Verifying a Parameterized Border Array in O(n1.5) Time

  • Tomohiro I.
  • Shunsuke Inenaga
  • Hideo Bannai
  • Masayuki Takeda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6129)


The parameterized pattern matching problem is to check if there exists a renaming bijection on the alphabet with which a given pattern can be transformed into a substring of a given text. A parameterized border array (p-border array) is a parameterized version of a standard border array, and we can efficiently solve the parameterized pattern matching problem using p-border arrays. In this paper we present an O(n 1.5)-time O(n)-space algorithm to verify if a given integer array of length n is a valid p-border array for an unbounded alphabet. The best previously known solution takes time proportional to the n-th Bell number \(\frac{1}{e} \sum_{k=0}^{\infty} \frac{k^{n}}{k!}\), and hence our algorithm is quite efficient.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baker, B.S.: Parameterized pattern matching: Algorithms and applications. Journal of Computer and System Sciences 52(1), 28–42 (1996)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amir, A., Farach, M., Muthukrishnan, S.: Alphabet dependence in parameterized matching. Information Processing Letters 49(3), 111–115 (1994)MATHCrossRefGoogle Scholar
  3. 3.
    Kosaraju, S.: Faster algorithms for the construction of parameterized suffix trees. In: Proc. FOCS 1995, pp. 631–637 (1995)Google Scholar
  4. 4.
    Hazay, C., Lewenstein, M., Sokol, D.: Approximate parameterized matching. ACM Transactions on Algorithms 3(3), Article No. 29 (2007)Google Scholar
  5. 5.
    Apostolico, A., Erdös, P.L., Lewenstein, M.: Parameterized matching with mismatches. Journal of Discrete Algorithms 5(1), 135–140 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    I, T., Deguchi, S., Bannai, H., Inenaga, S., Takeda, M.: Lightweight parameterized suffix array construction. In: Proc. IWOCA, pp. 312–323 (2009)Google Scholar
  7. 7.
    Idury, R.M., Schäffer, A.A.: Multiple matching of parameterized patterns. Theoretical Computer Science 154(2), 203–224 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Morris, J.H., Pratt, V.R.: A linear pattern-matching algorithm. Technical Report 40, University of California, Berkeley (1970)Google Scholar
  9. 9.
    I, T., Inenaga, S., Bannai, H., Takeda, M.: Counting parameterized border arrays for a binary alphabet. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 422–433. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Franek, F., Gao, S., Lu, W., Ryan, P.J., Smyth, W.F., Sun, Y., Yang, L.: Verifying a border array in linear time. J. Comb. Math. and Comb. Comp. 42, 223–236 (2002)MATHMathSciNetGoogle Scholar
  11. 11.
    Duval, J.P., Lecroq, T., Lefevre, A.: Border array on bounded alphabet. Journal of Automata, Languages and Combinatorics 10(1), 51–60 (2005)MATHMathSciNetGoogle Scholar
  12. 12.
    Duval, J.P., Lefebvre, A.: Words over an ordered alphabet and suffix permutations. Theoretical Informatics and Applications 36, 249–259 (2002)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    Schürmann, K.B., Stoye, J.: Counting suffix arrays and strings. Theoretical Computer Science 395(2-1), 220–234 (2008)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Clément, J., Crochemore, M., Rindone, G.: Reverse engineering prefix tables. In: Proc. STACS 2009, pp. 289–300 (2009)Google Scholar
  16. 16.
    Duval, J.P., Lecroq, T., Lefebvre, A.: Efficient validation and construction of border arrays and validation of string matching automata. RAIRO - Theoretical Informatics and Applications 43(2), 281–297 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gawrychowski, P., Jez, A., Jez, L.: Validating the Knuth-Morris-Pratt failure function, fast and online. In: Proc. CSR 2010 (to appear 2010)Google Scholar
  18. 18.
    Crochemore, M., Iliopoulos, C., Pissis, S., Tischler, G.: Cover array string reconstruction. In: Proc. CPM 2010 (to appear 2010)Google Scholar
  19. 19.
    Moore, D., Smyth, W., Miller, D.: Counting distinct strings. Algorithmica 23(1), 1–13 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

Personalised recommendations