Advertisement

Consensus String Problem for Multiple Regular Languages

  • Yo-Sub Han
  • Sang-Ki KoEmail author
  • Timothy Ng
  • Kai Salomaa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)

Abstract

The consensus string (or center string, closest string) of a set S of strings is defined as a string which is within a radius r from all strings in S. It is well-known that the consensus string problem for a finite set of equal-length strings is NP-complete. We study the consensus string problem for multiple regular languages. We define the consensus string of languages \(L_1, \ldots , L_k\) to be within distance at most r to some string in each of the languages \(L_1, \ldots , L_k\). We also study the complexity of some parameterized variants of the consensus string problem. For a constant k, we give a polynomial time algorithm for the consensus string problem for k regular languages using additive weighted finite automata. We show that the consensus string problem for multiple regular languages becomes intractable when k is not fixed. We also examine the case when the length of the consensus string is given as part of input.

Keywords

Consensus string problem Computational complexity Regular languages Edit-distance 

References

  1. 1.
    Amir, A., Landau, G.M., Na, J.C., Park, H., Park, K., Sim, J.S.: Efficient algorithms for consensus string problems minimizing both distance sum and radius. Theoret. Comput. Sci. 412(39), 5239–5246 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amir, A., Paryenty, H., Roditty, L.: On the hardness of the consensus string problem. Inf. Process. Lett. 113(10–11), 371–374 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calude, C.S., Salomaa, K., Yu, S.: Additive distances and quasi-distances between words. J. Univ. Comput. Sci. 8(2), 141–152 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Choffrut, C., Pighizzini, G.: Distances between languages and reflexivity of relations. Theoret. Comput. Sci. 286(1), 117–138 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, G.D., Honkala, I.S., Litsyn, S.N., Solé, P.: Long packing and covering codes. IEEE Trans. Inf. Theor. 43(5), 1617–1619 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deza, M.M., Deza, E.: Encyclopedia of Distances. Springer, Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Frances, M., Litman, A.: On covering problems of codes. Theor. Comput. Syst. 30(2), 113–119 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Graham, R.L., Sloane, N.J.A.: On the covering radius of codes. IEEE Trans. Inf. Theor. 31(3), 385–401 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gramm, J., Niedermeier, R., Rossmanith, P.: Exact solutions for closest string and related problems. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 441–453. Springer, Heidelberg (2001). doi: 10.1007/3-540-45678-3_38 CrossRefGoogle Scholar
  10. 10.
    Gramm, J., Niedermeier, R., Rossmanith, P.: Fixed-parameter algorithms for closest string and related problems. Algorithmica 37, 25–42 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Han, Y.-S., Ko, S.-K., Salomaa, K.: The edit-distance between a regular language and a context-free language. Int. J. Found. Comput. Sci. 24(7), 1067–1082 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Holzer, M., Kutrib, M.: Descriptional and computational complexity of finite automata—a survey. Inf. Comput. 209(3), 456–470 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Konstantinidis, S.: Computing the edit distance of a regular language. Inf. Comput. 205(9), 1307–1316 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)Google Scholar
  15. 15.
    Levenshtein, V.I.: Binary codes capable of correcting deletions, insertions, and reversals. Sov. Phys. Dokl. 10(8), 707–710 (1966)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ma, B., Sun, X.: More efficient algorithms for closest string and substring problems. SIAM J. Comput. 39(4), 1432–1443 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Manthey, B., Reischuk, R.: The intractability of computing the Hamming distance. Theoret. Comput. Sci. 337(1–3), 331–346 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Myers, G.: Approximately matching context-free languages. Inf. Process. Lett. 54, 85–92 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ng, T., Rappaport, D., Salomaa, K.: Quasi-distances and weighted finite automata. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 209–219. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-19225-3_18 CrossRefGoogle Scholar
  20. 20.
    Ng, T., Rappaport, D., Salomaa, K.: State complexity of neighbourhoods and approximate pattern matching. In: Potapov, I. (ed.) DLT 2015. LNCS, vol. 9168, pp. 389–400. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21500-6_31 CrossRefGoogle Scholar
  21. 21.
    Palopoli, L., Chen, Z.-Z., Ma, B., Wang, L.: A three-string approach to the closest string problem. J. Comput. Syst. Sci. 78(1), 164–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pighizzini, G.: How hard is computing the edit distance? Inf. Comput. 165(1), 1–13 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Povarov, G.: Descriptive complexity of the hamming neighborhood of a regular language. In: Proceedings of the 1st International Conference on Language and Automata Theory and Applications, pp. 509–520 (2007)Google Scholar
  24. 24.
    Shallit, J.: A Second Course in Formal Languages and Automata Theory, 1st edn. Cambridge University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  25. 25.
    Sim, J.S., Park, K.: The consensus string problem for a metric is NP-complete. J. Discret. Algorithms 1(1), 111–117 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stojanovic, N., Berman, P., Gumucio, D., Hardison, R., Miller, W.: A linear-time algorithm for the 1-mismatch problem. In: Dehne, F., Rau-Chaplin, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 126–135. Springer, Heidelberg (1997). doi: 10.1007/3-540-63307-3_53 CrossRefGoogle Scholar
  27. 27.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 41–110. Springer, Berlin (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yo-Sub Han
    • 1
  • Sang-Ki Ko
    • 2
    Email author
  • Timothy Ng
    • 3
  • Kai Salomaa
    • 3
  1. 1.Department of Computer ScienceYonsei UniversitySeoulRepublic of Korea
  2. 2.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  3. 3.School of ComputingQueen’s UniversityKingstonCanada

Personalised recommendations