1. An Algebraic Multigrid Solver for Analytical Placement With Layout Based Clustering Hongyu Chen, Chung-Kuan Cheng, Andrew B. Kahng, Bo Yao, Zhengyong Zhu University of California, San Diego Nan-Chi Chou, John F. MacDonald, Peter Suaris Mentor Graphics Corporation

2. Outline Analytical placement background Analytical placement background Algebraic multigrid solver Algebraic multigrid solver Layout based clustering Layout based clustering Experimental results Experimental results Conclusions Conclusions

3. Analytical Placement Successfully introduced and applied for 20 years Successfully introduced and applied for 20 years Minimize quadratic wire length Minimize quadratic wire length Basic formulation is a quadratic programming (QP) Basic formulation is a quadratic programming (QP) Optimal solution is given by the linear equation systems Optimal solution is given by the linear equation systems CX+d x =0 CY+d y =0

4. Matrix Solver Is the Key Need additional force to make cell distribution even Need additional force to make cell distribution even (Eisenmann & Johannes, DAC 1998) e x is based on cell density, need iterations to update e x is based on cell density, need iterations to update Key observation Key observation Linear equations with the same system matrix C is to be solved for many times A fast matrix solver is the key technology for analytical placement A fast matrix solver is the key technology for analytical placement

5. Ever Increasing Problem Dimensions ITRS roadmap ITRS roadmap –1 billion transistors integration in year 2007 –Millions of unknowns for the linear equation Number of transistors in a typical MPU/ASIC design (Millions) Year (source: ITRS Roadmap 2002 update)

6. Hierarchy Is the Way to Go Placement Placement –T.Chan, J. Cong, et al, Multilevel Optimization for Large-scale Circuit Placement, ICCAD 2000 –X.Hong, et al, CASH, ASP-DAC 2000 –Hsun Cheng Lee, et al, Multilevel Floorplanning for Large-Scale Building Block Design, DAC 2003 Graph partitioning/clustering Graph partitioning/clustering –G. Karypis and V. Kumar, hMETIS, 1998 –C.J.Alpert, et al, Multilevel Circuit Partitioning, 1998 Multigrid Multigrid –W.L.Briggs, V.E.Henson, S.F.McCormic, A Multigrid Tutorial, 2000 –K.Stüben, A Review of Algebraic Multigrid, 1999

7. AMG – a Good Candidate Algebraic MultiGrid – a new branch of the multigrid method Algebraic MultiGrid – a new branch of the multigrid method The method is scalable The method is scalable –O(n) time complexity theoretically. –SOR and PCG are O(nlogn). Guaranteed convergence and optimality Guaranteed convergence and optimality Overcome the restriction of classic multigrid, can be applied to linear systems without regular structure. Overcome the restriction of classic multigrid, can be applied to linear systems without regular structure. A hierarchy of linear systems are derived directly from the original equation. A hierarchy of linear systems are derived directly from the original equation.

8. AMG Idea (1): A Hierarchy of Problems Defined by a clustering scheme Defined by a clustering scheme 1 2 3 4 5 6  (0) A (0) X (0) =b (0) 2 1 4 3  (1) A (1) X (1) =b (1) 2 1  (2) A (2) X (2) =b (2) Original Problem

9. AMG Idea (2) : Solution Mapping Between Levels Coarse level node position is the average of fine level components Coarse level node position is the average of fine level components 2 1 4 3  (1) 2 1  (2) Interpolation : Interpolation : Restriction : Restriction :

10. AMG Idea (3): V-cycle Iterations Description of the algorithm in 2 levels Description of the algorithm in 2 levels A (1) X (1) = b (1)  (0)  (1) A (0) X (0) = b (0) Iterate => Start with Interpolation :  Restriction:  Iterate => Extend to multi-levels Extend to multi-levels Update R (0), b (1)

11. Derive Linear Systems for All Levels Galakin operation Galakin operation A is symmetric positive definite A is symmetric positive definite Easy operation with interpolation and restriction operators, only addition performed Easy operation with interpolation and restriction operators, only addition performed It introduces overhead to the solution process It introduces overhead to the solution process

12. Why Iterate at Coarse Levels ? Smaller problem => less CPU time Smaller problem => less CPU time Coarse level iterate is more efficient in reducing errors Coarse level iterate is more efficient in reducing errors –Iterations on each level are efficient only in reducing high frequency errors to that level –Restriction operation will transform low frequency errors into high frequency ones, which can be efficiently reduced at coarse level Idea of multigrid: Idea of multigrid: –Iterations to reduce high frequency error in each level, leave the low frequency error for the coarser level.

13. Aggregate-type of AMG Our approach turns out to be aggregate-type of AMG Our approach turns out to be aggregate-type of AMG –Interpolation: one fine level node depends only on one coarse level node. Pros Pros –Can be used with any clustering scheme –Simple derivation of the problems and simple solution mapping, less overhead Cons Cons –Piecewise linear interpolation, less effective.

14. Layout Based Clustering Motivation: identify which cells stay close together in the placement. Motivation: identify which cells stay close together in the placement. The clustering is needed for the AMG process. The clustering is needed for the AMG process. Layout based clustering Layout based clustering –Edge coarsening –Edge weight is according to the distance info –Collapse the edge with shortest distance first Alternatives Alternatives –hMETIS –Random

15. Distance Information of An Edge Distance information is calculated based on trials of placement Distance information is calculated based on trials of placement “Blown-up” placement “Blown-up” placement Starting point All cells at the same position Iterations on solving equations Snapshot of the placement when the standard deviation of cell coordinates is max

16. Deriving the Distance of An Edge Distance between cell i and j, d ij Distance between cell i and j, d ij d ij = d ij (ll) + d ij (lr) + d ij (ul) + d ij (ur) (1)(2) (3)(4)

17. Experimental Settings Two experiments Two experiments –Convergence comparison with SOR and PCG solvers SOR (  = 1.95 ) SOR (  = 1.95 ) PCG ( Incomplete Cholesky factorization precondition) PCG ( Incomplete Cholesky factorization precondition) –Comparison with different clustering algorithms Layout based clustering Layout based clustering hMETIS clustering hMETIS clustering Random clustering Random clustering Solve the system Ax=b once Solve the system Ax=b once Check relative error, e r =||X-X*|| ∞ / ||X*|| ∞ Check relative error, e r =||X-X*|| ∞ / ||X*|| ∞ Test cases from ISPD 2002. #cells 12k to 200k Test cases from ISPD 2002. #cells 12k to 200k

18. Comparison of the Solvers CPU time to converge to 10 -2 relative error CPU time to converge to 10 -2 relative error Case Number CPU Time (sec) AMG PCG SOR

19. Example Convergence Histories AMG solver converges fast at the beginning AMG solver converges fast at the beginning erer CPU Time (sec) AMG PCG SOR

20. Comparison of the Clustering Methods (1) CPU time to reach 10 -3 relative error CPU time to reach 10 -3 relative error Layout HMetis Random Case Number CPU Time (sec)

21. Layout HMetis Random Comparison of the Clustering Methods (2) Clustering time comparison Clustering time comparison CPU Time (sec) Case Number

22. Conclusions A hierarchical solver is important for large scale analytical placement problem A hierarchical solver is important for large scale analytical placement problem Algebraic multigrid method converges very fast at the beginning stage of iterations. It is a promising technology for solving linear equations in analytical placement Algebraic multigrid method converges very fast at the beginning stage of iterations. It is a promising technology for solving linear equations in analytical placement Layout based clustering scheme is useful for AMG solver Layout based clustering scheme is useful for AMG solver

23. Thank you !