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.ISAAC.2020.23
URN: urn:nbn:de:0030-drops-133678
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13367/
Dudek, Bartłomiej ;
Gawrychowski, Paweł
Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs
Abstract
Permutation σ appears in permutation π if there exists a subsequence of π that is order-isomorphic to σ. The natural algorithmic question is to check if σ appears in π, and if so count the number of occurrences. Only since very recently we know that for any fixed length k, we can check if a given pattern of length k appears in a permutation of length n in time linear in n, but being able to count all such occurrences in f(k)⋅ n^o(k/log k) time would refute the exponential time hypothesis (ETH). Together with practical applications in statistics, this motivates a systematic study of the complexity of counting occurrences for different patterns of fixed small length k. We investigate this question for k = 4. Very recently, Even-Zohar and Leng [arXiv 2019] identified two types of 4-patterns. For the first type they designed an ?̃(n) time algorithm, while for the second they were able to provide an ?̃(n^1.5) time algorithm. This brings up the question whether the permutations of the second type are inherently harder than the first type.
We establish a connection between counting 4-patterns of the second type and counting 4-cycles (not necessarily induced) in a sparse undirected graph. By designing two-way reductions we show that the complexities of both problems are the same, up to polylogarithmic factors. This allows us to leverage the work done on the latter to provide a reasonable argument for why there is a difference in the complexities for counting 4-patterns of the first and the second type. In particular, even for the seemingly simpler problem of detecting a 4-cycle in a graph on m edges, the best known algorithm works in ?(m^{4/3}) time. Our reductions imply that an ?(n^{4/3-ε}) time algorithm for counting occurrences of any 4-pattern of the second type in a permutation of length n would imply an exciting breakthrough for counting (and hence also detecting) 4-cycles. In the other direction, by plugging in the fastest known algorithm for counting 4-cycles, we obtain an algorithm for counting occurrences of any 4-pattern of the second type in ?(n^1.48) time.
BibTeX - Entry
@InProceedings{dudek_et_al:LIPIcs:2020:13367,
author = {Bart{\l}omiej Dudek and Pawe{\l} Gawrychowski},
title = {{Counting 4-Patterns in Permutations Is Equivalent to Counting 4-Cycles in Graphs}},
booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)},
pages = {23:1--23:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-173-3},
ISSN = {1868-8969},
year = {2020},
volume = {181},
editor = {Yixin Cao and Siu-Wing Cheng and Minming Li},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13367},
URN = {urn:nbn:de:0030-drops-133678},
doi = {10.4230/LIPIcs.ISAAC.2020.23},
annote = {Keywords: Permutations, pattern avoidance, counting cycles}
}
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Keywords: |
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Permutations, pattern avoidance, counting cycles |
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Collection: |
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31st International Symposium on Algorithms and Computation (ISAAC 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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04.12.2020 |
