Searching for Gapped Palindromes

  • Roman Kolpakov
  • Gregory Kucherov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5029)


We study the problem of finding, in a given word, all maximal gapped palindromes verifying two types of constraints, that we call long-armed and length-constrained palindromes. For both classes, we propose algorithms that run in time O(n + S), where S is the number of output palindromes. Both algorithms can be extended to compute biological gapped palindromes within the same time bound.


Equivalence Class Turing Machine Space Length Length Constraint Input Word 
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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  1. [ABCD03]
    Allouche, J.-P., Baake, M., Cassaigne, J., Damanik, D.: Palindrome complexity. Theor. Comput. Sci. 292(1), 9–31 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [ABG94]
    Apostolico, A., Breslauer, D., Galil, Z.: Parallel detection of all palindromes in a string. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 497–506. Springer, Heidelberg (1994)Google Scholar
  3. [BBD+03]
    Biedl, T., Buss, J., Demaine, E., Demaine, M., Hajiaghayi, M., Vinar, T.: Palindrome recognition using a multidimensional tape. Theor. Comput. Sci. 302(1-3), 475–480 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BG95]
    Breslauer, D., Galil, Z.: Finding all periods and initial palindromes of a string in parallel. Algorithmica 14, 355–366 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Col69]
    Cole, S.N.: Real-time computation by n-dimensional iterative arrays of finite-state machines. IEEE Transactions on Computers 18, 349–365 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Coo71]
    Cook, S.: Linear time simulation of deterministic two-way pushdown automata. In: Proceedings of the 5th World Computer Congress, IFIP 1971, Ljubljana, Yugoslavia, August 23-28, 1971, Vol. 1, pp. 75–80 (1971)Google Scholar
  7. [CR94]
    Crochemore, M., Rytter, W.: Text algorithms. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  8. [DLDL05]
    De Luca, A., De Luca, A.: Palindromes in Sturmian words. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 199–208. Springer, Heidelberg (2005)Google Scholar
  9. [DP99]
    Droubay, X., Pirillo, G.: Palindromes and Sturmian words. Theoret. Comput. Sci. 223, 73–85 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Dro95]
    Droubay, X.: Palindromes in the Fibonacci word. Information Processing Letters 55(4), 217–221 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Gal78]
    Galil, Z.: Palindrome recognition in real time by a multitape turing machine. Journal of Computer and System Sciences 16(2), 140–157 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [Gus97]
    Gusfield, D.: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  13. [KK00a]
    Kolpakov, R., Kucherov, G.: Finding repeats with fixed gap. In: Proceedings of the 7th International Symposium on String Processing and Information Retrieval (SPIRE), A Coruña, Spain, September 27-29, 2000, pp. 162–168. IEEE, Los Alamitos (2000)CrossRefGoogle Scholar
  14. [KK00b]
    Kolpakov, R., Kucherov, G.: On maximal repetitions in words. Journal of Discrete Algorithms 1(1), 159–186 (2000)MathSciNetGoogle Scholar
  15. [KK05]
    Kolpakov, R., Kucherov, G.: Identification of periodic structures in words. In: Berstel, J., Perrin, D. (eds.) Applied combinatorics on words, Encyclopedia of Mathematics and its Applications. Lothaire books, ch.8, vol. 104, pp. 430–477. Cambridge University Press, Cambridge (2005)Google Scholar
  16. [KMP77]
    Knuth, D., Morris, J., Pratt, V.: Fast pattern matching in strings. SIAM J. Comput. 6, 323–350 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [KS03]
    Kärkkäinen, J., Sanders, P.: Simple linear work suffix array construction. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 943–955. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. [LJDL07]
    Lu, L., Jia, H., Dröge, P., Li, J.: The human genome-wide distribution of DNA palindromes. Functional and Integrative Genomics 7(3), 221–227 (2007)CrossRefGoogle Scholar
  19. [Man75]
    Manacher, G.: A new linear-time “on-line” algorithm for finding the smallest initial palindrome of a string. Journ. ACM 22(3), 346–351 (1975)zbMATHCrossRefGoogle Scholar
  20. [PB02]
    Porto, A.H.L., Barbosa, V.C.: Finding approximate palindromes in strings. Pattern Recognition 35, 2581–2591 (2002)zbMATHCrossRefGoogle Scholar
  21. [Sli73]
    Slisenko, A.O.: Recognition of palindromes by multihead turing machines. In: Orverkov, V.P., Sonin, N.A. (eds.) Problems in the Constructive Trend in Mathematics VI, Proceedings of the Steklov Institute of Mathematics, vol. 129, pp. 30–202 (1973)Google Scholar
  22. [Sli81]
    Slissenko, A.: A simplified proof of real-time recognizability of palindromes on Turing machines. J. of Soviet Mathematics 15(1), 68–77 (1981); Russian original. In: Zapiski Nauchnykh Seminarov LOMI, vol. 68, pp. 123–139 (1977) Google Scholar
  23. [vdSSer]
    van de Snepscheut, J., Swenker, J.: On the design of some systolic algorithms. J. ACM 36(4), 826–840 (1989)zbMATHCrossRefGoogle Scholar
  24. [WGC+04]
    Warburton, P.E., Giordano, J., Cheung, F., Gelfand, Y., Benson, G.: Inverted repeat structure of the human genome: The X-chromosome contains a preponderance of large, highly homologous inverted repeats that contain testes genes. Genome Research 14, 1861–1869 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Roman Kolpakov
    • 1
  • Gregory Kucherov
    • 2
  1. 1.Moscow UniversityMoscowRussia
  2. 2.LIFL/CNRS/INRIAParc scientifique de la Haute BorneVilleneuve d’AscqFrance

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