1. Analytic Placement Optimization Using Nonlinear Methodologies
Jingwei Lu
UC San Diego
June 10, 2013
11/27/2018
UC San Diego

2. Outlines Introduction Formulation Density Function
Nonlinear Optimization
Parameter Setting
Experimental Results
Conclusion
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3. Introduction Synthesis Analysis ECO Placement Routing 11/27/2018
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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. Introduction Procedure: (All figure illustration use ADAPTEC1 [Nam05])
Initial Placement: Focus on wire length
Iterative Placement: Spread the cells with minimal wire length increment
Global Placement: Verify with detailed placement
initial placement
final placement
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6. Placement Flowchart
our focus
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7. Problem Formulation Obj: min HPWL Constraint: No overlapping
Nonlinear Programming:
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8. Density Function Analogy of charge force
Calculation via Fourier Transformation
Boundary Condition
Adjustment
Smoothing over discrete grids
Dummy cell insertion
Complexity
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9. Placement Density vs. Charge Force
Cell Instances
Electric Particles
Cell Location
(xi, yi)
Particle Location
(xi, yi)
Cell Area
(Ai)
Charge Quantity (qi)
Grid Density Equalization
Minimum AC Potential Energy N=Σqiψi
Placement Instance
Electrostatic System
Obj.: min W+λN
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10. 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

11. Poisson’s Equation Charge Density
Neumann boundary condition, is the outer norm vector.
Total charge and total potential are zero
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12. Numerical Solution: Boundary Conditions
Neumann
boundary condition
negative half-plane mirroring
[0,n-1]  [-n,n-1]
DCT expansion of density function
even & periodic density function
within [-n, n-1]
fast numerical solution using FFT
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13. 2D cosine series generation
Spectral Methods
frequency components
2D cosine series generation
2D DCT expansion
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14. From Poisson’s Equation
Spectral Methods
From Poisson’s Equation
Potential Function
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15. From Poisson’s Equation partial gradient generation
Spectral Methods
From Poisson’s Equation
partial gradient generation
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16. Smoothing Over Discrete Grids
Usage of Fast Fourier Transform
Smoothing function of triangular weight
Reflect the density cost for cell shifting
A one-dimension example is shown below
A(c,bi) : area contribution of cell c to grid bi
still analytic function
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17. Dummy Cell Insertion Initial solution
Placement: spread cells -> long wires
Placement with fillers
Placement (fillers removed)
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18. quality and efficiency improve by adding fillers (ADAPTEC1)
Dummy Cell Insertion
quality and efficiency improve
by adding fillers (ADAPTEC1)
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19. Complexity Analysis Complexity  O(m+n2logn) 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)
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20. Boundary Condition Verification
iteration 100
iteration 150
ADAPTEC1
128x128
The boundary force is zero!
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UC San Diego

21. Density Gradient (Global Effect)
iteration 3
ADAPTEC1
128x128
iteration 33
Global Information Aware
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22. Optimization Methods Line search for the step length
Locally quadratic approximation
Convergence  Hessian condition
Conjugate Gradient [Shewchuk94]
Lipschitz continuity of gradient
Locally strongly convex function
Convergence rate O(1/k2)
Nesterov’s method [Nesterov83]
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23. Conjugate Gradient (CG)
objective function:
f = W + λD
gradient vector
Iterative approach
Polak-Ribiere
search direction
line search
new solution
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24. escape from local optimal
Line Search
using golden
sectional search
optimal if
f is uni-modal
sub-optimal if
f is multi-modal
occasional
uphill climbing
escape from local optimal
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25. Nesterov’s Method function and gradient vector def. Lipschitz constant
objective function: f = W + λN
function and gradient vector
f(x) є C1,1(E)
def. Lipschitz constant
convex function inequality
maximum
step length
(1)
using Lipschitz constant
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26. Empirical Approximation
Dynamic approximation
Lipschitz const approximation
step length prediction
verification & backtrack
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27. wirelength parameter γ
grid dimension n
Lagrange multiplier λ
Parameter Setting
wirelength parameter γ
step length α
density overflow τ
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28. Grid Dimension (n×n) Tradeoff: accuracy v.s. complexity
n must be power of two
Required by FFT package [Ooura]
In our work: n is function of # cells
Simple & effective
Low penalty on efficiency
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29. Lagrange Multiplier (λ)
Initial value
Balance: wirelength and density force
Iterative update
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30. ρb’: density due to movable cells
Density Overflow (τ)
ρb’: density due to movable cells
Ab: area of bin
Ai: area of cell
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31. Wirelength Smoothing Parameter (γ)
τ: density overflow
wb: bin dimension
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32. Adaptive Step Length (α)
Conjugate Gradient
Nesterov’s method
Adaptive Step Length (α)
line search
Lipschitz approx
interval length adjustment
verification & backtrack
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UC San Diego

