Quadratic VLSI Placement

1. Quadratic VLSI Placement Manolis Pantelias

2. General • Various types of VLSI placement • Simulated-Annealing • Quadratic or Force-Directed • Min-Cut • Nonlinear Programming • Mix of the above • Quadratic placers • Try to minimize a quadratic wirelength objective function • Indirect measure of the wirelength but can be minimized quite efficiently • Results in a large amount of overlap among cells • Additional techniques needed

3. Proud R. S. Tsay, E. Kuh and C. P. Hsu, “Proud: A Sea-of-Gates Placement Algorithm”

4. Objective function • Don’t minimize the wirelength but the squared wirelength: • cij between module i and module j could be the number of nets connecting the two modules • Connectivity matrix C = [cij] • B = D – C • D is a diagonal matrix with: • Only the one-dimensional problem needs to be considered because of the symmetry between x and y

5. Electric Network Analogy • Objective function xTBx can be interpreted as the power dissipation of an n-node linear resistive network • Vector x corresponds to the voltage vector • x = [x1x2]T, x1 is of dimension m and is to be determined, x2 is due to the fixed I/O pads • B contains the conductance of the nodes • –bij is the conductance between node i and node j

6. Electric Network Analogy (Cont’d) • Placement problem is equivalent to that of choosing voltage vector for which power is a minimum • B11x1 + B12x2 = 0 • B21x1 + B22x2 = i2 • Solve for Ax1 = b • Where A = B11, b = - B12x2 • Solved with iterating method (successive over-relaxation or SOR)

7. Placement and Partitioning • How to avoid module overlap? • Solution: Iterative partitioning in a hierarchical way • Add module areas from left to right until roughly half of the total area, that defines partition-line • Make modules to the right of partition-line fixed, modules to the left of partition-line movable • Project fixed modules to center-line • Perform global placement in the left-plane (of center-line)

8. Placement and Partitioning (Cont’d) • Make all modules in the left-plane fixed, project to thecenter line • Make modules to the right of cut-line movables • Perform global placement in the right-plane • Proceed with horizontal cuts on each half • Continue until each block contains one and onlymodule

9. BGS Iteration • For a given hierarchy perform the previous partitioning more than once • Assume a vertical cut • y1 splits into y1a, y1b for each half • y1a* and y1b* are perturbed solution from y1a and y1b because of the partitioning process • 2-5 iterations give very good results at each hierarchy

10. Problems • A bad decision at a higher level affects the placement results of lower levels • Difficult to overcome a bad partitioning of a higher level • Backtracking?

11. KraftWerk Hans Eisenmann and F. Johannes, “Generic Global Placement and Floorplanning”

12. Force Directed Approach • Transform the placement problem to the classical mechanics problem of a system of objects attached to springs. • Analogies: • Module (Block/Cell/Gate) = Object • Net = Spring • Net weight = Spring constant. • Optimal placement = Equilibrium configuration

13. An Example Resultant Force

14. Force Calculation • Hooke’s Law: • Force = Spring Constant x Distance • Can consider forces in x- and y-direction separately: (xj, yj) F Fx (xi, yi) Fy

15. Problem Formulation • Equilibrium: Sj cij (xj - xi) = 0 for all module i. • However, trivial solution: xj = xi for all i, j. Everything placed on the same position! • Need to have some way to avoid overlapping. • Have connections to fixed I/O pins on the boundary of the placement region. • Push cells away from dense region to sparse region

16. Kraftwerk Approach • Iteratively solve the quadratic formulation: • Spread cells by additional forces: • Density-based force proposed • Push cells away from dense region to sparse region • The force exerted on a cell by another cell is repelling and proportional to the inverse of their distance. • The force exerted on a cell by a place is attracting and proportional to the inverse of their distance // equivalent to spring force // equilibrium

17. Some Details • Let fi be the additional force applied to cell i • The proportional constant k is chosen so that the maximum of all fiis the same as the force of a net with length K(W+H) • K is a user-defined parameter • K=0.2 for standard operation • K=1.0 for fast operation • Can be extended to handle timing, mixed block placement and floorplanning, congestion, heat-driven placement, incremental changes, etc.

18. Some Potential Problems of Kraftwerk • Convergence is difficult to control • Large K  oscillation • Small K  slow convergence • Example: Layout of a multiplier • Density-based force is expensive • to compute

19. FastPlace Natarajan Viswanathan and Chris Chu, “FastPlace: Efficient Analytical Placement using Cell Shifting, Iterative Local Refinement and a Hybrid Net Model”

20. FastPlace Approach • Framework: repeat Solve the convex quadratic program  Reduce wirelength by iterative heuristic  Spread the cells  until the cells are evenly distributed  • Special features of FastPlace: • Cell Shifting • Easy-to-compute technique  • Enable fast convergence • Hybrid Net Model • Speed up solving of convex QP  • Iterative Local Refinement • Minimize wirelength based on linear objective

21. Cell Shifting • Shifting of bin boundary • Apply to all rows and all columns independently Uniform Bin Structure Non-uniform Bin Structure • Shifting of cells linearly within each bin

22. Bin i Bin i Bin i+1 Bin i+1 k Ui j Ui+1 l OBi-1 OBi OBi+1 OBi-1 OBi+1 OBi k j l NBi Cell Shifting – Animation … NBi

23. Pseudo pin and Pseudo net Pseudo pin • Need to add forces to prevent cells from collapsing back • Done by adding pseudo pins and pseudo nets • Only diagonal and linear terms of the quadratic system need to be updated • Takes a single pass of O(n) time to regenerate matrix Q (which is common for both x and y problems) Pseudo net Pseudo pin Additional Force Pseudo net Target Position Original Position

24. V H H V Iterative Local Refinement • Iteratively go through all the cells one by one • For each cell, consider moving it in four directions by a certain distance • Compute a score for each direction based on • Half-perimeter wirelength (HPWL) reduction • Cell density at the source and destination regions • Move in the direction with highest positive score (Do not move if no positive score) • Distance moved (H or V) is decreasing over iterations • Detailed placement is handled by the same heuristic

25. Clique, Star and Hybrid Net Models • Runtime is proportional to # of non-zero entries in Q • Each non-zero entry in Q corresponds to one 2-pin net • Traditionally, placers model each multi-pin net by a clique • Hybrid Net Model is a mix of Clique and Star Models Star Node Hybrid Model Clique Model Star Model

26. Equivalence of Clique and Star Models • Lemma: By setting the net weights appropriately, clique and star net models are equivalent. • Proof: When star node is at equilibrium position, total forces on each cell are the same for clique and star net models. Star Node Weight = γW Weight = γ kW for a k-pin net Clique Model Star Model

27. Comparison • FastPlace is fast compared to Kraftwerk (based on published data) • 20-25x faster • With cell shifting technique can converge in around 20 iterations. • KraftWerk may need hundreds of iterations to converge. • 10% better in wirelength • hybrid net model