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.STACS.2019.30
URN: urn:nbn:de:0030-drops-102694
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10269/
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Gawrychowski, Pawel ; Manea, Florin ; Serafin, Radoslaw

Fast and Longest Rollercoasters

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LIPIcs-STACS-2019-30.pdf (0.7 MB)

Abstract

For k >= 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(n log n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,...,n}). They have also shown that every sequence of n >=slant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least n/(2(k-1)) - 3k/2, and gave an O(nk log n)-time (respectively, O(n k log log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,...,n}).
In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log^2 n)-time, that is, subquadratic even for large values of k <= n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega(n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

BibTeX - Entry

@InProceedings{gawrychowski_et_al:LIPIcs:2019:10269,
  author =	{Pawel Gawrychowski and Florin Manea and Radoslaw Serafin},
  title =	{{Fast and Longest Rollercoasters}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10269},
  doi =		{10.4230/LIPIcs.STACS.2019.30},
  annote =	{Keywords: sequences, alternating runs, patterns in permutations}
}

Keywords: sequences, alternating runs, patterns in permutations
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019

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