33. Experiment Setup ISPD 2005 benchmark suite [Nam05] Placement density
GSRC bookshelf format
iterative density distribution
Placement density
ρt=1.00 for Conjugate Gradient method
ρt=0.97 for Nesterov’s method
Seven state-of-the-art placers
Covering three categories: quadratic, min-cut, nonlinear
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UC San Diego

34. * from published results
Table of Results
* from published results
Wirelength (e6)
Placers
Our work
Min-Cut
Quadratic
Non-linear
Benchmarks
Nesterov (LC)
CG (LS)
Capo10.5
FastPlace3.0
RQL*
SimPL*
APlace2
NTUplace3
mPL6
ADAPTEC1
76.46
87.80
78.34
77.82
77.73
78.35
80.29
77.93
ADAPTEC2
87.69
85.57
102.66
93.47
88.51
90.36
95.70
90.18
92.04
ADAPTEC3
196.27
202.16
234.27
213.48
210.96
208.95
218.52
233.77
214.16
ADAPTEC4
186.15
185.63
204.33
196.88
188.86
187.4
209.28
215.02
193.89
BIGBLUE1
93.04
91.64
106.58
96.23
94.98
97.42
100.02
98.65
96.80
BIGBLUE2
146.48
145.54
161.68
154.89
150.03
145.78
153.75
158.27
152.34
BIGBLUE3
323.49
337.39
403.36
369.19
323.09
339.78
411.59
346.33
344.10
BIGBLUE4
800.95
805.90
871.29
834.04
797.66
808.22
945.77
829.09
829.44
Avg. Diff
0.00%
0.38%
14.93%
6.38%
1.95%
2.75%
11.64%
8.39%
4.77%
CPU (minutes)
Placers
Our work
Min-Cut
Quadratic
Non-linear
Benchmarks
Nesterov (LC)
CG (LS)
Capo10.5
FastPlace3.0
APlace2
NTUplace3
mPL6
ADAPTEC1
2.91
9.17
48.33
2.92
48.88
7.17
23.27
ADAPTEC2
4.47
12.67
61.63
4.13
68.07
8.22
24.75
ADAPTEC3
21.06
45.40
133.43
9.53
186.67
18.53
73.97
ADAPTEC4
14.66
34.33
141.85
8.75
209.60
23.53
71.03
BIGBLUE1
5.38
23.63
77.90
4.57
64.05
14.30
30.05
BIGBLUE2
7.31
30.83
150.15
8.00
136.43
35.10
79.00
BIGBLUE3
32.69
107.75
373.87
21.05
289.78
38.77
104.63
BIGBLUE4
43.08
165.00
730.42
40.13
779.22
106.08
238.82
Avg. Mult.
1.00
3.28
13.73
0.81
14.09
2.24
5.88
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UC San Diego

35. Quality & CPU Comparison
* from published results
our work
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our work
UC San Diego

