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 = × 10.32
Practical Problems in VLSI Physical Design

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