I. Floorplanning with Fixed Modules

1. I. Floorplanning with Fixed Modules • Fixed modules only, no rotation allowed • m1 (4,5), m2 (3,7), m3 (6,4), m4 (7,7) Practical Problems in VLSI Physical Design

2. ILP Formulation Practical Problems in VLSI Physical Design

3. Non-Overlapping Constraints (cont) Practical Problems in VLSI Physical Design

4. Additional Constraints Practical Problems in VLSI Physical Design

5. Solutions • Using GLPK we get the following solutions: Practical Problems in VLSI Physical Design

6. Final Floorplan • Why the non-optimality? • Due to linear approximation of area objective (= y*) • Chip width/height constraints also affected • In fact, our ILP solution (y* = 12) is optimal under these conditions. Practical Problems in VLSI Physical Design

7. II. Floorplanning with Rotation • Fixed modules, rotation allowed • Fixed modules: m1 (4,5), m2 (3,7), m3 (6,4), m4 (7,7) • Need 4 more binary variables for rotation: z1, z2, z3, z4 • We use M = max{W,H} = 23 Practical Problems in VLSI Physical Design

8. ILP Formulation Practical Problems in VLSI Physical Design

9. Non-Overlapping Constraints (cont) Practical Problems in VLSI Physical Design

10. Non-Overlapping Constraints (cont) Practical Problems in VLSI Physical Design

11. Additional Constraints Practical Problems in VLSI Physical Design

12. Solutions • Using GLPK we get the following solutions: Practical Problems in VLSI Physical Design

13. III. Floorplanning with Flexible Modules • 2 Fixed modules: • m1 (4,5), m2 (3,7) (rotation allowed) • 2 Flexible modules: • m3: area = 24, aspect ratio [0.5, 2] • m4: area = 49, aspect ratio [0.3, 2.5] Practical Problems in VLSI Physical Design

14. Linear Approximation Practical Problems in VLSI Physical Design

15. Linear Approximation (cont) Practical Problems in VLSI Physical Design

16. Upper Bound of Chip Dimension Practical Problems in VLSI Physical Design

17. Non-Overlap Constraint Practical Problems in VLSI Physical Design

18. Non-Overlap Constraint (cont) Practical Problems in VLSI Physical Design

19. More Constraints Practical Problems in VLSI Physical Design

20. Solutions Practical Problems in VLSI Physical Design

21. Comparison • Fixed modules only = 12 × 12 • Rotation allowed = 11 × 11 • Flexible modules used = 10.46 × 10.32 Practical Problems in VLSI Physical Design

22. Approximation Error and Overlap • Due to linear approximation • Approximated area of m3 = 3.46 × 5.2 = 17.99 (actually 24) • Approximated area of m4 = 3.83 × 7.32 = 28.04 (actually 49) • Real area of m3 = 3.46 × 6.94 = 24 • Real area of m4 = 3.83 × 12.79 = 49 • Floorplan area increases, overlap occurs Practical Problems in VLSI Physical Design