36. Conclusion Background introduction & problem formulation
Poisson’s equation for potential coupling
Fast numerical solution using spectral methods
Flat nonlinear algorithm using conjugate gradient and Nesterov’s method
Experiments and results on ISPD05 bookshelf format
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UC San Diego

37. Future Work Placement quality improvement
Preconditioning for convergence acceleration
Stochastic process to complement analytic algorithm
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 [Kim11], timing [Chan09], thermal [Cong11]
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UC San Diego

38. References
[Adya03] S. N. Adya, I. L. Markov, and P. G. Villarrubia. On Whitespace and Stability in Mixed-Size Placement. In ICCAD, pages 311–318, 2003.
[Caldwell00] A. E. Caldwell, A. B. Kahng, and I. L. Markov. Can Recursive Bisection Alone Produce Routable Placements? In DAC, pages 477–482, 2000.
[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.
[Cong08] J. Cong, G. Luo, and E. Radke. Highly Efficient Gradient Computation for Density-Constrained Analytical Placement. IEEE TCAD, 27(12):2133–2144, 2008.
[Cong09] J. Cong and Y. Zou, Paralell Multi-level Analytical Global Placement on Graphics Processing Unit, in ICCAD, pages , 2009
[Cong11] J. Cong, G. Luo and Y. Shi, Thermal-Aware Cell and Through-Silicon-Via Co-Placement for 3D ICs, in DAC, pages , 2011.
[Eisenmann98] H. Eisenmann and F. M. Johannes. Generic Global Placement and Floorplanning. In DAC, pages 269–274, 1998.
[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
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39. References
[Kim10] M.-C. Kim, D.-J. Lee, and I. L. Markov. SimPL: An Effective Placement Algorithm. In ICCAD, pages 649–656, 2010.
[Moreland03] K. Moreland and E. Angel. The FFT on a GPU. Graphics Hardware, 2003.
[Nam05] G.-J. Nam, C. J. Alpert, P. Villarrubia, B. Winter, and M. Yildiz. The ISPD2005 Placement Contest and Benchmark Suite. In ISPD, pages 216–220, 2005.
[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.
[Shewchuk94] J. Shewchuk. An Introduction to the Conjugate Gradient Method without the Agonizing Pain. In Technical Report CMU-CS-TR , Carnegie Mellon University, 1994.
[Skollermo75] G. Skollermo. A Fourier Method for the Numerical Solution of Poisson’s Equation. Mathematics of Computation, 29(131):697–711, 1975.
[Viswanathan07a] N. Viswanathan et al. RQL: Global Placement via Relaxed Quadratic Spreading and Linearization. In DAC, pages 453–458, 2007.
[Viswanathan07b] N. Viswanathan, M. Pan, and C. Chu. FastPlace3.0: A Fast Multilevel Quadratic Placement Algorithm with Placement Congestion Control, in ASPDAC, pages 135–140, 2007.
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40. Q & A
Thank you
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41. Backup Slides
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42. Graph Problem Abstraction
G=(V,E,R)
hyper-graph:
placement instance V
node set:
cells & macros E
edge set: nets R
placement region
Global routing remains a fundamental and important research problem in VLSI physical design.
The quality of routing has direct impact on the overall circuit performance
Low-quality routing solution will cause serious problem to the timing closure
Routing failure will loopback to placement and reduce design efficiency.
As a result, High-performance routing approaches are quite necessary to the back-end
Electronic design automation
ISPD hosts two international contests on global router design optimization, they have
Published benchmark suite, which are generated from placed real integrated circuits.
Lots of research works have been done based on their formulation.
State-of-the-art approaches are mostly based on integer linear programming, or
Negotiation-based Iterative rip-up and rerouting.
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43. Uniform Grid Decomposition
original instance
1024x1024
256x256
64x64
Uniform Grid Decomposition
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44. Grid Density Equalization
original density distribution
after 5 iterations
after 10 iterations
after 15 iterations
Grid Density Equalization
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45. Algorithm Categorization
Stochastic
Min-Cut
Quadratic
Simulated annealing
Timberwolf [Sechen86]
Circuit partitioning
Capo [Caldwell00]
Dragon[Taghavi05]
Fengshui [Agnihorti05]
Nonlinear
Linear system solution
SimPL [Kim10],
RQL [Viswanathan07a]
FastPlace [Viswanathan07b]
Nonlinear optimization
Aplace[Kahng05]
NTUplace[Chen08]
mPL [Chan06]
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46. Analysis of Prior Methods
Stochastic
Best performance at small case (<10K cells)
Longest runtime, low scalability to large case
Min-cut
Unrecoverable quality loss due to incorrect partitioning
Quality loss due to naïve choice at early iteration
Quadratic
Inaccurate quadratic wirelength modeling
Density force modulation remains a problem
Nonlinear
High-order smoothing function  complexity
Cell clustering  reduced search space & quality loss
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47. Wirelength Modeling
HPWL
LSE WA
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48. Wirelength Error Bound
LSE WA
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49. Existing Problems in Prior PDEs
Poisson’s equation
Ill-definition induces non-unique solution [Chan05]
Performance instability
Can be improved by introducing linear terms
Helmholtz equation
Unique PDE solution with linear terms
Quality is sensitive to the choice of ε [Cong07]
Robustness degradation
Gaussian smoothing
Convolution with Gaussian equation
High complexity, computationally intensive
Only applied to fixed blocks [Chen08]
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50. PDE Solutions & Problems
Green’s function [Eisenmann98] [Cong08]
Computationally intensive
Low extensibility to large design & high granularity
Finite difference method [chan05] [Chan06]
Still only local information aware
Slow variation propagation in global scale
Low convergence rate & quality loss
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UC San Diego

