Abstract
A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence  increasing or decreasing  has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as an xmonotone polygonal path for which every maximal subpath, with positive or negativeslope edges, has at least three vertices. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximumlength (not necessarily contiguous) subsequence that is a rollercoaster. It was conjectured that every sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, while the best known lower bound is Omega(n/log n). In this paper we prove this conjecture. Our proof is constructive and implies a lineartime algorithm for computing a rollercoaster of this length. Extending the O(n log n)time algorithm for computing a longest increasing subsequence, we show how to compute a maximumlength rollercoaster within the same time bound. A maximumlength rollercoaster in a permutation of {1,...,n} can be computed in O(n log log n) time.
The search for rollercoasters was motivated by orthogeodesic pointset embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A topview caterpillar is one of maximum degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every nvertex topview caterpillar on every set of 25/3(n+4) points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is O(n log n). We also show that such a drawing can be obtained in linear time, when the points are given in sorted order.
BibTeX  Entry
@InProceedings{biedl_et_al:LIPIcs:2018:9022,
author = {Therese Biedl and Ahmad Biniaz and Robert Cummings and Anna Lubiw and Florin Manea and Dirk Nowotka and Jeffrey Shallit},
title = {{Rollercoasters and Caterpillars}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {18:118:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9022},
URN = {urn:nbn:de:0030drops90224},
doi = {10.4230/LIPIcs.ICALP.2018.18},
annote = {Keywords: sequences, alternating runs, patterns in permutations, caterpillars}
}
Keywords: 

sequences, alternating runs, patterns in permutations, caterpillars 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 