1. EE5780 Advanced VLSI Computer-Aided Design
VLSI Placement
Prof. Shiyan Hu
Office: EERC 518

2. Objectives Placement Problem Formulation Placement Technique
Motivation
Definition
Placement Technique
General flow
Quadratic program based wirelength optimization
FM Partitioning

3. Placement Is Critical
VLSI physical design is to design a layout satisfying the timing target
Need to compute the physical locations of gates
Interconnect delay is the dominating factor and it is proportional to wirelength

4. Placement Solution
Placement is to determine the physical locations of all gates such that the gates are not overlapped and the total wirelength is minimized

5. Different Placements
Bad
Placement
Good
Placement

6. Problem Definition Input Output
Blocks (standard cells and macros) B1, ... , Bn
The locations of some blocks are fixed
Shapes and pin positions for each block Bi
Nets N1, ... , Nm
Output
Coordinates (xi , yi) for block Bi
The total wire length is minimized
Subject to non-overlapping constraint

7. General Flow of Placement Technique
Wirelength optimization without considering non-overlapping constraint
Adding constraints
Partitioning to reduce overlaps

8. Wirelength Optimization
Write down the placement problem as an analytical mathematical problem (analytical placement)
Quadratic placement
Optimization target is to minimize the sum of wirelength
Precisely sum of squared wirelength, also called cost
Wirelength minimization problem can be formulated as a quadratic program
It can be solved efficiently

9. 2 gates to place and 2 pins are fixed
Example (I)
x=100
x=200 x1 x2
2 gates to place and 2 pins are fixed

10. Example (II) x1 x2 x=100 x=200 Interpretation of matrices A and B
The diagonal values A[i,i] correspond to the number of connections to xi
The off diagonal values A[i,j] are -1 if object i is connected to object j, 0 otherwise
The values B[i] correspond to the sum of the locations of fixed objects connected to object i
This approach can be easily extended to the two dimensional optimization

11. Exercise 1
Form the quadratic program and then the linear system for the following example
x=200
x=500 x1 x2 x3

12. Exercise 2 How do we solve the linear system? How large is it?
It is a sparse linear system which can be efficiently solved using standard linear system solver such as the iterative conjugate gradient based approach.

13. Exercise 3
We choose to use square of wirelength in optimization target since the delay is a quadratic function of wirelength, but is it right target?

14. Exercise 4 When there are no fixed pins, what would the solution be?
Even with the fixed pins, there are still a lot of overlaps since in the quadratic program, non-overlapping constraint is not considered.

15. The Placement Technique
Quadratic program minimize the squared wirelength
Partitioning reduce the overlap among gates

16. The Solution of Initial Quadratic Program

17. Partitioning Gates into Regions
After partitioning, restrict that the gates to the one side of the cut line can only be placed in that region
The purpose of partitioning is to sparsify the placement, so it should try to balance the number of gates in each region
Interconnects crossing the cut line should be few to reduce wirelength
The FM partitioning technique will be described soon

18. Restrict Region During Wirelength Optimization
In the quadratic program for each region, add the additional constraint that the center of gravity of gates is the center of region.
Center of Gravities

19. Exercise 5
Suppose that in the example of Exercise 1, the cutline is in the middle of the region and gate 1 and gate 2 are grouped into the left region. Form the quadratic program with the additional constraint.
x=200
x=500 x1 x2 x3

20. (b) Placement with 4 regions
Placement Process
(a) Placement with 1 region
(b) Placement with 4 regions
(c) Final placements

21. Partitioning Formulation
Given a set of interconnected gates, to generate two sets that are of roughly equal size, and such that the number of nets connecting the two sets is minimized.

22. FM Partitioning FM partitioning is a heuristic
- each object is assigned a
gain, which is the amount of change in cut crossings that will occur if an object is moved from its current partition into the other partition.
- objects are put into a sorted
gain list
- set the current region
- at the current region, the object with the highest gain is selected and moved.
- the moved object is locked
- the gains of touched objects are
recomputed
gain lists are sorted again
set the current region to the other region -1 2 -1 -2 -2 -1 1 -1 1

23. FM Partitioning -1 2 -1 -2 -2 -1 1 -1 1

24. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -1 1 -1 1

25. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -1 1 1 -1

26. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -1 1 1 -1

27. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

28. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

29. FM Partitioning -1 -2 -2 -2 -1 -2 -2 -2 1 -1 -1 -1

30. FM Partitioning -1 -2 -2 -2 1 -2 -2 -2 -2 1 -1 -1 -1

31. FM Partitioning -1 -2 -2 -2 1 -2 -2 -2 -2 1 -1 -1 -1

32. FM Partitioning -1 -2 -2 -2 1 -2 -2 -2 1 -2 -1 -1 -1

33. FM Partitioning -1 -2 -2 -2 1 -2 -2 -1 -2 -2 -3 -1 -1

34. FM Partitioning -1 -2 -2 1 -2 -2 -1 -2 -2 -2 -3 -1 -1

35. FM Partitioning -1 -2 -2 1 -2 -2 -1 -2 -2 -2 -3 -1 -1

36. FM Partitioning -1 -2 -2 -1 -2 -2 -2 -1 -2 -2 -2 -3 -1 -1

37. (b) Placement with 4 regions
Summary
Placement Problem
Minimize wirelength
Subject to the non-overlapping constraint
Placement Technique
General flow
Quadratic program based wirelength optimization
FM Partitioning
(a) Placement with 1 region
(b) Placement with 4 regions
(c) Final placements