CSE 242A Integrated Circuit Layout Automation

1. CSE 242A Integrated Circuit Layout Automation Lecture: Partitioning Winter 2009 Chung-Kuan Cheng

2. Outlines • Motivations • Formulations • 2-way partitioning, multi-way partitioning, multilevel partitioning, replication cut, clustering. • Net Modeling • Algorithms • Optimal Methods: Special cases, branch and bound, • Heuristic Methods: group migration, network flow, clustering, simulated annealing, genetic approaches

3. Motivation • Huge designs • 100 Millions Trans • Design • Analysis • Engineering Change Orders • Good partitioning makes difference on design quality

4. Motivation: Applications • Physical Hierarchy • Divide and Conquer • Project decomposition • Complexity reduction of each level • Emulation • Hardware & Software codesign • Management design resource

5. Formulations: • Two Way Partitioning • Random two-way partition • N-pin net to be cut b a a a b a b b

6. Min-Cut s t Theorem: There exists an optimal linear placement s.t. X and are separated. Trend:

7. Min-Cut • Opt->Linear Placement • Trend: X=V S S

8. Bisection

9. Ratio Cut

10. Multi-Way Partitioning • K-Way Partitioning X1 X2 Xk

11. Cluster Ratio Cut

12. Multi-Level Partitioning • K-Level Partitioning • Li<= Xi<= Ui • Min Connection Cost • E(Ti) <= Ci • External connection cost <= threshold at level i

13. Generic Binary Tree • L <= Xi <=U • Level of Node = longest path to leaves • Connection cost of node i at level L

14. Replication Cut X Y R Y R X R C(X,Y)+C(Y,X)+C(Y,R)+C(X,R)

15. Performance-Driven Partitioning • Need an incremental timing analysis to reflect the performance fast • Replication helps

16. Retiming (Pipelined process) • Allocate one clock cycle for interpartition communication • System performance is dominated by Loops:

18. Clustering • K-Way Partitioning • K>>10 • Obj strongly depends on applications • Performance driven • Obj min max # cuts between registers • Complexity reduction Ei external connection Ii internal connection

19. Net Modeling • Shifting: For each shift, we update k, n-k. The cost of the net changes only when k = 0, 1 or n-k =0, 1 • Two pin net clique • 2/k total weight k-1 • 1/(k-1) S.K.

20. Net Modeling: Loop Model • Suppose relative positions of pins are given, we can use a loop model • The model remains correct if any two adjacent (in order) pins swap

21. Net Modeling: Hypernet Model (Flow Approach)

22. Optimal Methods: Branch & Bound Prune the branches when size constraint is violated Partial cost >= existing cost For U=L=|V|/2 # combinations = |V| ! / (|V/2)! (|V|/2)! With an elegant implementation |V| <= 60 is feasible

23. Optimal Methods: Serial & Parallel Graph • Dynamic Programming on Series-Parallel graph G(V, E, s, t) • C(a, i, j) a: s, t on different sides • C(b, i, j) b: s, t on the left side, left side has i nodes, right side has j nodes

24. Optimal Methods: Serial & Parallel Graph (Cont) • Dynamic Programming on Series-Parallel graph G(V, E, s, t) • C(a, i, j) a: s, t on different sides • C(b, i, j) b: s, t on the left side, left side has i nodes, right side has j nodes

25. Heuristic Methods • Group Migration • Kernighan & Lin • Fiducccia-Matheyses • Programming • Network Flow • Replication Cut • Clustering

26. Group MigrationKermighan & Lin Bisection • Cost Ci: change of #cuts by moving node i to the other side • Heapsort nodes in each partition according to Ci Repeat Repeat Choose among the top k the best pair to swap Update the cost, lock the moved nodes Until all nodes are locked Find the best sequence to swap Until no more improvement

28. Group Migration • Hill climbing to jump over local optimal solutions. • Locking mechanism to avoid repeated moves.

29. Fiduccia-Mattheyses • No swapping, move a single node each time • Replace the heap with an array

30. High Order Gains

31. Probabilistic Model

32. Move all nodes according to a single net Adv: move more nodes a the same operation. Good for multi-way, hierarchical where cost function dominated by the way to handle the nets Dis: Complicate

33. Simulated Annealing

34. Simulated Annealing s, nexts: configurations, T: real, count: integer • Begin • S= random initial configuration • T=T0 • Repeat • Count= 0 • Repeat • Count= count+1 • Nexts= generate(s) • If c(nexts)<= c(s) or f(c(s),c(nexts),T)> random(0,1) • Then s= nexts • Until equilibrium(count,s,T) • T= update(T) • Unitl frozen(T) • End

35. Programming

36. Improving the lower bound

37. Network Flow Approaches

39. Multiple Commodity Flow

41. B constraints can be used to replace A

42. Weighted Cluster Ratio Cut

43. Replication Cut

44. R R R S T S T S T

45. Replication Graph

46. Replication Graph

50. Heuristic Flow Approaches