Counting the Orderings for Multisets in Consecutive Ones Property and PQ-Trees

  • Giovanni Battaglia
  • Roberto Grossi
  • Noemi Scutellà
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

A binary matrix satisfies the consecutive ones property (C1P) if its columns can be permuted such that the 1s in each row of the resulting matrix are consecutive. Equivalently, a family of setsF = {Q1,...,Qm}, where Qi ⊆ R for some universe R, satisfies the C1P if the symbols in R can be permuted such that the elements of each set Qi ∈ F occur consecutively, as a contiguous segment of the permutation of R’s symbols. Motivated by combinatorial problems on sequences with repeated symbols, we consider the C1P version on multisets and prove that counting the orderings (permutations) thus generated is #P-complete. We prove completeness results also for counting the permutations generated by PQ-trees (which are related to the C1P), thus showing that a polynomial-time algorithm is unlikely to exist when dealing with multisets and sequences with repeated symbols.

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References

  1. 1.
    Amir, A., Apostolico, A., Landau, G.M., Satta, G.: Efficient text fingerprinting via Parikh mapping. J. Discrete Algorithms 1(5-6), 409–421 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arora, S., Barak, B.: Computational Complexity A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Booth, K.S.: PQ-tree algorithms. Ph.D. thesis, Univ. of California (December 1975)Google Scholar
  4. 4.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. JCSS 13(3), 335–379 (1976)MathSciNetMATHGoogle Scholar
  5. 5.
    Christof, T., Oswald, M., Reinelt, G.: Consecutive ones and a betweenness problem in computational biology. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 213–228. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Engel, K.: Sperner theory. Cambridge University Press, New York (1997)CrossRefMATHGoogle Scholar
  7. 7.
    Eres, R., Landau, G.M., Parida, L.: A combinatorial approach to automatic discovery of cluster-patterns. In: Benson, G., Page, R.D.M. (eds.) WABI 2003. LNCS (LNBI), vol. 2812, pp. 139–150. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Fulkerson, D.R., Gross, D.A.: Incidence matrices and interval graphs. Pacific J. Math. 15(3), 835–855 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. Freeman, NY (1979)MATHGoogle Scholar
  10. 10.
    Ghosh, S.P.: File organization: the consecutive retrieval property. Commun. ACM 15(9), 802–808 (1972)CrossRefMATHGoogle Scholar
  11. 11.
    Jain, M., Myers, E.W.: Algorithms for computing and integrating physical maps using unique probes. In: RECOMB 1997, pp. 151–161 (1997)Google Scholar
  12. 12.
    Landau, G.M., Parida, L., Weimann, O.: Gene proximity analysis across whole genomes via PQ-trees. Journal of Computational Biology 12(10), 1289–1306 (2005)CrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.M.: Computational complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  14. 14.
    Parida, L.: Statistical significance of large gene clusters. Journal of Computational Biology 14(9), 1145–1159 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Giovanni Battaglia
    • 1
  • Roberto Grossi
    • 1
  • Noemi Scutellà
    • 2
  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly
  2. 2.List SpAPisaItaly

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