Multilevel Generalized Forcedirected Method for Circuit Placement

1
Multilevel Generalized Force-directed Method for
Circuit Placement

• Tony Chan1, Jason Cong2, Kenton Sze1
• 1UCLA Mathematics Department
• 2UCLA Computer Science Department

This work is partially supported by SRC, NSF,
and ONR.
2
Outline

• A Brief History of mPL
• Recent Progress in Analytical Placement
• Our new contributions and enhancements mPL5
• Generalization of force-directed method (GFD)
• More accurate approximation of half-perimeter
  wirelength
• More accurate computation of cell spreading
  forces
• Systematic scaling of the cell spreading forces
• Multilevel implementation of GFD
• Overview of mPL multilevel framework
• mPL5 framework
• Conclusions

3
Relative Wirelength
A Brief History of mPL

• mPL 1.1
• FC-Clustering
• added partitioning to legalization

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

UNIFORM CELL SIZE

• mPL 2.0
• RDFL relaxation
• primal-dual netlist pruning

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

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

NON-UNIFORM CELL SIZE

• mPL 5.0
• Multilevel Force-Directed

year
2002
2003
2000
2001
2004
4
Recent Progress on Analytical Placement

• Force-directed method Eisenmann and Johannes 98
• Efficient spreading force computation using a
  fast Poisson solver
• Interleave with quadratic placement
• Limitations
• Inaccurate objective function
• Require ad hoc tuning of forces for good
  convergence
• Aplace Kahng and Wang 04
• More accurate approximation to half-perimeter
  wirelength
• Log-sum-exp Naylor. et al 01
• Solving the non-linear optimization problem in a
  multilevel framework
• Limitations
• Local smoothing of density functions
• Penalty formulation lumps all constraints
  together

5
Basic Formulation of Our Approach

• Minimize the half-perimeter wirelength subject to
  even density constraint

6
Choices of Wirelength Objective Functions
7
Bin based Density Formulation

• Average bin density
• Equality constraint
• Average bin density utilization ratio
• However, density function is highly non-smooth

8
Smoothing Density Function

• Smoothing operator
• Larger epsilon
• More local smoothing
• Slow convergence
• Smaller epsilon
• More global smoothing
• Faster convergence

9
Smoothed Constrained WL Minimization Problem

• Minimize smooth objective wirelength subject to
  smooth density function

10
Solving Density Constrained WL Minimization

• Using the Uzawa algorithm, we iteratively solve
• can be viewed as generalized force
• Advantages
• Individual scaling factor at each bin
• Systematic updates of these scaling factors
• No Hessian inversion is required

11
Summary of Generalized Force-directed (GFD)
Algorithm

• If initial solution not given
• Use unconstrained quadratic minimizer
• Set stopping criterion
• Iteratively solve
• Poisson equation to get forces
• Updating the scaling factor (Lagrange multiplier)
  for forces based on the smoothed density
• The nonlinear equation by stabilized fixed point
  iteration

12
Important Ingredients of GFD

• Use of accurate objective functions
• Optimization-based bin-density constraint
  formulation
• Global smoothing of density function
• Use of Uzawa algorithm enables
• Systematic bin-level adjustment of force-scaling
  factors
• Convergence to a well defined solution via
  fixed-point iteration
• Applying multilevel optimization can lead to
  better runtime and wirelength

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Overview of mPL multilevel framework

• Coarsening build a hierarchy of problem
  approximations by First Choice clustering
• Relaxation improve the placement at each level
  by iterative optimization
• Interpolation transfer coarse-level solution to
  adjacent, finer level (AMG declustering)
• Multilevel Flow multiple traversals over
  multiple hierarchies (V-cycle variations)

