Faster and simpler algorithms for finding large patterns in permutations

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length $n$ contains a given pattern of length $k$. In this work we give two new algorithms for this well-studied problem, one whose running time is $n^{{0.44k+o(k)}$,} and one whose running time is the better of $O(1.6181^n)$ and $n^{k/2+o(k)}$. These results improve the earlier best bounds of Ahal and Rabinovich (2000), and Bruner and Lackner (2012), and are the fastest algorithms for the problem when $k = \Omega(\log n)$. When $k = o(\log n)$, the parameterized algorithm of Guillemot and Marx (2013) dominates. Our second algorithm uses polynomial space and is significantly simpler than all previous approaches with comparable running times, including an $n^{k/2+o(k)}$ algorithm proposed by Guillemot and Marx. Our approach can be summarized as follows: "for every matching of the even-valued entries of the pattern, try to match all odd-valued entries left-to-right". For the special case of patterns that are Jordan-permutations, we show an improved, subexponential running time.