Rollercoasters and Caterpillars

by   Therese Biedl, et al.

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 n distinct real numbers contains a rollercoaster of length at least n/2 for n>7, while the best known lower bound is Ω(n/ n). 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 O(n n)-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of {1,...,n} can be computed in O(n n) time. The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of 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 n-node top-view caterpillar on every set of 25/3n 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 n). We also show that such a drawing can be obtained in linear time, provided that the points are given in sorted order.


page 1

page 2

page 3

page 4


Fast and Longest Rollercoasters

For k≥ 3, a k-rollercoaster is a sequence of numbers whose every maximal...

Computing Bend-Minimum Orthogonal Drawings of Plane Series-Parallel Graphs in Linear Time

A planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph ...

Optimal Orthogonal Drawings of Planar 3-Graphs in Linear Time

A planar orthogonal drawing Γ of a planar graph G is a geometric represe...

Acute Tours in the Plane

We confirm the following conjecture of Fekete and Woeginger from 1997: f...

A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

Given a triangulation of a point set in the plane, a flip deletes an edg...

Optimal Algorithm for the Planar Two-Center Problem

In this paper, we consider the planar two-center problem: Given a set S ...

An Optimal Algorithm to Compute the Inverse Beacon Attraction Region

The beacon model is a recent paradigm for guiding the trajectory of mess...

Please sign up or login with your details

Forgot password? Click here to reset