A.B. Kahng, Ion I. Mandoiu University of California at San Diego, USA A.Z. Zelikovsky

1. Highly Scalable Algorithms for Rectilinear and Octilinear Steiner Trees A.B. Kahng, Ion I. Mandoiu University of California at San Diego, USA A.Z. Zelikovsky Georgia State University, USA Supported in part by MARCO GSRC and Cadence Design Systems, Inc.

2. Outline • Single net routing problem • Problem definition • Previous work • Motivation for highly scalable heuristics • The batched greedy algorithm • High-level algorithm • Efficient generation of triples • Efficient bottleneck-edge computation • Experimental results and conclusions • Single net routing problem • Problem definition • Previous work • Motivation for highly scalable heuristics • The batched greedy algorithm • High-level algorithm • Efficient generation of triples • Efficient bottleneck-edge computation • Experimental results and conclusions

3. Steiner points Rectilinear (Manhattan) routing Octilinear (X) routing Single Net Routing Problem Given: set of terminals in the plane Find: minimum length interconnection  Rectilinear/Octilinear Minimum Steiner Trees (RMST/OMST)

4. Why Minimum Steiner Tree Routing? • Advantages • Minimum routing area • Minimum total capacitance • Reduced power consumption • Steiner tree routing appropriate for • Non-critical nets • Physically small instances

5. Previous Work on Steiner Tree Problem • Long history • Euclidean version [Gauss 1836] • Rectilinear version [Hanan 1966] • Octilinear version [Sarrafzadeh&Wong 1992] • Steiner tree problem in graphs [Hakimi/Levin 1971] • Fundamental results • All versions are NP-hard [Karp 1972, GJ77] • Minimum Spanning Tree (MST) gives good approximation • Always within factor 2 of optimum [3 papers, 1979-1981] • Within factor 1.5 in rectilinear plane [Hwang 1976]

6. Previous RSMT/OSMT Algorithms • Exact algorithms • GeoSteiner [Warme,Winter&Zachariasen] • Approximation algorithms • Zelikovsky 1993, Berman&Ramaier 1994, Hougardy&Promel 1999, Rajagopalan&Vazirani 1999, Robins&Zelikovsky 2000, … • Best-performing RSMT heuristics (within ~0.5% of optimum) • Iterated 1-Steiner heuristic [Kahng&Robins 1992] • Edge-based heuristic [Borah,Owens&Irwin 1999] • Iterated Rajagopalan-Vazirani [Mandoiu,Vazirani&Ganley 2000] • Not practical for tens of thousands of terminals!

7. Motivation for Highly Scalable Heuristics • Most nets are small (<20 terminals)… • But nets with >104 terminals become increasingly common • Scan-enable nets in large designs using full-scan test • All flip-flops need to receive the scan-enable signal • Nets with pre-routes and non-zero terminal dimensions • Each terminal represented by set of electrically equivalent pins RSMT/OSMT instances with up to 105 pins

8. Requirements for Highly Scalable RSMT/OSMT Heuristics • Linear memory • Sub-quadratic runtime • Solutions within ~0.5% of optimum • Previous Steiner tree heuristics do not meet first two requirements

9. Outline • Single net routing problem • Problem definition • Previous work • Motivation for highly scalable heuristics • The batched greedy algorithm • High-level algorithm • Efficient generation of triples • Efficient bottleneck-edge computation • Experimental results and conclusions

10. GAIN = - Triple Contraction • Connect 3 terminals (=triple) using shortest connection • Remove longest edge on each of the 2 formed cycles

11. High-level Algorithm Greedy Triple-Contraction Algorithm[Zelikovsky 1993]: • Compute MST of terminals • While there exist triples with positive gain, do: • Find triple with maximum gain • Contract triple: remove longest edges, replace triple with 2 zero-cost edges • Output MST of terminals and centers of contracted triples • Expensive to compute max-gain triple in Step 2 • Best implementation uses complex dynamic MST datastructures • We use a batched implementation • Find positive-gain triples • Contract triples in descending order of gain without recomputing gains • A triple is selected if 2 longest edges not used by previous triples

12. Efficient Generation of Triples • O(n3) triples overall • Use geometry to avoid generating all of them! • [Fossmeier,Kaufmann&Zelikovsky 1997]:sufficient to consider a set of O(n) triples with • Empty interior ( no other terminal in bounding box) • Positive gain • We use a practical O(nlogn) divide-and-conquer algorithm to compute a superset of size O(n logn) • Some triples may have negative gain • Eliminated after gain computation • Some triples may be non-empty • Can be removed, but too few to justify the extra testing time

13. Divide-and-conquer for Empty Triples

14. Divide-and-conquer for Empty Triples

15. Divide-and-conquer for Empty Triples

16. Divide-and-conquer for Empty Triples

17. Efficient Bottleneck-edge Computation Bottleneck edges needed for computing triple gains • We give a new data structure • O(logn) time per bottleneck-edge query • O(n logn) pre-processing (O(n) if edges already sorted) • Much more practical than methods based on least-common-ancestors Gain evaluation in O(logn) time per triple  O(n log2n) total time for the batched greedy algorithm Given: tree T, vertices u,v Find: most expensive edge on path between u and v

18. Outline • Single net routing problem • Problem definition • Previous work • Motivation for highly scalable heuristics • The batched greedy algorithm • High-level algorithm • Efficient generation of triples • Efficient bottleneck-edge computation • Experimental results and conclusions

19. Experimental Setup • Testcases • Random nets with 100 to 500,000 terminals • 100 samples for each size • Nets extracted from recent designs (330 to 34,000 terminals) • Compared algorithms • Batched greedy O(n log2n) • MST [Guibas&Stolfi 1983] O(n logn) • Prim-based heuristic [Rohe 2001] O(nlog2n) • Edge-based heuristic of [Borah,Owens&Irwin 1999] O(n2) • GeoSteiner 4.0, beta version [Nielsen,Winter&Zachariasen 2002]

20. Quality: Random Rectilinear Tests • BatchGreedy quality slightly better than Edge-based, 1% better than Prim-based • Within 0.7% of optimum computed by GeoSteiner

21. CPU Time: Random Rectilinear Tests • BatchGreedy highly scalable, practical runtime up to 105 terminals • Edge-Based impractical for >104 terminals

22. CPU Time: Rectilinear Industry Tests • 34k terminals: 24 seconds BatchGreedy vs. 86 minutes Edge-based

23. Quality: Rectilinear Industry Tests • BatchGreedy up to 1% better than Prim-Based, within 0.5% of GeoSteiner • Slightly better than Edge-Based in half of the cases

24. CPU Time: Octilinear Industry Tests • 34k terminals: 25 seconds BatchGreedy vs. 17.5 hours Edge-based • Octilinear Prim-Based not available

25. Quality: Octilinear Industry Testcases • BatchGreedy slightly better than Edge-Based, within 0.5% of GeoSteiner

26. Conclusions • Presented new rectilinear/octilinear Steiner tree heuristic • Highly-scalable • O(n) memory, O(nlog2n) runtime, with small hidden constants • High-quality • Better quality than O(n2) edge-based heuristic • Within 0.7% of optimum computed by GeoSteiner • Ongoing extensions • Via costs • Preferred directions • Routing obstacles

27. Thank You for Your Attention! Further details on our work are available on the group’s website, http://vlsicad.ucsd.edu