1. Multilevel Generalized Force-directed Method for Circuit Placement Tony Chan 1, Jason Cong 2, Kenton Sze 1 1 UCLA Mathematics Department 2 UCLA Computer Science Department This work is partially supported by SRC, NSF, and ONR.

2. UCLA VLSICAD LAB2 Outline u u A Brief History of mPL u Recent Progress in Analytical Placement u 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 u Multilevel implementation of GFD  Overview of mPL multilevel framework  mPL5 framework u Conclusions

3. UCLA VLSICAD LAB3 Relative Wirelength mPL 1.0 [ICCAD00] Recursive ESC clustering NLP at coarsest level Goto discrete relaxation Slot Assignment legalization Domino detailed placement year 20002001 20022003 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

4. UCLA VLSICAD LAB4 Recent Progress on Analytical Placement u Force-directed method [Eisenmann and Johannes 98]  Efficient spreading force computation using a fast Poisson solver  Interleave with quadratic placement  Limitations: Inaccurate objective function Inaccurate objective function Require ad hoc tuning of forces for good convergence Require ad hoc tuning of forces for good convergence u Aplace [Kahng and Wang 04]  More 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 framework  Limitations: Local smoothing of density functions Local smoothing of density functions Penalty formulation lumps all constraints together Penalty formulation lumps all constraints together

5. UCLA VLSICAD LAB5 Basic Formulation of Our Approach u Minimize the half-perimeter wirelength subject to even density constraint:

6. UCLA VLSICAD LAB6 Choices of Wirelength Objective Functions HPWL Log-Sum-Exp QuadraticLp-norm

7. UCLA VLSICAD LAB7 Bin based Density Formulation u Average bin density u Equality constraint  Average bin density = utilization ratio u However, density function is highly non-smooth 1 1 3 2 43 2 m n v6v6 v5v5 v4v4 v3v3 v2v2 v1v1 v7v7 = a 13 (v 7 ) = fractional area of cell v 7 in bin B 13

8. UCLA VLSICAD LAB8 Smoothing Density Function u Smoothing operator: u Larger epsilon  More local smoothing  Slow convergence u Smaller epsilon  More global smoothing  Faster convergence

9. UCLA VLSICAD LAB9 Smoothed Constrained WL Minimization Problem u Minimize smooth objective wirelength subject to smooth density function:

10. UCLA VLSICAD LAB10 Solving Density Constrained WL Minimization u Using the Uzawa algorithm, we iteratively solve u can be viewed as “generalized force” u Advantages:  Individual scaling factor at each bin  Systematic updates of these scaling factors  No Hessian inversion is required

11. UCLA VLSICAD LAB11 Summary of Generalized Force-directed (GFD) Algorithm u If initial solution not given:  Use unconstrained quadratic minimizer u Set stopping criterion u 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. UCLA VLSICAD LAB12 Important Ingredients of GFD u Use of accurate objective functions u Optimization-based bin-density constraint formulation u Global smoothing of density function u Use of Uzawa algorithm enables:  Systematic bin-level adjustment of force-scaling factors  Convergence to a well defined solution via fixed-point iteration u Applying multilevel optimization can lead to better runtime and wirelength

13. UCLA VLSICAD LAB13 Overview of mPL multilevel framework u Coarsening: build a hierarchy of problem approximations by First Choice clustering u Relaxation: improve the placement at each level by iterative optimization u Interpolation: transfer coarse-level solution to adjacent, finer level (AMG declustering) u Multilevel Flow: multiple traversals over multiple hierarchies (V-cycle variations)

14. UCLA VLSICAD LAB14 mPL5 Framework Level at which GFD is applied Level 3 Level 2 Level 1 C C I I C+I I I C Coasening I Interpolation Keep coarsening until # cells less than 500

15. UCLA VLSICAD LAB15 Improvement by Our Multilevel Framework Improvement by multilevel GFD over flat GFD Circuit % WL improved % runtime reduced Ibm0113%42% Ibm0537%43% Ibm1023%59% Ibm1520%67% Ibm1831%66% Average24.8%55.4% Experiments carried out on ISPD2004 FastPlace IBM benchmarks.