14
mPL5 Framework

• Keep coarsening until cells less than 500

15
Improvement by Our Multilevel Framework
Experiments carried out on ISPD2004 FastPlace IBM
benchmarks.
16
Comparison on Standard Cell Designs
Experiments carried out on ISPD2004 FastPlace IBM
benchmarks.
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Scalability Comparison
mPL5-fast is slightly more scalable than
FastPlace1.0
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Comparison on Mixed-Size Placement Benchmarks
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
IBM02
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
IBM10
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 Benchmarks
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mPL5 placement on ICCAD2004 Mixed-size IBM02
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Conclusions

• mPL5 is a highly scalable multilevel placer based
  on bin-density constrained optimization
  formulation
• Provides a mathematically sound foundation for
  force-directed methods
• mPL5 produces the best wirelength with
  competitive runtime on both standard cell and
  mixed-size designs.
• 3 to 9 shorter WL on standard cell designs
• 9 to 18 shorter WL on mixed size designs
• compared the best-known academic placers

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Acknowledgement

• Financially supported by SRC, NSF, and ONR.
• Thank Min Xie for implementation of detailed
  placement
• Thank Joseph Shinnerl and Min Xie for valuable
  discussions
• Thank Chris Chu and Natarajan Viswanathan for
  providing ISPD04 FastPlace IBM benchmarks.

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End of the Presentation
Thank you!
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Overview of Force Directed Method

• Force directed method in Kraftwerk Eisenmann and
  Johannes 98
• Minimize quadratic wirelength solve Ax0 b
• Compute forces (fk) acting on cells based on the
  current density
• Iteratively solve A(?x) - b c1 fk xk1 xk
  c2 ?x.
• Assume forces are zero at infinity.

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Fast Poisson solver

• Solve the Poisson equation using a Neumann
  boundary condition, i.e., assume forces pointing
  outside the chip region are zero
• Diagonalized by discrete cosine matrix C
  (C-1CT), where the ij-th entry is given by
• Matrix-vector product Cx or CTx can be computed
  in O(n log n)
• Eigenvalues can be computed explicitly.
• Solution is ready

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Coarsening by Recursive Aggregation First Choice
Aggregation Karypis, 1999
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Adjustable Vertex Affinity for Reggregation

• First V-cycle affinity

• Next V-cycle affinity (distance is incorporated)

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AMG-based Linear Interpolation A. Brandt 1986
constant
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Discretization for Poisson Equation under Neumann
Boundary Condition
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Computation of Forces Acting on Cells

• Divide the placement region into m x n bins
• Computed the density in each bin
• Solve the Poisson equation to get smoothed
  density for each bin
• Compute the force acting on each bin by forward
  difference of the smoothed density
• Cells get the forces acting on the bin where they
  lie in.

m
3
2
1
n
1
4
3
2
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Generalized Force-directed Algorithm (GFD)
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Stabilized Fixed Point Iteration for Solving
Nonlinear equation

• Solve f(x) 0, where f(x) is nonlinear, we
  iteratively solve
• Guarantee convergence for small
• Placement is saved initially. If is not small
  enough for convergence, placement is restored and
  is decreased by a certain ratio lt 1.
• Too small implies slow convergence.

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Stopping Criterions for GFD

• Percentage of non-zero density bins (95 97)
• Assume total cells area nearly equal to core area
• Percentage of bin overflow (7-10)
• Fast to evaluate
• Practical

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Comparisons between Aplace and GFD

• Aplace Kahng and Wang 04
• Minimize log-sum-exp Naylor. et al 01 objective
• Subject to equal bin/grid density constraint
• Smooth the density function locally by a
  bell-shaped function Naylor. et al 01
• Penalty method to solve the equality constrained
  problem
• GFD
• Minimize log-sum-exp Naylor. et al 01 objective
• Subject to equal bin/grid density constraint
• Smooth the density function by solving a Poisson
  equation (more global)
• Uzawa algorithm to solve the equality constrained
  problem

37
Comparisons of Different Approximations to
Half-perimeter Wirelength
Experiments carried out on ISPD2004 FastPlace IBM
benchmarks. Relative global placement wirelength
and runtime is reported.