Rollercoasters and Caterpillars
Therese Biedl, Ahmad Biniaz, Robert Cummings, Anna Lubiw, Florin, Manea, Dirk Nowotka, and Jeffrey Shallit

TL;DR
This paper proves a conjecture that every sequence of n distinct real numbers contains a long rollercoaster subsequence of at least half its length, and provides efficient algorithms to find such subsequences and related graph drawings.
Contribution
It establishes the minimum length of rollercoasters in sequences, proves the conjecture, and develops linear-time algorithms for their computation, with applications to graph drawing.
Findings
Proved every sequence of n distinct numbers has a rollercoaster of length at least ⌈n/2⌉.
Developed a linear-time algorithm to find maximum-length rollercoasters.
Applied the result to efficiently draw top-view caterpillars with orthogonal paths.
Abstract
A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is 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 a polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three points. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length subsequence that is a rollercoaster. It was conjectured that every sequence of distinct real numbers contains a rollercoaster of length at least for , while the best known lower bound is . In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the -time algorithm for…
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