The Complexity of Pattern Matching for 321-Avoiding and Skew-Merged Permutations

@article{Albert2016TheCO,
  title={The Complexity of Pattern Matching for 321-Avoiding and Skew-Merged Permutations},
  author={Michael H. Albert and Marie-Louise Lackner and Martin Lackner and Vincent Vatter},
  journal={Discret. Math. Theor. Comput. Sci.},
  year={2016},
  volume={18}
}
The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $\pi$ and $\tau$ are $321$-avoiding; the second is applicable if $\pi$ and $\tau$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $\pi$ and $n$ the length of $\tau$. 

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