Counting Parameterized Border Arrays for a Binary Alphabet

  • Tomohiro I
  • Shunsuke Inenaga
  • Hideo Bannai
  • Masayuki Takeda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5457)


The parameterized pattern matching problem is a kind of pattern matching problem, where a pattern is considered to occur in a text when there exists a renaming bijection on the alphabet with which the pattern can be transformed into a substring of the text. A parameterized border array (p-border array) is an analogue of a border array of a standard string, which is also known as the failure function of the Morris-Pratt pattern matching algorithm. In this paper we present a linear time algorithm to verify if a given integer array is a valid p-border array for a binary alphabet. We also show a linear time algorithm to compute all binary parameterized strings sharing a given p-border array. In addition, we give an algorithm which computes all p-border arrays of length at most n, where n is a a given threshold. This algorithm runs in time linear in the number of output p-border arrays.


Linear Time Pattern Match String Match Linear Time Algorithm Constant Symbol 
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. 1.
    Baker, B.S.: Parameterized pattern matching: Algorithms and applications. Journal of Computer and System Sciences 52(1), 28–42 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baker, B.S.: A program for identifying duplicated code. Computing Science and Statistics 24, 49–57 (1992)Google Scholar
  3. 3.
    Fredriksson, K., Mozgovoy, M.: Efficient parameterized string matching. Information Processing Letters 100(3), 91–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Shibuya, T.: Generalization of a suffix tree for RNA structural pattern matching. Algorithmica 39(1), 1–19 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amir, A., Farach, M., Muthukrishnan, S.: Alphabet dependence in parameterized matching. Information Processing Letters 49(3), 111–115 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Baker, B.S.: Parameterized pattern matching by Boyer-Moore-type algorithms. In: Proc. 6th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1995), pp. 541–550 (1995)Google Scholar
  7. 7.
    Kosaraju, S.R.: Faster algorithms for the construction of parameterized suffix trees. In: Proc. 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp. 631–637 (1995)Google Scholar
  8. 8.
    Hazay, C., Lewenstein, M., Tsur, D.: Two dimensional parameterized matching. In: Apostolico, A., Crochemore, M., Park, K. (eds.) CPM 2005. LNCS, vol. 3537, pp. 266–279. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Hazay, C., Lewenstein, M., Sokol, D.: Approximate parameterized matching. ACM Transactions on Algorithms 3(3), Article No. 29 (2007)Google Scholar
  10. 10.
    Apostolico, A., Erdös, P.L., Lewenstein, M.: Parameterized matching with mismatches. Journal of Discrete Algorithms 5(1), 135–140 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Apostolico, A., Giancarlo, R.: Periodicity and repetitions in parameterized strings. Discrete Applied Mathematics 156(9), 1389–1398 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deguchi, S., Higashijima, F., Bannai, H., Inenaga, S., Takeda, M.: Parameterized suffix arrays for binary strings. In: Proc. The Prague Stringology Conference 2008 (PSC 2008), pp. 84–94 (2008)Google Scholar
  13. 13.
    Idury, R.M., Schäffer, A.A.: Multiple matching of parameterized patterns. Theoretical Computer Science 154(2), 203–224 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Aho, A.V., Corasick, M.J.: Efficient string matching: An aid to bibliographic search. Communications of the ACM 18(6), 333–340 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Morris, J.H., Pratt, V.R.: A linear pattern-matching algorithm. Technical Report Report 40, University of California, Berkeley (1970)Google Scholar
  16. 16.
    Franek, F., Gao, S., Lu, W., Ryan, P.J., Smyth, W.F., Sun, Y., Yang, L.: Verifying a border array in linear time. J. Combinatorial Math. and Combinatorial Computing 42, 223–236 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duval, J.P., Lecroq, T., Lefevre, A.: Border array on bounded alphabet. Journal of Automata, Languages and Combinatorics 10(1), 51–60 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Moore, D., Smyth, W., Miller, D.: Counting distinct strings. Algorithmica 23(1), 1–13 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Manber, U., Myers, G.: Suffix arrays: a new method for on-line string searches. SIAM J. Computing 22(5), 935–948 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Blumer, A., Blumer, J., Haussler, D., Ehrenfeucht, A., Chen, M.T., Seiferas, J.: The smallest automaton recognizing the subwords of a text. Theoretical Computer Science 40, 31–55 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Baeza-Yates, R.A.: Searching subsequences (note). Theoretical Computer Science 78(2), 363–376 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duval, J.P., Lefebvre, A.: Words over an ordered alphabet and suffix permutations. Theoretical Informatics and Applications 36, 249–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bannai, H., Inenaga, S., Shinohara, A., Takeda, M.: Inferring strings from graphs and arrays. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 208–217. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. 24.
    Schürmann, K.B., Stoye, J.: Counting suffix arrays and strings. Theoretical Computer Science 395(2-1), 220–234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lyndon, R.C., Schützenberger, M.P.: The equation a M = b N c P in a free group. Michigan Math. J. 9(4), 289–298 (1962)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Tomohiro I
    • 1
  • Shunsuke Inenaga
    • 2
  • Hideo Bannai
    • 1
  • Masayuki Takeda
    • 1
  1. 1.Department of InformaticsKyushu UniversityJapan
  2. 2.Graduate School of Information Science and Electrical EngineeringKyushu UniversityFukuokaJapan

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