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A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

Abstract

The NP-complete Permutation Pattern Matching problem asks whether a k-permutation P is contained in a n-permutation T as a pattern. This is the case if there exists an order-preserving embedding of P into T. In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of \({\mathcal O}(1.79^{\mathsf {run}(T)}\cdot n\cdot k)\), where \(\mathsf {run}(T)\) denotes the number of alternating runs of T. This algorithm is particularly well-suited for instances where T has few runs, i.e., few ups and downs. Moreover, since \(\mathsf {run}(T)<n\), this can be seen as a \({\mathcal O}(1.79^{n}\cdot n\cdot k)\) algorithm which is the first to beat the exponential \(2^n\) runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixed-parameter tractability result is not possible for \(\mathsf {run}(P)\).

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Notes

  1. 1.

    The reason why we do not set \(b_{\mathsf {run}(P)}=\lfloor \mathsf {run}(F(\mathsf {run}(P))/2\rfloor \) is a rather technical one: \(F(\mathsf {run}(P))\) may end with a run up if the last run in P is a run up and may end with a run down if the last run in P is a run down. This would lead to unwanted collisions concerning the Index function and consequently would prohibit the proof of Lemma 5.

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Acknowledgments

We would like to thank the anonymous reviewers for their feedback and suggestions leading to numerous improvements of this paper. The first author was supported by the Austrian Science Foundation FWF, Grant P25337-N23, the second author by the Austrian Science Foundation FWF, Grant P25518-N23 and Y698.

Author information

Correspondence to Marie-Louise Bruner.

Additional information

A preliminary version of this paper [9] appeared in the proceedings of the 13th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2012.

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Bruner, M., Lackner, M. A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs. Algorithmica 75, 84–117 (2016). https://doi.org/10.1007/s00453-015-0013-y

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Keywords

  • Permutation patterns
  • Parameterized complexity
  • Pattern matching
  • Fixed-parameter tractability
  • Exponential algorithms
  • Combinatorics