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European Symposium on Algorithms

ESA 2002: Algorithms — ESA 2002 pp 550–562Cite as

Efficient Implementation of a Minimal Triangulation Algorithm

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

Abstract

LB-triang, an algorithm for computing minimal triangulations of graphs, was presented by Berry in 1999 [1], and it gave a new characterization of minimal triangulations. The time complexity was conjectured to be O(nm), but this has remained unproven until our result. In this paper we present and prove an O(nm) time implementation of LB-triang, and we call the resulting algorithm LB-treedec. The data structure used to achieve this time bound is tree decomposition. We also report from practical runtime tests on randomly generated graphs which indicate that the expected behavior is even better than the proven bound.

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References

  1. A. Berry. A wide-range efficient algorithm for minimal triangulation. In Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999.

    Google Scholar 

  2. A. Berry, J-P. Bordat, and P. Heggernes. Recognizing weakly triangulated graphs by edge separability. Nordic Journal of Computing, 7:164–177, 2000.

    MATH  MathSciNet  Google Scholar 

  3. J.R. S. Blair, P. Heggernes, and J.A. Telle. Making an arbitrary filled graph minimal by removing fill edges. In R. Karlsson and A. Lingas, editors, Algorithm Theory-SWAT’ 96, pages 173–184. Springer Verlag, 1996. Lecture Notes in Computer Science 1097.

    Google Scholar 

  4. J.R. S. Blair, P. Heggernes, and J.A. Telle. A practical algorithm for making filled graphs minimal. Theoretical Computer Science, 250:125–141, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.R. S. Blair and B.W. Peyton. An introduction to chordal graphs and clique trees. In J.A. George, J.R. Gilbert, and J.W.H. Liu, editors, Graph Theory and Sparse Matrix Computations, pages 1–30. Springer Verlag, 1993. IMA Volumes in Mathematics and its Applications, Vol. 56.

    Google Scholar 

  6. H.L. Bodlaender. A tourist guide through treewidth. Acta Cybernetica, 11:1–21, 1993.

    MATH  MathSciNet  Google Scholar 

  7. E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In R. H. Möhring, editor, Graph Theoretical Concepts in Computer Science-WG’ 97, pages 132–143. Springer Verlag, 1997. Lecture Notes in Computer Science 1335.

    Chapter  Google Scholar 

  8. F. Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Combin. Theory Ser. B, 16:47–56, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  9. C-W. Ho and R. C. T. Lee. Counting clique trees and computing perfect elimination schemes in parallel. Inform. Process. Lett., 31:61–68, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Kloks, D. Kratsch, and J. Spinrad. On treewidth and minimum fill-in of asteroidal triple-free graphs. Theoretical Computer Science, 175:309–335, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  11. C. G. Lekkerkerker and J.C. Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51:45–64, 1962.

    MATH  MathSciNet  Google Scholar 

  12. T. Ohtsuki. A fast algorithm for finding an optimal ordering in the vertex elimination on a graph. SIAM J. Comput., 5:133–145, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Parra. Structural and algorithmic aspects of chordal graph embeddings. PhD thesis, Technische Universität Berlin, 1996.

    Google Scholar 

  14. B.W. Peyton. Minimal orderings revisited. SIAM J. Matrix Anal. Appl., 23(1):271–294, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. J. Rose, R.E. Tarjan, and G. S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput., 5:266–283, 1976.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Informatics, University of Bergen, N-5020, Bergen, Norway

    Pinar Heggernes & Yngve Villanger

Authors
  1. Pinar Heggernes
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  2. Yngve Villanger
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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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Heggernes, P., Villanger, Y. (2002). Efficient Implementation of a Minimal Triangulation Algorithm. 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_49

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  • DOI: https://doi.org/10.1007/3-540-45749-6_49

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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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