Faster and Simpler Minimal Conflicting Set Identification

(Extended Abstract)
  • Aïda Ouangraoua
  • Mathieu Raffinot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7354)

Abstract

Let \({\cal C}\) be a finite set of n elements and \({\cal R}=\{r_1,r_2, \ldots , r_m\}\) a family of m subsets of \({\cal C}\). A subset \({\cal X}\) of \({\cal R}\) satisfies the Consecutive Ones Property (C1P) if there exists a permutation P of \({\cal C}\) such that each r i in \({\cal X}\) is an interval of P. A Minimal Conflicting Set (MCS) \({\cal S} \subseteq{\cal R}\) is a subset of \({\cal R}\) that does not satisfy the C1P, but such that any of its proper subsets does. In this paper, we present a new simpler and faster algorithm to decide if a given element \(r \in{\cal R}\) belongs to at least one MCS. Our algorithm runs in O(n 2 m 2 + nm 7), largely improving the current O(m 6 n 5 (m + n)2 log(m + n)) fastest algorithm of [Blin et al, CSR 2011]. The new algorithm is based on an alternative approach considering minimal forbidden induced subgraphs of interval graphs instead of Tucker matrices.

Keywords

Fast Algorithm Form Versus Vertex Versus Interval Graph Internal Vertex 
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.

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References

  1. 1.
    Bergeron, A., Blanchette, M., Chateau, A., Chauve, C.: Reconstructing Ancestral Gene Orders Using Conserved Intervals. In: Jonassen, I., Kim, J. (eds.) WABI 2004. LNCS (LNBI), vol. 3240, pp. 14–25. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Blin, G., Rizzi, R., Vialette, S.: A Faster Algorithm for Finding Minimum Tucker Submatrices. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 69–77. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Blin, G., Rizzi, R., Vialette, S.: A Polynomial-Time Algorithm for Finding a Minimal Conflicting Set Containing a Given Row. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 373–384. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Chauve, C., Haus, U.-U., Stephen, T., You, V.P.: Minimal Conflicting Sets for the Consecutive Ones Property in Ancestral Genome Reconstruction. In: Ciccarelli, F.D., Miklós, I. (eds.) RECOMB-CG 2009. LNCS, vol. 5817, pp. 48–58. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Chauve, C., Tannier, E.: A methodological framework for the reconstruction of contiguous regions of ancestral genomes and its application to mammalian genomes. PLoS Comput. Biol. 4(11), 11 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dom, M.: Algorithmic aspects of the consecutive-ones property. Bulletin of the Eur. Assoc. for Theor. Comp. Science (EATCS) 98, 27–59 (2009)MathSciNetMATHGoogle Scholar
  7. 7.
    Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)MathSciNetMATHGoogle Scholar
  8. 8.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific (2004)Google Scholar
  9. 9.
    Ouangraoua, A., Raffinot, M.: Faster and Simpler Minimal Conflicting Set Identification. In: Kärkkäinen, J., Stoye, J. (eds.) CPM 2012. LNCS, vol. 7354, pp. 41–55. Springer, Heidelberg (2012)Google Scholar
  10. 10.
    Stoye, J., Wittler, R.: A unified approach for reconstructing ancient gene clusters. IEEE/ACM Trans. Comput. Biol. Bioinf. 6(3), 387–400 (2009)CrossRefGoogle Scholar
  11. 11.
    Tucker, A.C.: A structure theorem for the consecutive 1s property. Journal of Combinatorial Theory. Series B 12, 153–162 (1972)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aïda Ouangraoua
    • 1
  • Mathieu Raffinot
    • 2
  1. 1.INRIA Lille, LIFL - Université Lille 1Villeneuve d’AscqFrance
  2. 2.CNRS/LIAFA, Université Paris Diderot - Paris 7France

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