2019/02/23 by László Kozma, Kozma, László · 1 citation
Computer Science · Engineering · #Algorithms and Data Compression #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1902.08809
openalex publication_date 2019/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 n0.44k+o(k), and one whose running time is the better of O(1.6181n) and nk/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 = Ω(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 nk/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.