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Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

  • Tomasz Kociumaka
  • Jakub Radoszewski
  • Wojciech Rytter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)

Abstract

We introduce efficient data structures for an indexing problem in non-standard stringology — jumbled pattern matching. Moosa and Rahman [J. Discr. Alg., 2012] gave an index for jumbled pattern matching for the case of binary alphabets with \(O(\frac{n^2}{\log^2 n})\)-time construction. They posed as an open problem an efficient solution for larger alphabets. In this paper we provide an index for any constant-sized alphabet. We obtain the first o(n 2)-space construction of an index with o(n) query time. It can be built in O(n 2) time. Precisely, our data structure can be implemented with O(n 2 − δ ) space and O(m (2σ − 1)δ ) query time for any δ > 0, where m is the length of the pattern and σ is the alphabet size (σ = O(1)). We also break the barrier of quadratic construction time for non-binary constant alphabet simultaneously obtaining poly-logarithmic query time.

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References

  1. 1.
    Avgustinovich, S.V., Glen, A., Halldórsson, B.V., Kitaev, S.: On shortest crucial words avoiding Abelian powers. Discrete Applied Mathematics 158(6), 605–607 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Blanchet-Sadri, F., Kim, J.I., Mercaş, R., Severa, W., Simmons, S.: Abelian square-free partial words. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 94–105. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Blanchet-Sadri, F., Simmons, S.: Avoiding Abelian powers in partial words. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 70–81. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: On table arrangements, scrabble freaks, and jumbled pattern matching. In: Boldi, P. (ed.) FUN 2010. LNCS, vol. 6099, pp. 89–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: Algorithms for jumbled pattern matching in strings. Int. J. Found. Comput. Sci. 23(2), 357–374 (2012)MATHCrossRefGoogle Scholar
  6. 6.
    Burcsi, P., Cicalese, F., Fici, G., Lipták, Z.: On approximate jumbled pattern matching in strings. Theory Comput. Syst. 50(1), 35–51 (2012)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cassaigne, J., Richomme, G., Saari, K., Zamboni, L.Q.: Avoiding Abelian powers in binary words with bounded Abelian complexity. Int. J. Found. Comput. Sci. 22(4), 905–920 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cicalese, F., Fici, G., Lipták, Z.: Searching for jumbled patterns in strings. In: Holub, J., Žďárek, J. (eds.) Proceedings of the Prague Stringology Conference 2009, Czech Technical University in Prague, Czech Republic, pp. 105–117 (2009)Google Scholar
  9. 9.
    Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for Abelian periods. Bulletin of the EATCS 89, 167–170 (2006)MathSciNetMATHGoogle Scholar
  10. 10.
    Crochemore, M., Iliopoulos, C., Kociumaka, T., Kubica, M., Pachocki, J., Radoszewski, J., Rytter, W., Tyczyński, W., Waleń, T.: A note on efficient computation of all Abelian periods in a string. Information Processing Letters 113(3), 74–77 (2013)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Currie, J.D., Aberkane, A.: A cyclic binary morphism avoiding Abelian fourth powers. Theor. Comput. Sci. 410(1), 44–52 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Currie, J.D., Visentin, T.I.: Long binary patterns are Abelian 2-avoidable. Theor. Comput. Sci. 409(3), 432–437 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Upper and lower bounds for finding connected motifs in vertex-colored graphs. J. Comput. Syst. Sci. 77(4), 799–811 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fici, G., Lecroq, T., Lefebvre, A., Prieur-Gaston, É.: Computing Abelian periods in words. In: Holub, J., Žďárek, J. (eds.) Proceedings of the Prague Stringology Conference 2011, Czech Technical University in Prague, Czech Republic, pp. 184–196 (2011)Google Scholar
  15. 15.
    Fici, G., Lecroq, T., Lefebvre, A., Prieur-Gaston, E., Smyth, W.: Quasi-linear time computation of the abelian periods of a word. In: Holub, J., Žďárek, J. (eds.) Proceedings of the Prague Stringology Conference 2012, Czech Technical University in Prague, Czech Republic, pp. 103–110 (2012)Google Scholar
  16. 16.
    Fredman, M.L., Komlós, J., Szemerédi, E.: Storing a sparse table with O(1) worst case access time. J. ACM 31(3), 538–544 (1984)MATHCrossRefGoogle Scholar
  17. 17.
    Gagie, T., Hermelin, D., Landau, G.M., Weimann, O.: Binary jumbled pattern matching on trees and tree-like structures. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 517–528. Springer, Heidelberg (2013)Google Scholar
  18. 18.
    Kociumaka, T., Radoszewski, J., Rytter, W.: Fast algorithms for abelian periods in words and greatest common divisor queries. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs, vol. 20, pp. 245–256. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  19. 19.
    Lacroix, V., Fernandes, C.G., Sagot, M.-F.: Motif search in graphs: Application to metabolic networks. IEEE/ACM Trans. Comput. Biology Bioinform. 3(4), 360–368 (2006)CrossRefGoogle Scholar
  20. 20.
    Moosa, T.M., Rahman, M.S.: Indexing permutations for binary strings. Inf. Process. Lett. 110(18-19), 795–798 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Moosa, T.M., Rahman, M.S.: Sub-quadratic time and linear space data structures for permutation matching in binary strings. J. Discrete Algorithms 10, 5–9 (2012)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tomasz Kociumaka
    • 1
  • Jakub Radoszewski
    • 1
  • Wojciech Rytter
    • 1
    • 2
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Faculty of Mathematics and Computer ScienceCopernicus UniversityToruńPoland

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