1. Advanced Interconnect Optimizations

2. Timing Driven Buffering Problem Formulation Given –A Steiner tree –RAT at each sink –A buffer type –RC parameters –Candidate buffer locations Find buffer insertion solution such that the slack at the driver is maximized

3. Candidate Buffering Solutions

4. Candidate Solution Characteristics Each candidate solution is associated with –v i : a node –c i : downstream capacitance –q i : RAT v i is a sink c i is sink capacitance v is an internal node

5. Van Ginneken’s Algorithm Candidate solutions are propagated toward the source Dynamic Programming

6. Solution Propagation: Add Wire c 2 = c 1 + cx q 2 = q 1 – rcx 2 /2 – rxc 1 r: wire resistance per unit length c: wire capacitance per unit length (v 1, c 1, q 1 ) (v 2, c 2, q 2 ) x

7. 7 Solution Propagation: Insert Buffer c 1b = C b q 1b = q 1 – R b c 1 C b : buffer input capacitance R b : buffer output resistance (v 1, c 1, q 1 ) (v 1, c 1b, q 1b )

8. Solution Propagation: Merge c merge = c l + c r q merge = min(q l, q r ) (v, c l, q l )(v, c r, q r )

9. Solution Propagation: Add Driver q 0d = q 0 – R d c 0 = slack min R d : driver resistance Pick solution with max slack min (v 0, c 0, q 0 ) (v 0, c 0d, q 0d )

10. 10 Example of Merging Left candidates Right candidates Merged candidates

11. Solution Pruning Two candidate solutions –(v, c 1, q 1 ) –(v, c 2, q 2 ) Solution 1 is inferior if –c 1 > c 2 : larger load –and q 1 < q 2 : tighter timing

12. Pruning When Insert Buffer They have the same load cap C b, only the one with max q is kept

13. 13 Generating Candidates (1) (2) (3) From Dr. Charles Alpert

14. 14 Pruning Candidates (3) (a) (b) Both (a) and (b) “look” the same to the source. Throw out the one with the worst slack (4)

15. 15 Candidate Example Continued (4) (5)

16. 16 Candidate Example Continued After pruning (5) At driver, compute which candidate maximizes slack. Result is optimal.

17. 17 Merging Branches Right Candidates Left Candidates

18. 18 Pruning Merged Branches Critical With pruning

19. Basic Data Structure (c 1, q 1 )(c 2, q 2 )(c 3, q 3 ) Sorted list such that c 1 < c 2 < c 3 If there is no inferior candidates q 1 < q 2 < q 3 Worse load cap Better timing

20. 20 Prune Solution List (c 1, q 1 )(c 2, q 2 )(c 3, q 3 ) Increasing c q 1 < q 2 ? (c 4, q 4 ) q 3 < q 4 ? Y N Prune 2 q 1 < q 3 ? q 2 < q 3 ? Y q 3 < q 4 ? Y Prune 3 q 1 < q 4 ? N Prune 3 N N Prune 4 N q 2 < q 4 ?

21. 21 Pruning In Merging (c l1, q l1 ) (c l2, q l2 ) (c l3, q l3 ) (c r1, q r1 ) (c r2, q r2 ) q l1 < q l2 < q r1 < q l3 < q r2 Merged candidates (c l1 +c r1, q l1 ) (c l2 +c r1, q l2 ) (c l3 +c r1, q r1 ) (c l3 +c r2, q l3 ) (c l1, q l1 ) (c l2, q l2 ) (c l3, q l3 ) (c r1, q r1 ) (c r2, q r2 ) (c l1, q l1 ) (c l2, q l2 ) (c l3, q l3 ) (c r1, q r1 ) (c r2, q r2 ) (c l1, q l1 ) (c l2, q l2 ) (c l3, q l3 ) (c r1, q r1 ) (c r2, q r2 ) Left candidates Right candidates

22. Van Ginneken Complexity Generate candidates from sinks to source Quadratic runtime –Adding a wire does not change #candidates –Adding a buffer adds only one new candidate –Merging branches additive, not multiplicative –Linear time solution list pruning