51. Density Scaling for Fixed Macros
GP solution w/o density scaling
GP solution w/ density scaling
white margin around macros
global even distribution,
less quality penalty
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52. wirelength and energy functions conjugate orthogonalization
Analysis
wirelength and energy functions
quadratic
approximation
highly nonlinear
f= W+λD
conjugate orthogonalization
easy to lose
conjugacy
convergence
rate estimation
time-varying
Hessian matrix Hf,
hard to approx
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53. Analysis unconstrained optimization constraint relaxation (λ)
convex objective
function
non-convex function f= W+λD
convergence
rate estimation
time-varying Lipschitz constant
L = func(λ,t)
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54. Density Smoothing Effect
original density
Bell-shaped smoothing
global smoothing
two-level smoothing
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UC San Diego

55. Existing Problems in Prior PDEs
Poisson’s equation
Ill-definition induces non-unique solution [Chan05]
Performance instability
Can be improved by introducing linear terms
Helmholtz equation
Unique PDE solution with linear terms
Quality is sensitive to the choice of ε [Cong07]
Robustness degradation
Gaussian smoothing
Convolution with Gaussian equation
High complexity, computationally intensive
Only applied to fixed blocks [Chen08]
11/27/2018
UC San Diego

56. Original Electrostatic Equilibrium
Original Charge Distr.
(in 1D illustration)
Even distribution at the boundary
Converged Charge Distr.
(using electric force)
But, still overlap
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57. 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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58. Breakdown of Node Set V fixed movable original inserted Vm Vf std cell
macros Vd
dark nodes
Vfc
fillers Vm
std cell
original
inserted
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59. Boundary Condition Verification
iteration 15
iteration 30
ADAPTEC1
128x128
The boundary force is zero!
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UC San Diego

60. Density Gradient w/ Global Effect
whitespace at quadrant 2
whitespace at quadrant 4
Global Information Aware
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UC San Diego

61. Nesterov’s Method gradient vector Iterative approach
objective function:
f = W + λD
gradient vector
Iterative approach no
step length search ?
new solution &
parameters
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UC San Diego

62. Step Length & Lipschitz Constant
f(x) є C1,1(E)
function inequality
step length search
(1)
using Lipschitz constant
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UC San Diego