1. ePlace: Electrostatics based Placement
CK Cheng
UC San Diego
2/28/2013
9/22/2018
UC San Diego

2. Outline Introduction Flow of Placement Statement of Problem
Wire Length and Density Approximation: Electrostatic Analogy
Nonlinear Optimization
Nesterov’s method
preconditioning
Macro legalization
Annealing-directed block shifting
Experiments and results
Conclusion and future work
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UC San Diego

3. Introduction Synthesis Analysis ECO Placement Routing 9/22/2018
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4. Introduction Placement: NP complete problem.
Analytic Placement: Placement having a first derivative at all points of placement domain.
Heuristic Approach: Mathematic derivation with empirical tuning.
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UC San Diego

5. ePlace Flow
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6. MFNPL Algorithm One-stage approach Nonlinear optimization
Simultaneous placement of cell & macro
Based on our existing placement prototype [Lu13]
Nonlinear optimization
Solver: Nesterov’s method (c.f. conjugate gradient)
Steplength: dynamic prediction of Lipschitz constant
Convergence: backtracking  closer to local optimum
Mixed size: preconditioning on Hessian matrix
resolve gap between macro and cell
Macro legalization
Annealing: direction of local macro shifting
Process flow: nested loop of parameter adjustment
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UC San Diego

7. Cell Placement
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8. 𝝍𝒊: electric potential
Statement of Problem
Objective: min HPWL
Constraint: no density violation
Density penalty: potential energy
Objective function: Obj. + Density penalty
𝑩: grid set
𝒒 𝒊 : charge, cell area
𝝍𝒊: electric potential
𝝀: penalty factor

9. Wire Length Approximation [Kahng05, chan06, Chen08]
High-order density and wirelength function
Exponential function to extract boundary pins
High accuracy & complexity
Line search to locate local optimum
Repeated objective evaluation  low efficiency
Multi-level netlist & grid coarsening
Suboptimal clustering  quality degradation

10. Density Approximation: Electrostatic Analogy
Electrostatic System
Cell Density
Charge Density
Density Penalty
Potential Energy
Cell Instances
Electric Particles
Placement Instance
Density Gradient
Electric Field

11. Original Distr. w/o direct current (DC) globally even distribution
Modified System
Original Distr. w/o direct current (DC)
globally even distribution
Converged Distr.
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UC San Diego

12. cell placement charge density field distribution potential energy
ADAPTEC1
128x128
charge density
cell placement
charge density
electric field
potential distr
field distribution
potential energy
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UC San Diego

13. Neumann boundary condition total charge density: zero
Poisson’s Equation
ρ(x,y): charge density
Neumann boundary condition
: the outer norm vector
total charge density: zero
even density distr.
total potential: zero
unique solution

14. 2D cosine series generation
Density Expression
2D cosine series generation
2D DCT expansion
Potential Function
Field Function

15. Boundary Condition Verification
iteration 100
iteration 150
ADAPTEC1
128x128
The boundary force is zero!

16. Runtime Density Gradient
iteration 3
ADAPTEC1
128x128
iteration 33
Global Information Aware

17. Complexity Analysis m: total movable nodes
standard cells & filler cells
Density computation  O(m)
n: each dimension of 2-D grids
Uniform decomposition
2D fast Fourier transform  O(n2logn)
total complexity: O(mlogm)
m=O(n2): # grid ≈ # cells
O(m+n2logn) = O(mlogm)

18. Nesterov’s Method: Problems of CG & Line Search
Runtime bottleneck
CPU breakdown from ADAPTEC1 of 512x512 grid
Line search: % of PL, 63.22% of GP
Zero gradient point?
May actually beyond the search interval …
Previous solution: step size prediction
Empirical method [Chen08]: results oriented
Our systematic solution: Nesterov’s method
Runtime Lipschitz constant prediction
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UC San Diego

19. Nesterov’s Method function and gradient vector def. Lipschitz constant
f(x) є C1,1(E)
def. Lipschitz constant
convex function inequality
step length limit?
prediction
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UC San Diego

20. Nesterov’s Method gradient vector Iterative approach
objective function:
f = W + λD
gradient vector
Iterative approach
step length search
new solution &
parameters
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UC San Diego

21. Dynamic Steplength Prediction
Lipschitz const. prediction
step length prediction
known solution from k & k-1 iterations
known gradient from k & k-1 iterations
zero computation overhead
However, аk may not be accurate enough…
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UC San Diego

22. Correction: Steplength Backtracking
initial prediction
temporary solution
reference prediction
final prediction & solution
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UC San Diego

23. Steplength Backtracking
Correction on mis-prediction of steplength
Get closer towards local optimum
Improve convergence of nonlinear optimization
Before: density starts oscillation at ~15%
After: density down to ~2.2%
Average of 16 MMS cases
Set stopping threshold as 1.0%
Limited overhead on efficiency
Zero overhead if 1st backtrack is passed
Average # backtracks: 1.037
Average of 16 MMS cases, < 4% CPU overhead on GP
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UC San Diego

24. Preconditioning
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25. Preconditioning Local quadratic approximation
Nonlinear problem: Hessian matrix
Approximation by diagonal matrix
Complexity: O(n2)  O(n)
Feasible on modern IC design
Hessian inversion
Reduce condition number
clustering eigenvalues & eigenvectors together
Ideal case: one step to converge
Single eigenvalue & eigenvector
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UC San Diego

26. Preconditioning A generalized solution to mixed-size design Limitation
Equalize cell & macro in placement perspective
Resolve physical difference
Balance gradient magnitude
Limitation
Non-convex cost function  not positive definite
Negative eigenvalue  motion mis-direction
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UC San Diego

