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An Exact Algorithm for the Uniformly-Oriented Steiner Tree Problem

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Algorithms — ESA 2002 (ESA 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2461))

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Abstract

An exact algorithm to solve the Steiner tree problem for uniform orientation metrics in the plane is presented. The algorithm is based on the two-phase model, consisting of full Steiner tree (FST) generation and concatenation, which has proven to be very successfulf or the rectilinear and Euclidean Steiner tree problems. By applying a powerful canonicalf orm for the FSTs, the set of optimal solutions is reduced considerably. Computational results both for randomly generated problem instances and VLSI design instances are provided. The new algorithm solves most problem instances with 100 terminals in seconds, and problem instances with up to 10000 terminals have been solved to optimality.

It has not been proved that the problem is NP-hard for every value of λ, but we strongly conjecture this to be the case.

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References

  1. J.E. Beasley. OR-Library: Distributing Test Problems by Electronic Mail. Journal of the Operational Research Society, 41:1069–1072, 1990.

    Article  Google Scholar 

  2. M. Brazil. Steiner Minimum Trees in Uniform Orientation Metrics. In D.-Z. Du and X. Cheng, editor, Steiner Trees in Industries, pages 1–27. Kluwer Academic Publishers, 2001.

    Google Scholar 

  3. M. Brazil, D.A. Thomas, and J. F. Weng. Minimum Networks in Uniform Orientation Metrics. SIAM Journal on Computing, 30:1579–1593, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Brazil, D.A. Thomas, and J. F. Weng. Forbidden Subpaths for Steiner Minimum Networks in Uniform Orientation Metrics. Networks, 39:186–202, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Brazil, D.A. Thomas, J. F. Weng, and M. Zachariasen. Canonical Forms and Algorithms for Steiner Trees in Uniform Orientation Metrics. In preparation.

    Google Scholar 

  6. M.R. Garey and D. S. Johnson. The Rectilinear Steiner Tree Problem is NPComplete. SIAM Journal on Applied Mathematics, 32(4):826–834, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  7. F.K. Hwang, D.S. Richards, and P. Winter. The Steiner Tree Problem. Annals of Discrete Mathematics 53. Elsevier Science Publishers, Netherlands, 1992.

    Google Scholar 

  8. Y.Y. Li, S. K. Cheung, K. S. Leung, and C. K. Wong. Steiner Tree Constructions in λ3-Metric. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 45(5):563–574, 1998.

    Article  Google Scholar 

  9. Y.Y. Li, K. S. Leung, and C. K. Wong. Efficient Heuristics for Orientation Metric and Euclidean Steiner Tree Problems. Journal of Combinatorial Optimization, 4:79–98, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  10. K. Mehlhorn and S. Näher. LEDA-A Platform for Combinatorial and Geometric Computing. Max-Planck-Institut für Informatik, Saarbrücken, Germany, http://www.mpi-sb.mpg.de/LEDA/leda.html, 1996.

  11. B.K. Nielsen, P. Winter, and M. Zachariasen. On the Location of Steiner Points in Uniformly-Oriented Steiner Trees. Information Processing Letters, 83:237–241, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Sarrafzadeh and C.K. Wong. HierarchicalS teiner Tree Construction in Uniform Orientations. IEEE Transactions on Computer-Aided Design, 11:1095–1103, 1992.

    Article  Google Scholar 

  13. D. M. Warme, P. Winter, and M. Zachariasen. Exact Algorithms for Plane Steiner Tree Problems: A Computational Study. In D.-Z. Du, J.M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 81–116. Kluwer Academic Publishers, Boston, 2000.

    Google Scholar 

  14. D. M. Warme, P. Winter, and M. Zachariasen. GeoSteiner 3.1. Department of Computer Science, University of Copenhagen (DIKU), http://www.diku.dk/geosteiner/, 2001.

  15. P. Winter and M. Zachariasen. Euclidean Steiner Minimum Trees: An Improved Exact Algorithm. Networks, 30:149–166, 1997.

    Google Scholar 

  16. M. Zachariasen. Rectilinear Full Steiner Tree Generation. Networks, 33:125–143, 1999.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen, DK-2100, Copenhagen Ø, Denmark

    Benny K. Nielsen, Pawel Winter & Martin Zachariasen

Authors
  1. Benny K. Nielsen
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  2. Pawel Winter
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  3. Martin Zachariasen
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Editor information

Editors and Affiliations

  1. Fakultät II: Mathematik und Naturwissenschaften, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Rolf Möhring

  2. Department of Mathematics and Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Rajeev Raman

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© 2002 Springer-Verlag Berlin Heidelberg

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Nielsen, B.K., Winter, P., Zachariasen, M. (2002). An Exact Algorithm for the Uniformly-Oriented Steiner Tree Problem. In: Möhring, R., Raman, R. (eds) Algorithms — ESA 2002. ESA 2002. Lecture Notes in Computer Science, vol 2461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45749-6_66

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  • DOI: https://doi.org/10.1007/3-540-45749-6_66

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44180-9

  • Online ISBN: 978-3-540-45749-7

  • eBook Packages: Springer Book Archive

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