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Efficient Neighborhood Encoding for Interval Graphs and Permutation Graphs and O(n) Breadth-First Search

  • Christophe Crespelle
  • Philippe Gambette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5874)

Abstract

In this paper we address the problem of designing O(n) space representations for permutation and interval graphs that provide the neighborhood of any vertex in O(d) time, where d is its degree. To that purpose, we introduce a new parameter, called linearity, that would solve the problem if bounded for the two classes. Surprisingly, we show that it is not. Nevertheless, we design representations with the desired property for the two classes, and we implement the Breadth-First Search algorithm in O(n) time for permutation graphs; thereby lowering the complexity of All Pairs Shortest Paths and Single Source Shortest Path problems for the class.

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References

  1. 1.
    Boldi, P., Vigna, S.: The webgraph framework I: compression techniques. In: WWW 2004, pp. 595–602. ACM, New York (2004)CrossRefGoogle Scholar
  2. 2.
    Boldi, P., Vigna, S.: Codes for the world wide web. Internet Mathematics 2(4), 407–429 (2005)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  4. 4.
    Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: STOC 1984, pp. 135–143 (1984)Google Scholar
  5. 5.
    Gavoille, C., Peleg, D.: The compactness of interval routing. SIAM Journal on Discrete Mathematics 12(4), 459–473 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goldberg, P.W., Golumbic, M.C., Kaplan, H., Shamir, R.: Four strikes against physical mapping of DNA. Journal of Computational Biology 2(1), 139–152 (1995)CrossRefGoogle Scholar
  7. 7.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, 2nd edn., vol. 57. Elsevier, Amsterdam (2004)Google Scholar
  8. 8.
    Harel, D.: A linear time algorithm for the lowest common ancestors problem (extended abstract). In: FOCS 1980, pp. 308–319 (1980)Google Scholar
  9. 9.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Roberts, F.S.: Representations of Indifference Relations. PhD thesis, Stanford University (1968)Google Scholar
  11. 11.
    Spinrad, J.P.: Efficient graph representations. Fields Institute Monographs, vol. 19. American Mathematical Society (2003)Google Scholar
  12. 12.
    Turan, G.: On the succinct representation of graphs. Discr. Appl. Math. 8, 289–294 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vuillemin, J.: A unifying look at data structures. Commun. ACM 23(4), 229–239 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Wang, R., Lau, F.C.M., Zhao, Y.: Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices. Discr. Appl. Math. 155(17), 2312–2320 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christophe Crespelle
    • 1
  • Philippe Gambette
    • 2
  1. 1.CNRS - Univ. Paris 6 
  2. 2.LIRMMUniv. Montpellier 2 - CNRS 

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