Finding and counting permutations via CSPs
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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^k/4 + o(k), and a polynomial-space algorithm whose running time is the better of O(1.6181^n) and O(n^k/2 + 1). These results improve the earlier best bounds of n^0.47k + o(k) and O(1.79^n) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k ∈Ω(n). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f(k) · n^o(k/k) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes.
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