License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2019.1
URN: urn:nbn:de:0030-drops-114627
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11462/
Berendsohn, Benjamin Aram ;
Kozma, László ;
Marx, Dániel
Finding and Counting Permutations via CSPs
Abstract
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 in Omega(log{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/log{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.
BibTeX - Entry
@InProceedings{berendsohn_et_al:LIPIcs:2019:11462,
author = {Benjamin Aram Berendsohn and L{\'a}szl{\'o} Kozma and D{\'a}niel Marx},
title = {{Finding and Counting Permutations via CSPs}},
booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
pages = {1:1--1:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-129-0},
ISSN = {1868-8969},
year = {2019},
volume = {148},
editor = {Bart M. P. Jansen and Jan Arne Telle},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2019/11462},
URN = {urn:nbn:de:0030-drops-114627},
doi = {10.4230/LIPIcs.IPEC.2019.1},
annote = {Keywords: permutations, pattern matching, constraint satisfaction, exponential time}
}
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Keywords: |
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permutations, pattern matching, constraint satisfaction, exponential time |
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Collection: |
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14th International Symposium on Parameterized and Exact Computation (IPEC 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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04.12.2019 |
