Querying and Embedding Compressed Texts

  • Yury Lifshits
  • Markus Lohrey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)


The computational complexity of two simple string problems on compressed input strings is considered: the querying problem (What is the symbol at a given position in a given input string?) and the embedding problem (Can the first input string be embedded into the second input string?). Straight-line programs are used for text compression. It is shown that the querying problem becomes P-complete for compressed strings, while the embedding problem becomes hard for the complexity class \(\Theta^{p}_{2}\).


Polynomial Time Pattern Match Input String Longe Common Subsequence Polynomial Size 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amir, A., Benson, G., Farach, M.: Let sleeping files lie: Pattern matching in Z-compressed files. J. Comput. Syst. Sci 52(2), 299–307 (1996)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beaudry, M., McKenzie, P., Péladeau, P., Thérien, D.: Finite monoids: From word to circuit evaluation. SIAM J. Comput. 26(1), 138–152 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berman, P., Karpinski, M., Larmore, L.L., Plandowski, W., Rytter, W.: On the complexity of pattern matching for highly compressed two-dimensional texts. J. Comput. Syst. Sci. 65(2), 332–350 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Charikar, M., Lehman, E., Lehman, A., Liu, D., Panigrahy, R., Prabhakaran, M., Sahai, A., Shelat, A.: The smallest grammar problem. IEEE Trans. Inf. Theory 51(7), 2554–2576 (2005)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Downey, R.G., Fellows, M.R.: Parametrized Complexity. Springer, Heidelberg (1999)Google Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP–completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  7. 7.
    Gasieniec, L., Gibbons, A., Rytter, W.: Efficiency of fast parallel pattern searching in highly compressed texts. In: Kutyłowski, M., Wierzbicki, T., Pacholski, L. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 48–58. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Gasieniec, L., Karpinski, M., Plandowski, W., Rytter, W.: Efficient algorithms for Lempel-Ziv encoding (extended abstract). In: Karlsson, R., Lingas, A. (eds.) SWAT 1996. LNCS, vol. 1097, pp. 392–403. Springer, Heidelberg (1996)Google Scholar
  9. 9.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  10. 10.
    Gushfield, D.: Algorithms on Strings, Trees, and Sequences. Cambridge University Press, Cambridge (1999)Google Scholar
  11. 11.
    Karloff, H.J., Ruzzo, W.L.: The iterated mod problem. Inf. Comput. 80(3), 193–204 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Markey, N., Schnoebelen, P.: A PTIME-complete matching problem for SLP-compressed words. Inf. Process. Lett. 90(1), 3–6 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Miyazaki, M., Shinohara, A., Takeda, M.: An improved pattern matching algorithm for strings in terms of straight-line programs. In: Hein, J., Apostolico, A. (eds.) CPM 1997. LNCS, vol. 1264, pp. 1–11. Springer, Heidelberg (1997)Google Scholar
  15. 15.
    Navarro, G.: Regular expression searching on compressed text. J. Discrete Algorithms 1(5–6), 423–443 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  17. 17.
    Plandowski, W.: Testing equivalence of morphisms on context-free languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  18. 18.
    Plandowski, W., Rytter, W.: Complexity of language recognition problems for compressed words. In: Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa, pp. 262–272. Springer, Heidelberg (1999)Google Scholar
  19. 19.
    Rytter, W.: Algorithms on compressed strings and arrays. In: Bartosek, M., Tel, G., Pavelka, J. (eds.) SOFSEM 1999. LNCS, vol. 1725, pp. 48–65. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  20. 20.
    Rytter, W.: Compressed and fully compressed pattern matching in one and two dimensions. Proceedings of the IEEE 88(11), 1769–1778 (2000)CrossRefGoogle Scholar
  21. 21.
    Rytter, W.: Application of Lempel-Ziv factorization to the approximation of grammar-based compression. Theor. Comput. Sci. 302(1–3), 211–222 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Rytter, W.: Grammar compression, LZ-encodings, and string algorithms with implicit input. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 15–27. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Wagner, K.W.: More complicated questions about maxima and minima, and some closures of NP. Theor. Comput. Sci. 51, 53–80 (1987)MATHCrossRefGoogle Scholar
  24. 24.
    Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Inf. Theory 23(3), 337–343 (1977)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yury Lifshits
    • 1
  • Markus Lohrey
    • 2
  1. 1.Steklov Institut of MathematicsSt.PetersburgRussia
  2. 2.FMIUniversität StuttgartGermany

Personalised recommendations