Advertisement

The T-join Problem in Sparse Graphs: Applications to Phase Assignment Problem in VLSI Mask Layout

  • Piotr Berman
  • Andrew B. Kahng
  • Devendra Vidhani
  • Alexander Zelikovsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)

Abstract

Given a graph G with weighted edges, and a subset of nodes T, the T-join problem asks for a minimum weight edge set A such that a node u is incident to an odd number of edges of A iff uT. We describe the applications of the T-join problem in sparse graphs to the phase assignment problem in VLSI mask layout and to conformal refinement of finite element meshes. We suggest a practical algorithm for the Tjoin problem. In sparse graphs, this algorithm is faster than previously known methods. Computational experience with industrial VLSI layout benchmarks shows the advantages of the new algorithm.

Keywords

Planar Graph Perfect Match Dual Graph Sparse Graph Conflict Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. V. Aho, J. E. Hopcroft and J. D. Ulman, The Design and Analysis of Computer Algorithms, Addison Wesley, Reading, MA, 1974.zbMATHGoogle Scholar
  2. 2.
    F. Barahona, “Planar multicommodity flows, max cut and the Chinese postman problem”, In W. Cook and P. D. Seymour, eds., Polyhedral Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1 (1990), pp. 189–202.Google Scholar
  3. 3.
    P. Berman, A. B. Kahng, D. Vidhani, H. Wang and A. Zelikovsky, “Optimal Phase Conflict Removal for Layout of Dark Field Alternating Phase Shifting Masks”, Proc. ACM/IEEE Intl. Symp. on Physical Design, 1999, to appear.Google Scholar
  4. 4.
    W. J. Cook, W. H. Cunningham, W. R. Pulleyblank and A. Shrijver, Combinatorial Optimization, Willey Inter-Science, New York, 1998.zbMATHGoogle Scholar
  5. 5.
    W. Cook and A. Rohe, “Computing Minimum-Weight Perfect Matchings”, http://www.or.uni-bonn.de/home/rohe/matching.html, manuscript, August, 1998.
  6. 6.
    H. N. Gabow and R. E. Tarjan, “Faster scaling algorithms for general graph matching problems”, Journal of the ACM 38 (1991) 815–853.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M. X. Goemans and D. P. Williamson, “A general approximation technique for constrained forest problems”, SIAM Journal on Computing 24 (1995) 296–317.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    F. O. Hadlock, “Finding a Maximum Cut of a Planar Graph in Polynomial Time”, SIAM J. Computing 4(3) (1975), pp. 221–225.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    A. B. Kahng and H. Wang, “Toward Lithography-Aware Layout: Preliminary Litho Notes”, manuscript, July 1997.Google Scholar
  10. 10.
    A. B. Kahng, H. Wang and A. Zelikovsky, “Automated Layout and Phase Assignment Techniques for Dark Field Alternating PSM”, SPIE 11th Annual BACUS Symposium on Photomask Technology, SPIE 1604 (1998), pp. 222–231.Google Scholar
  11. 11.
    M. D. Levenson, N. S. Viswanathan and R. A. Simpson, “Improving Resolution in Photolithography with a Phase-Shifting Mask”, IEEE Trans. on Electron Devices ED-29(11) (1982), pp. 1828–1836.CrossRefGoogle Scholar
  12. 12.
    R. J. Lipton and R. E. Tarjan, “A separator theorem for planar graphs”, SIAM J. Appl. Math., 36 (1979), pp. 177–189.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A. Moniwa, T. Terasawa, K. Nakajo, J. Sakemi and S. Okazaki, “Heuristic Method for Phase-Conict Minimization in Automatic Phase-Shift Mask Design”, Jpn. J. Appl. Phys. 34 (1995), pp. 6584–6589.CrossRefGoogle Scholar
  14. 14.
    M. Müller-Hannemann and K. Weihe, “Improved Approximations for Minimum Cardinality Quadrangulations of Finite Element Meshes”, Proc. ESA’97, Graz, Austria, pp. 364–377.Google Scholar
  15. 15.
    K. Ooi, K. Koyama and M. Kiryu, “Method of Designing Phase-Shifting Masks Utilizing a Compactor”, Jpn. J. Appl. Phys. 33 (1994), pp. 6774–6778.CrossRefGoogle Scholar
  16. 16.
    G. I. Orlova and Y. G. Dorfman, “Finding the Maximum Cut in a Graph”, Engr. Cybernetics 10 (1972), pp. 502–506.MathSciNetzbMATHGoogle Scholar
  17. 17.
    SIA, The National Technology Roadmap for Semiconductors, Semiconductor Industry Association, December 1997.Google Scholar
  18. 18.
    D. P. Williamson and M. X. Goemans, “Computational experience with an approximation algorithm on large-scale Euclidean matching instances”, INFORMS Journal of Computing, 8 (1996) 29–40.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Piotr Berman
    • 1
  • Andrew B. Kahng
    • 2
  • Devendra Vidhani
    • 2
  • Alexander Zelikovsky
    • 3
  1. 1.Dept. of Computer Science and EngineeringPennsylvania State UniversityUniversity Park
  2. 2.Department of Computer ScienceUniversity of California at Los AngelesLos Angeles
  3. 3.Department of Computer ScienceGeorgia State University, University PlazaAtlanta

Personalised recommendations