Palindromic Decompositions with Gaps and Errors

  • Michał Adamczyk
  • Mai Alzamel
  • Panagiotis Charalampopoulos
  • Costas S. Iliopoulos
  • Jakub RadoszewskiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10304)


Identifying palindromes in sequences has been an interesting line of research in combinatorics on words and also in computational biology, after the discovery of the relation of palindromes in the DNA sequence with the HIV virus. Efficient algorithms for the factorization of sequences into palindromes and maximal palindromes have been devised in recent years. We extend these studies by allowing gaps in decompositions and errors in palindromes, and also imposing a lower bound to the length of acceptable palindromes.

We first present an algorithm for obtaining a palindromic decomposition of a string of length n with the minimal total gap length in time \(\mathcal {O}(n \log {n} \cdot g)\) and space \(\mathcal {O}(n \cdot g)\), where g is the number of allowed gaps in the decomposition. We then consider a decomposition of the string in maximal \(\delta \)-palindromes (i.e. palindromes with \(\delta \) errors under the edit or Hamming distance) and g allowed gaps. We present an algorithm to obtain such a decomposition with the minimal total gap length in time \(\mathcal {O}(n \cdot (g+\delta ))\) and space \(\mathcal {O}(n\cdot g)\).


Dynamic Programming Time Algorithm Edit Distance Edit Operation Length Constraint 
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.


  1. 1.
    Alatabbi, A., Iliopoulos, C.S., Rahman, M.S.: Maximal palindromic factorization. In: Stringology, pp. 70–77 (2013)Google Scholar
  2. 2.
    Apostolico, A., Breslauer, D., Galil, Z.: Parallel detection of all palindromes in a string. Theor. Comput. Sci. 141(1), 163–173 (1995). CrossRefzbMATHGoogle Scholar
  3. 3.
    Breslauer, D., Galil, Z.: Finding all periods and initial palindromes of a string in parallel. Algorithmica 14(4), 355–366 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Crochemore, M., Hancart, C., Lecroq, T.: Algorithms on Strings. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Crochemore, M., Rytter, W.: Jewels of Stringology. World Scientific, Singapore (2003)zbMATHGoogle Scholar
  6. 6.
    Droubay, X.: Palindromes in the Fibonacci word. Inf. Process. Lett. 55(4), 217–221 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Droubay, X., Pirillo, G.: Palindromes and Sturmian words. Theor. Comput. Sci. 223(1–2), 73–85 (1999).–6 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fici, G., Gagie, T., Kärkkäinen, J., Kempa, D.: A subquadratic algorithm for minimum palindromic factorization. J. Discret. Algorithms 28(C), 41–48 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frid, A., Puzynina, S., Zamboni, L.: On palindromic factorization of words. Adv. Appl. Math. 50(5), 737–748 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fujishige, Y., Nakamura, M., Inenaga, S., Bannai, H., Takeda, M.: Finding gapped palindromes online. In: Mäkinen, V., Puglisi, S.J., Salmela, L. (eds.) IWOCA 2016. LNCS, vol. 9843, pp. 191–202. Springer, Cham (2016). doi: 10.1007/978-3-319-44543-4_15 CrossRefGoogle Scholar
  11. 11.
    Galil, Z.: Real-time algorithms for string-matching and palindrome recognition. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, pp. 161–173. ACM (1976).
  12. 12.
    Galil, Z., Seiferas, J.: A linear-time on-line recognition algorithm for “palstar”. J. ACM 25(1), 102–111 (1978). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gupta, S., Prasad, R., Yadav, S.: Searching gapped palindromes in DNA sequences using dynamic suffix array. Indian J. Sci. Technol. 8(23), 1 (2015)CrossRefGoogle Scholar
  14. 14.
    Gusfield, D.: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  15. 15.
    I, T., Sugimoto, S., Inenaga, S., Bannai, H., Takeda, M.: Computing palindromic factorizations and palindromic covers on-line. In: Kulikov, A.S., Kuznetsov, S.O., Pevzner, P. (eds.) CPM 2014. LNCS, vol. 8486, pp. 150–161. Springer, Cham (2014). doi: 10.1007/978-3-319-07566-2_16 Google Scholar
  16. 16.
    Knuth, D.E., Morris Jr., J.H., Pratt, V.R.: Fast pattern matching in strings. SIAM J. Comput. 6(2), 323–350 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kolpakov, R., Kucherov, G.: Searching for gapped palindromes. Theor. Comput. Sci. 410(51), 5365–5373 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kosolobov, D., Rubinchik, M., Shur, A.M.: Palk is linear recognizable online. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015. LNCS, vol. 8939, pp. 289–301. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46078-8_24 Google Scholar
  19. 19.
    Manacher, G.: A new linear-time “on-line” algorithm for finding the smallest initial palindrome of a string. J. ACM (JACM) 22(3), 346–351 (1975)CrossRefzbMATHGoogle Scholar
  20. 20.
    Rubinchik, M., Shur, A.M.: EERTREE: an efficient data structure for processing palindromes in strings. In: Lipták, Z., Smyth, W.F. (eds.) IWOCA 2015. LNCS, vol. 9538, pp. 321–333. Springer, Cham (2016). doi: 10.1007/978-3-319-29516-9_27 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Michał Adamczyk
    • 1
  • Mai Alzamel
    • 2
  • Panagiotis Charalampopoulos
    • 2
  • Costas S. Iliopoulos
    • 2
  • Jakub Radoszewski
    • 1
    • 2
    Email author
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Department of InformaticsKing’s College LondonLondonUK

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