The Associative-Skew Clock Routing Problem

1. The Associative-Skew Clock Routing Problem In memory of Mr. Patrick Catapano, Jr. of Motorola Corporation and his contributions to modern chip implementation methodology Yu Chen (UCLA) Andrew B. Kahng (UCLA) Gang Qu (UCLA) Alexander Zelikovsky (Georgia State) Supported in part by grants from Cadence Design Systems, Inc. and the MARCO (SRC/DARPA) Gigascale Silicon Research Center, and by a GSU research initiation grant.

2. Introduction • Zero-Skew clock routing • Associative-Skew Problem • Example: potential gain • 3 composing methods: • optimal RSMT • optimal joining • optimal merging • improves over GDME • Testbed: random sink sets • Experiments • Conclusion

3. Model • Set of sinks S = s1, …, sn and source s0 • Sink delay = t(s0, si) • Skew between two sinks skew (si , sj) = |t(s0 , si )- t(s0 , sj )| • Skew of a tree T = maximum skew between sinks • Objective: minimize wirelength, such that skews between certain (all) pairs of sinks are small

4. Relevant Clock Routing • Zero skew over ALL sinks • Zero-Skew Tree (ZST) literature • Deferred-Merge Embedding (DME) • optimal with the given topology • Greedy-DME: finds high-quality topology • Almost zero skew over ALL sinks • Bounded-Skew Tree (BST) literature • solvable with DME-like approaches • Zero skew over SUBSETS of sinks • zero skew inside each subset • no skew constraints between subsets

5. Associative-Skew Problem • Given: Set of sinks Spartitioned into disjoint subsets S = S1 …  Sk • Construct: a Steiner tree connecting all sinks with a root S0 to achieve • zero skew within each subset Si • unconstrained skew between subsets • Real-world: bounded global skew • skew in subsets < 1 gate delay • latch-to-latch “0-level” datapaths • 0-level paths quite small • skew between subsets < 12 gate delays

6. Potential Wirelength Gain • Potential gain is logarithmic: • each subset has single sink, | Si | =1; all sinks on a segment • standard Steiner minimum tree may be log k times shorter than zero-skew tree 1

7. Potential Wirelength Gain • Potential gain is logarithmic: • each subset has single sink, | Si | =1; all sinks on a segment • standard Steiner minimum tree may be log k times shorter than zero-skew tree 1 2/7 4/7 1/7

8. Heuristic H0 • All our heuristics combine GDME solutions for each sink subset into single tree with common root • H0 = k-Greedy DME: • Greedy DME for each subset Si • join the roots with any rectilinear Steiner minimum tree heuristic • Can do better: connect root of B to any point in A H0 H1 A A B B

9. Heuristic H1 • H1 = H0 but connect root from any internal point • For each ordered pair of ZST’s A = ZST(Si), B = ZST(Sj), find optimal joining = shortest edge from a point in A to root(B) • Full directed graph G: nodes = ZST(Si), arcs = optimal joinings • Find optimal branching (directed MST) in G • O(k2) - Tarjan (1977), Camerini-Fratta-Maffioli (1979) Directed graph G with optimal branching

10. Better Merging • Potential gain of total wirelength of B A A B B • Given: Given two ZST’s A and B and insertion delay offset w = skew between sinks in A and B from root(A) • Find: min cost tree T(A,B,w) • rooted at root(A) • containing A • offset w between path delays of sinks in A and sinks in B

11. Optimal Slice Merging Algorithm • For each v in B • Find merge (v) = merging cost of v to A with offset w • Find adj(v) = adjustedcost of root(B)-v-path • gain(v) = adj(v)-merge(v) • Sum of gains over slice = gain of slice • Find optimal slice by preorder/postorder traversal r(A) A u merge(v) = cost of thick edge adj(v) = cost of dashed edges divided by 2L, L = edge level in B slice = roots of subtrees of B joined to points of A B u` r(B) v s2 s s1 T(A,B,w)

12. Wirelength Gain over GDME (-2,2) (2,2) (-2,2) (2,2) (1,1) (-1,1) (1,1) (-1,1) Root(0,0) Root(0,0) (1,-1) (-1,-1) (1,-1) (-1,-1) (-2,-2) (2,-2) (-2,-2) (2,-2) Optimal Slice Merging Greedy DME with offset Gain for this instance: 28%

13. Testbed: Random Sink Sets Class 1: “Shift” Class 2: “Ring”

14. Experimental Results (“Shift”) Experimental data for n-point sets randomly generated in K shifted square regions Time1 refers to H2 runtimes; Time2refer to GDME runtimes

15. Experimental Results (“Shift”) Runtime improvement since GDME is nonlinear

16. Experimental Results (“Shift”) H2 is losing to GDME with offsets

17. Experimental Results (“Ring”) Runtime improvement since GDME is nonlinear

18. Experimental Results (“Ring”) Average wirelength improvement 7%

19. Conclusions & Future Work • We introduced the Associative-Skew Problem • We suggested a series of heuristics • outperform GDME for separated sink subsets • GDME is good otherwise • Open issues: • find a consistently better heuristic • solve the ASP with limited-height (sub)trees