A Theoretical Study on Wire Length Estimation Algorithms for Placement with Opaque Blocks

1. A Theoretical Study on Wire Length Estimation Algorithms forPlacement with Opaque Blocks Tan Yan*, Shuting Li Yasuhiro Takashima, Hiroshi Murata The University of Kitakyushu * Now with University of Illinois at Urbana-Champaign

2. Motivation • “Opaque” blocks makes HPWL inexact • Because of IP blocks, analog blocks, memory module… • Lead to timing violation, unroutable nets…

3. Motivation—cont’d • Exact wire length estimation for Block Placement • the obstacle-avoiding shortest path length • Time complexity: O(n)? O(n2)? O(nlogn)?... • Time complexity is almost the same as HPWL! • Already proposed in Computational Geometry • However • Not well-known in CAD community • Need interpretation to be applicable to CAD!

4. Our Contribution • We restate the results in [P.J.de Rezende ’85] & [M.J.Atallah ’91] • Simplify the discussion (with Block Placement notions) • CAD-oriented language • Tailor the theory to fit into Physical Design background

5. Problem Formulation • Input: • Block location • Pin location (on block boundaries) • ABLR relations * (obtainable from Sequence Pair, etc) • Output: • Rectilinear block-avoiding shortest path length for every 2-pin net • = Minimal Wire Length (MWL)

6. Assumption • 2-pin net • s on S, t on T • S≠ T • S is left-to T • ys ≤ yt t S T s

7. Locus

8. Theorem 1 • MWL = HPWL ↔ RU locus of s goes below or through t • Proof omitted

9. AB-region

10. Lemma 2 • There exists an MWL routing inside the AB-region

11. Horizontal Visibility Graph (HVG)

12. MWL = shortest path length • Lemma 4: There exists a path (s,t) on the visibility graph that corresponds to an MWL routing. • Only linear number of edges, but still captures MWL!

13. Visibility graph of a placement

14. The overall flow and so on …

15. Time complexity • M = # of blocks, N = # of nets • Building visibility graph: • O(M logM) • Estimating one net: • O(M) • Total: • O(M logM + NM) • Shortest path on channel graph takes O(NM2)

16. Use LUT to enhance the speed • No path between two vertices? (a2b2) • Need to judge whether RU locus above t ? • How to find out A & B promptly?

17. Two lemmas: • Lemma 5: Two vertices s and t on visibility graph. If there is no path between them, then MWL = HPWL • Lemma 6: If t is above s’s RU locus and there exists a shortest path between them, then its length = HPWL. • MWL(a,b) = HPWL • ShortestPath(c,d) = MWL (c,d) = HPWL

18. Theorem 3 • The MWL of any two vertices on the visibility graph can be obtained by shortest path algorithm: • Shortest path exists, MWL = path length • Otherwise, MWL = HPWL

19. How it works Lookup table MWL = shortest path length No path! MWL = HPWL And so on…

20. Time complexity • Building LUT: • O(M2) • Estimating one net: • O(1) • Total: • O(M2+N) • Almost the same as HPWL!

21. Future works • Integration of routing congestion • Extension to handle multi-pin nets • Application to global router • Experimental study

22. Thank you! Q & A

23. Proof of Theorem 1 • MWL = HPWL ↔ RU locus of s goes below or through t

24. Proof of Lemma 2 • There exists an MWL routing completely inside AB-region

25. Proof of Lemma 4 • There exists a path p from s to t on HVG that corresponds to an MWL routing.

26. Proof of Lemma 6 • If t is above s’s RU locus and there exists a shortest path between them, then its length = HPWL.