16. UCLA VLSICAD LAB16 Comparison on Standard Cell Designs Experiments carried out on ISPD2004 FastPlace IBM benchmarks.

17. UCLA VLSICAD LAB17 Scalability Comparison mPL5-fast is slightly more scalable than FastPlace1.0

18. UCLA VLSICAD LAB18 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.

19. UCLA VLSICAD LAB19 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

20. UCLA VLSICAD LAB20 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

21. UCLA VLSICAD LAB21 Results on PEKO Benchmarks

22. UCLA VLSICAD LAB22 mPL5 placement on ICCAD2004 Mixed-size IBM02

23. UCLA VLSICAD LAB23 Conclusions u mPL5 is a highly scalable multilevel placer based on bin- density constrained optimization formulation u Provides a mathematically sound foundation for force- directed methods u 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

24. UCLA VLSICAD LAB24 Acknowledgement u Financially supported by SRC, NSF, and ONR. u Thank Min Xie for implementation of detailed placement u Thank Joseph Shinnerl and Min Xie for valuable discussions u Thank Chris Chu and Natarajan Viswanathan for providing ISPD04 FastPlace IBM benchmarks.

25. UCLA VLSICAD LAB25 End of the Presentation Thank you!

26. UCLA VLSICAD LAB26 Overview of Force Directed Method u Force directed method in Kraftwerk [Eisenmann and Johannes 98]  Minimize quadratic wirelength: solve Ax 0 = b  Compute forces (f k ) acting on cells based on the current density;  Iteratively solve A(∆x) - b = c 1 * f k ; x k+1 = x k + c 2 * ∆x.  Assume forces are zero at infinity.

27. UCLA VLSICAD LAB27 Fast Poisson solver u Solve the Poisson equation using a Neumann boundary condition, i.e., assume forces pointing outside the chip region are zero u Diagonalized by discrete cosine matrix C (C -1 =C T ), where the ij-th entry is given by u Matrix-vector product Cx or C T x can be computed in O(n log n) u Eigenvalues can be computed explicitly. u Solution is ready:

28. UCLA VLSICAD LAB28 Coarsening by Recursive Aggregation First Choice Aggregation [Karypis, 1999] 1 3/2 1 1/2 1 Transform the hypergraph to clique model graph using the weight 1/(|e|-1) Match each vertex with a neighboring vertex with which it shares the most total hyperedge weight

29. UCLA VLSICAD LAB29 Adjustable Vertex Affinity for Reggregation u First V-cycle affinity: u Next V-cycle affinity (distance is incorporated):

30. UCLA VLSICAD LAB30 AMG-based Linear Interpolation [A. Brandt 1986] interpolation constant AMG cluster Next finer level cells Within each cluster, select the one with maximum degree as C-point; others are considered as F-points C-point

31. UCLA VLSICAD LAB31 Discretization for Poisson Equation under Neumann Boundary Condition

32. UCLA VLSICAD LAB32 Computation of Forces Acting on Cells u Divide the placement region into m x n bins u Computed the density in each bin u Solve the Poisson equation to get smoothed density for each bin u Compute the force acting on each bin by forward difference of the smoothed density u Cells get the forces acting on the bin where they lie in. 1 1 3 2 43 2 m n v6v6 v5v5 v4v4 v3v3 v2v2 v1v1 v7v7 1 1 3 2 43 2 m n

33. UCLA VLSICAD LAB33 Generalized Force-directed Algorithm (GFD)

34. UCLA VLSICAD LAB34 Stabilized Fixed Point Iteration for Solving Nonlinear equation u Solve f(x) = 0, where f(x) is nonlinear, we iteratively solve: u Guarantee convergence for small u Placement is saved initially. If is not small enough for convergence, placement is restored and is decreased by a certain ratio < 1. u Too small implies slow convergence.

35. UCLA VLSICAD LAB35 Stopping Criterions for GFD u Percentage of non-zero density bins (95 – 97%)  Assume total cells area nearly equal to core area u Percentage of bin overflow (7-10%) u Fast to evaluate u Practical

36. UCLA VLSICAD LAB36 Comparisons between Aplace and GFD u 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 u 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. UCLA VLSICAD LAB37 Comparisons of Different Approximations to Half- perimeter Wirelength Log-sum-expLp-norm(p=32)Quadratic Circuit WL, runtime Ibm01 1.00, 1.00 1.05, 1.71 1.73, 0.81 Ibm05 1.00, 1.00 1.06, 1.74 1.49, 1.17 Ibm10 1.00, 1.00 1.03, 2.07 1.47, 0.72 Ibm15 1.00, 1.00 1.04, 1.84 1.61, 0.80 Ibm18 1.00, 1.00 1.07, 1.78 1.77, 0.97 Average GP 1.00, 1.00 1.05, 1.83 1.61, 0.89 Average DP 1.00, 1.00 1.03, 1.28 1.21. 1.43 Experiments carried out on ISPD2004 FastPlace IBM benchmarks. Relative global placement wirelength and runtime is reported.