1. VLSI Quadratic Placement
EE4271 VLSI Design, Fall 2016
VLSI Quadratic Placement

2. Problem formulation Input: Output:
Blocks (standard cells and macros) B1, ... , Bn
Shapes and Pin Positions for each block Bi
Nets N1, ... , Nm
Output:
Coordinates (xi , yi ) for block Bi.
The total wirelength as an estimation of timing is minimized.

3. Placement Can Make A Difference
Random Initial
Placement
Final
Placement

4. Given a set of interconnected blocks, produce two sets that
Partitioning:
Given a set of interconnected blocks, produce two sets that
are of equal size, and such that the number of nets
connecting the two sets is minimized.

5. FM Partitioning:
The more crossings along the cut, the longer the total wirelength. The idea is to reduce the crossings.
Initial Random Placement
After Cut 1
After Cut 2

6. FM Partitioning: Moves are made based on object gain.
Object Gain: The amount of change in cut crossings
that will occur if an object is moved from
its current partition into the other partition -1 2
- each object is assigned a
gain
- objects are put into a sorted
gain list
- the object with the highest gain
is selected and moved.
- the moved object is "locked"
- gains of "touched" objects are
recomputed
- gain lists are resorted -1 -2 -2 -1 1 -1 1

7. FM Partitioning: -1 2 -1 -2 -2 -1 1 -1 1

8. -1 -2 -2 -2 -1 -2 -2 -1 1 -1 1

9. -1 -2 -2 -2 -1 -2 -2 -1 1 1 -1

10. -1 -2 -2 -2 -1 -2 -2 -1 1 1 -1

11. -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

12. -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

13. -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

14. -1 -2 -2 -2 1 -2 -2 -2 -2 1 -1 -1 -1

15. -1 -2 -2 -2 1 -2 -2 -2 -2 1 -1 -1 -1

16. -1 -2 -2 -2 1 -2 -2 -2 1 -2 -1 -1 -1

17. -1 -2 -2 -2 1 -2 -2 -1 -2 -2 -3 -1 -1

18. -1 -2 -2 1 -2 -2 -1 -2 -2 -2 -3 -1 -1

19. -1 -2 -2 1 -2 -2 -1 -2 -2 -2 -3 -1 -1

20. -1 -2 -2 -1 -2 -2 -2 -1 -2 -2 -2 -3 -1 -1

21. Analytical Placement
Write down the placement problem as an analytical mathematical problem
Quadratic placement
Sum of squared wire length is quadratic in the cell coordinates.
So the wirelength minimization problem can be formulated as a quadratic program.
It can be proved that the quadratic program is convex, hence polynomial time solvable

22. Example:
x=100
x=200 x1 x2

23. Quadratic Placement Form quadratic placement problem
Perform partitioning to reduce overlap

24. Solution of the Original QP

25. Partitioning
Use FM to cut.

26. Applying the Idea Recursively
Perform the quadratic placement on each region with additional constraints that the center of gravities should be in the center of regions.
Center of Gravities

27. (b) Global placement with 4 region
Process
(a) Global placement with 1 region
(b) Global placement with 4 region
(c) Final placements

28. Summary Definition of VLSI quadratic placement problem
Partitioning technique