A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

  • Marie-Louise Bruner
  • Martin Lackner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


The NP-complete Permutation Pattern Matching problem asks whether a permutation P can be matched into a permutation T. A matching is an order-preserving embedding of P into T. We present a fixed-parameter algorithm solving this problem with an exponential worst-case runtime of \(\mathcal{O}^*(1.79^{\sf{run}(T)})\), where run(T) denotes the number of alternating runs of T. This is the first algorithm that improves upon the \(\mathcal{O}^*(2^n)\) runtime required by brute-force search without imposing restrictions on P and T. Furthermore we prove that – under standard complexity theoretic assumptions – such a fixed-parameter tractability result is not possible for run(P).


Dynamic Programming Algorithm Matching Function Representative Element Permutation Graph Pattern Avoidance 
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  1. 1.
    Ahal, S., Rabinovich, Y.: On complexity of the subpattern problem. SIAM J. Discrete Math. 22(2), 629–649 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Albert, M., Aldred, R., Atkinson, M., Holton, D.: Algorithms for Pattern Involvement in Permutations. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 355–367. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    André, D.: Étude sur les maxima, minima et séquences des permutations. Ann. Sci. École Norm. Sup. 3(1), 121–135 (1884)Google Scholar
  4. 4.
    Bona, M.: Combinatorics of permutations. Discrete Mathematics and Its Applications. Chapman & Hall/CRC (2004)Google Scholar
  5. 5.
    Bose, P., Buss, J.F., Lubiw, A.: Pattern matching for permutations. Information Processing Letters 65(5), 277–283 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bruner, M.L., Lackner, M.: A fast algorithm for permutation pattern matching based on alternating runs. CoRR (2012)Google Scholar
  7. 7.
    Chang, M.S., Wang, F.H.: Efficient algorithms for the maximum weight clique and maximum weight independent set problems on permutation graphs. Information Processing Letters 43(6), 293–295 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized complexity theory. Springer, Heidelberg (2006)Google Scholar
  9. 9.
    Guillemot, S., Vialette, S.: Pattern Matching for 321-Avoiding Permutations. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 1064–1073. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Ibarra, L.: Finding pattern matchings for permutations. Information Processing Letters 61(6), 293–295 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kitaev, S.: Patterns in Permutations and Words. Springer, Heidelberg (2011)zbMATHCrossRefGoogle Scholar
  12. 12.
    Knuth, D.E.: The Art of Computer Programming. Fundamental Algorithms, vol. I. Addison-Wesley (1968)Google Scholar
  13. 13.
    Levene, H., Wolfowitz, J.: The covariance matrix of runs up and down. The Annals of Mathematical Statistics 15(1), 58–69 (1944)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Mäkinen, E.: On the longest upsequence problem for permutations. International Journal of Computer Mathematics 77(1), 45–53 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics And Its Applications. Oxford University Press (2006)Google Scholar
  16. 16.
    Schensted, C.: Longest increasing and decreasing subsequences. Classic Papers in Combinatorics, pp. 299–311 (1987)Google Scholar
  17. 17.
    Simion, R., Schmidt, F.W.: Restricted permutations. European Journal of Combinatorics 6, 383–406 (1985)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marie-Louise Bruner
    • 1
  • Martin Lackner
    • 2
  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyAustria
  2. 2.Institute of Information SystemsVienna University of TechnologyAustria

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