27. Macro Legalization Difference with most prior works General framework
Post global placement
Sufficient logic and physical information
Good starting point  small design space
Direct annealing on macro shifting
c.f. annealing on macro sequence / representation
General framework
Combined cost function
Wirelength, cell density, macro overlap
Incremental cost update
Dynamic parameter adjustment
Annealing temperature, motion diameter, penalty factor
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UC San Diego

28. Macro Legalization Flow
outer: adjusting penalty factor (λ) in cost function
inner: adjusting annealing parameters (temperature T, diameter R)
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29. Layout Comparison: adaptec1 (gif)
cell & macro co-placement
macro legalization
initial placement
cell detail placement
cell-only placement
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UC San Diego

30. state-of-the-art mixed-size placers
Experiment Setup
MMS benchmark suite [Yan09]
GSRC bookshelf format, based on ISPD 2005 & 2006 benchmarks
Free originally fixed macros, insert terminals to preserve ASIC structure
Up to 2.5M cells & 3.7K movable macros, well represent modern design complexity
No rotation or flipping of macros (For now)
Four state-of-the-art placers
Covering two cutting-edge categories: constructive (floorplan-guided) and one-stage (simultaneous macro & cell placement)
categories
constructive
one-stage
state-of-the-art mixed-size placers
FLOP [Yan09]
mPL6 [Chan06]
Capo10.5 [Roy06]
NTUplace3-unified [Hsu12]
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UC San Diego

31. Quality & CPU Comparison
* from published results
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UC San Diego

32. Conclusion Nonlinear optimization Annealing-based macro legalization
Nesterov’s method, Lipschitz prediction
Steplength backtracking, Hessian preconditioning
Annealing-based macro legalization
Direct cell shifting
Nested framework on parameter adjustment
Experiments on MMS benchmark suite
Improved quality and efficiency
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UC San Diego

33. Future Work Extensibility towards parallel platform
GPU architecture and distributed system
Parallel FFT for density gradient [Moreland03]
Parallel wirelength gradient generation [Cong09]
Extensibility towards other design objectives
Routability-driven placement [Kim11]
Timing-driven placement [Chan09]
3D-IC analytic placement [Cong09b]
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UC San Diego

34. References
[Chan06] T. F. Chan, J. Cong, J. R. Shinnerl, K. Sze, and M. Xie. mPL6: Enhanced Multilevel Mixed-Size Placement, in ISPD, pages 212–214, 2006.
[Chan09] T. F. Chan, J. Cong, and E. Radke, A Rigorous Framework for Convergent Net Weighting Schemes in Timing-Driven Placement, in ICCAD, pages , 2009. 
[Chen08] T.-C. Chen et al. NTUPlace3: An Analytical Placer for Large-Scale Mixed-Size Designs with Preplaced Blocks and Density Constraint, IEEE TCAD, 27(7):1228–1240, 2008.
[Chen08b] T.-C. Chen, P.-H. Yuh, Y.-W. Chang, F.-J. Huang, and D. Liu, “MP-trees: A packing-based macro placement algorithm for modern mixed-size designs,” IEEE Trans. Comput.-Aided Des., vol. 27, no. 9, pp. 1621–1634, Sep
[Chen08c] H.-C. Chen, Y.-L. Chung, Y.-W. Chang, and Y.-C. Chang, “Constraint graph-based macro placement for modern mixed-size circuit designs,” in ICCAD, 2008, pp. 218–223
[Cong09] J. Cong and Y. Zou, Parallel Multi-level Analytical Global Placement on Graphics Processing Unit, in ICCAD, pages , 2009
[Cong09b] J. Cong and G. Luo. A Multilevel Analytical Placement for 3D ICs, in ASPDAC, pp , 2009.
[Hsu11] M.-K. Hsu, Y.-W. Chang and V. Balabanov. TSV-Aware Analytical Placement for 3D IC Designs, in DAC, pages , 2011.
[Kahng05] A. B. Kahng, S. Reda, and Q. Wang. Architecture and Details of a High Quality, Large-Scale Analytical Placer. In ICCAD, pages 890–897, 2005
[Lu13] J.Lu et al., FFTPL: An Analytic Placement Algorithm Using Fast Fourier Transform for Density Equalization, in ASICON 2013, to appear.
[Moreland03] K. Moreland and E. Angel. The FFT on a GPU. Graphics Hardware, 2003.
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UC San Diego

35. References
[Naylor01] W. C. Naylor, R. Donelly, and L. Sha. Non-Linear Optimization System and Method for Wire Length and Delay Optimization for an Automatic Electric Circuit Placer. In US Patent , 2001.
[Nesterov83] Y. E. Nesterov. A Method of Solving a Convex Programming Problem with Convergence Rate O(1/k2). In Soviet Math, 27(2), 1983.
[Ooura] General-Purpose FFT Package.
[Pan05] M. Pan, N. Viswanathan, and C. Chu. An Efficient and Effective Detailed Placement Algorithm. In ICCAD, pages 48–55, 2005.
[Roy06] J. A. Roy, S. N. Adya, D. A. Papa and I. L. Markov, ``Min-cut Floorplacement''  IEEE TCAD 25(7): pp , 2006.
[Shewchuk94] J. Shewchuk. An Introduction to the Conjugate Gradient Method without the Agonizing Pain. In Technical Report CMU-CS-TR , Carnegie Mellon University, 1994.
[Yan09] J. Z. Yan, N. Viswanathan, and C. Chu, “Handling complexities in modern large-scale mixed-size placement,” in DAC, 2009, pp. 436–441.
[Yao05] B. Yao, H. Chen, C.-K. Cheng, N.-C. Chou, L.-T. Liu, and P. Suaris, Unified quadratic programming approach for mixed mode placement, in ISPD, 2005, pp. 193–199.
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UC San Diego

36. Q & A
Thank you
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UC San Diego