1. Unified Quadratic Programming Approach for Mixed Mode Placement Bo Yao, Hongyu Chen, Chung-Kuan Cheng, Nan-Chi Chou*, Lung-Tien Liu*, Peter Suaris* CSE Department University of California, San Diego *Mentor Graphics Corporation

2. Outline Introduction to the mixed mode placement Unified cost function DCT based cell density cost Experimental results Conclusions

3. Mixed Mode Placement Common design needs Mixed signal designs (analog and RF parts are macros) Mixed signal designs (analog and RF parts are macros) Hierarchical design style Hierarchical design style IP blocks IP blocks Memory blocks Memory blocks Challenges for placement Huge amount of components Huge amount of components Heterogeneous module sizes/shapes Heterogeneous module sizes/shapes Memory IP Analog

4. Previous Works on Mixed Mode Placement Combined floorplanning and std. cell placement Capo (Markov, ISPD 02, ICCAD 2003) Capo (Markov, ISPD 02, ICCAD 2003) Multi-level annealing placement mPG-MS (Cong, ASPDAC 2003) mPG-MS (Cong, ASPDAC 2003) Partitioning based approaches Feng Shui (Madden, ISPD 04) Feng Shui (Madden, ISPD 04) Force-directed / analytical approaches Kraftwork (Eisenmann and Johannas, DAC 98 ) Kraftwork (Eisenmann and Johannas, DAC 98 ) FastPlace(Chu, ISPD 04) FastPlace(Chu, ISPD 04) APlace (Kahng, ISPD 04, ICCAD 04) APlace (Kahng, ISPD 04, ICCAD 04)

5. UPlace: Optimization Flow Analytical Placement Analytical Placement Discrete Optimization Discrete Optimization Detailed Placement Detailed Placement

6. Unified Cost Function Combined object function for global placement DP: Penalties for un-even cell densities DP: Penalties for un-even cell densities WL: Wire length cost function WL: Wire length cost function Quadratic analytical placement Quadratic analytical placement WL = 1/2x T Qx+px +1/2y T Qy+py Bounding box wire length for discrete optimization Bounding box wire length for discrete optimization

7. Cell Density Common strategy Partition the placement area into N by N rooms Partition the placement area into N by N rooms Cell density matrix D = {d ij } d ij is the total cell area in room (i,j) d ij is the total cell area in room (i,j) A

8. DCT: Cell Density in Frequency Domain 2-D Discrete Cosine Transform (DCT) Cell density matrix D => Frequency matrix F = {f ij } where f ij is the weight of density pattern (i,j)

9. Properties of Frequency Matrix Each f uv is the weight of frequency (u,v) Inverse DCT recovers the cell density … … … (0,0)(1,0)(3,0) (0,1) (1,1) (0,3) (3,3) …

10. … … … (0,0)(1,0)(3,0) (0,1) (1,1) (0,3) (3,3) … Frequency Matrix: An Example Density matrix D and frequency matrix F

11. Properties of DCT Cell density energy  d ij 2 =  f ij 2 Cell perturbation and frequency matrix Uniform density  f ij = 0

12. Density Cost of a Placement Weighted sum of f ij 2 Higher weight for lower frequency

13. Approximate the density cost with a quadratic function DP = ½ a i x i 2 + b i x i +c i DP = ½ a i x i 2 + b i x i +c i Make DP convex Make DP convex a i >= 0 to ensure a i >= 0 to ensure Matrix form DP = ½ x T Ax+Bx DP = ½ x T Ax+Bx A = diag(a 1, a 2, …, a n ), A = diag(a 1, a 2, …, a n ), B=(b 1, b 2, …, b n ) T B=(b 1, b 2, …, b n ) T Approximation of the Density Cost xixi DP x-  x x+  DP xixi x-  x x+  a i > 0 a i = 0

14. UPlace: Minimize Combined Objective Function Combine quadratic objectives WL +  DP WL +  DP WL = ½ x T Qx+px WL = ½ x T Qx+px DP = ½ x T Ax+Bx DP = ½ x T Ax+Bx Solve linear equation for each minimization (Q +  A)x = -p -  B (Q +  A)x = -p -  B Use Lagrange relaxation to reduce cell congestion  (k+1) =  (k) +  (k) * DP  (k+1) =  (k) +  (k) * DP  0 = 0,  0 = Const  0 = 0,  0 = Const  (k+1) =  (k) * , 0<   1  (k+1) =  (k) * , 0<   1

15. Discrete Optimization Try  -distance moves in four directions. Pick the best position. Try  -distance moves in four directions. Pick the best position. Sweep all the cells in each iteration Sweep all the cells in each iteration AAAAA

16. Legalization/Detailed Placement – Zone Refinement One cell a time, ceiling -> floor One cell a time, ceiling -> floor Two directional alternation Two directional alternation A

17. Experimental Results: Wire length Normalized Wire Length

18. Experimental results: CPU time CPU (Min)

19. UPlace: Placement Results IBM-02

20. Conclusions We propose a unified cost function for global optimization, which provides good trade-offs between wire length minimization and cell spreading. We introduce a DCT based cell density calculation method, and a quadratic approximation. The unified placement approach generates promising results on mixed mode designs.

21. Thank You !