation

Our efficient computation

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Penalty

i i

x x ∂ − ≈ ∂ Δ

Penalty Function Penalty Function

Penalty

i i

x w ∂ − = ∂

Twice Smoothed Density Twice Smoothed Density

• Avoid exploring the high dimensional penalty function

Avoid exploring the high dimensional penalty function

• Reduce the computational time by a factor of

Reduce the computational time by a factor of n n

Comparison with the mPL Implementation Comparison with the mPL Implementation

• mPL implementation

mPL implementation

• Our efficient computation

Our efficient computation

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Penalty

i i

A x x ∂ − ≈ ⋅ ∂ Δ Penalty

i i

x w ∂ − = ∂

Twice Smoothed Density Twice Smoothed Density

• Essentially the same for small cells with proper scaling

Essentially the same for small cells with proper scaling A A

• Our method is directly applicable to large macros

Our method is directly applicable to large macros

Smoothed Density Smoothed Density (with operator H) (with operator H) (with operator H (with operator H1/2

1/2)

)

Experimental Results on IBM Experimental Results on IBM-

• HB+ Benchmarks

HB+ Benchmarks

In average, wirelength is 13% shorter than SCAMPI [Ng, Markov, et al.] And 15% shorter than mPL6 [Chan, Cong, et al.]

NTUplace [Chen, Chang, et al.] as the detailed placer

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Summary & Future Work Summary & Future Work

• We proposed an efficient gradient computation method

We proposed an efficient gradient computation method

• Applicable to a wide range of global smoothing techniques

Applicable to a wide range of global smoothing techniques

• Reduce the runtime by a factor of

Reduce the runtime by a factor of n n compared to a naive method compared to a naive method

• Achieved good quality mixed

Achieved good quality mixed-

• size placement results

size placement results

• Future work

Future work

• Apply the computation to test other iterative methods

Apply the computation to test other iterative methods

• Such as augmented Lagrangian method

Such as augmented Lagrangian method

• Extend the nonlinear programming framework to handle complex

Extend the nonlinear programming framework to handle complex

• bjective and constraints
• bjective and constraints
• such as power density constraints

such as power density constraints

• and TS via constraints in 3D placement

and TS via constraints in 3D placement

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Thank You! Thank You!

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Backup Slide Backup Slide

• An equivalent expression

An equivalent expression

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( ) ( )

2 1 1 ( , ) 1 1 ( , ) ( , ) ( , ) ( , )

• /2

( , ) ( , ) ( , ; ) ( , ) 2 ( , ) ( , ) ( ', ') ( , , ', ') ' ' ( , ) ( , , / 2, ') '

k k k k

x y k k x y x y k a b x y x y k k y k k y h

μ D u v C u v dudv P x y μ x x D u v μ D u v C u v dudv x D u v G u v u v du dv D u v x x G u v x w v dv ∂ − ∂ ≈ ∂ ∂ ∂ = − ∂ ∂ ∂ = ∂ ∂ = +

∫ ∫ ∫ ∫ ∫ ∫

r r r r r r r r

) ) r r ) ) ) )

( )

( )

/2 /2

• /2

' /2 /2 ( , )

• /2

' /2

( , , / 2, ') ' ( , ; ) 2 ( ', ') ( ', ') '

k k k k k k k k k k k k k k

h y h k k y h u x w y h x y y h u x w k

G u v x w v dv P x y μ μ D u v C u v dv x

+ + = − + = −

− − ∂ = − ∂

∫ ∫ ∫

r r

r r ) ) ) )

Leibniz Integral Rule

( )

( , )

2 ( ', ') ( ', ')

x y

μ D u v C u v −

r r

) ) ) )

v’ u’

( , ; )

k

P x y μ x ∂ = − ∂ r r