A fast algorithm for permutation pattern matching based on alternating runs

Authors:
Marie-Louise Bruner;Martin Lackner
Affiliations:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Austria;Institute of Information Systems, Vienna University of Technology, Austria
Venue:
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Year:
2012

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Abstract

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).