Rollercoasters with Plateaus

TL;DR

This paper introduces algorithms for detecting, counting, and enumerating maximum length plateau-$k$-rollercoasters, a new type of subsequence with alternating runs of weakly increasing and decreasing sequences allowing plateaus, in strings.

Contribution

It defines plateau-$k$-rollercoasters with runs containing repeated elements and provides efficient algorithms for their detection, counting, and enumeration in single and multiple strings.

Findings

Algorithms run in $O(n \sigma k)$ time for single strings.

Counting maximum length plateau-$k$-rollercoasters is efficient.

All maximum length rollercoasters can be enumerated with linear delay.

Abstract

In this paper we investigate the problem of detecting, counting, and enumerating (generating) all maximum length plateau-$k$-rollercoasters appearing as a subsequence of some given word (sequence, string), while allowing for plateaus. We define a plateau-$k$-rollercoaster as a word consisting of an alternating sequence of (weakly) increasing and decreasing \emph{runs}, with each run containing at least $k$ \emph{distinct} elements, allowing the run to contain multiple copies of the same symbol consecutively. This differs from previous work, where runs within rollercoasters have been defined only as sequences of distinct values. Here, we are concerned with rollercoasters of \emph{maximum} length embedded in a given word $w$, that is, the longest rollercoasters that are a subsequence of $w$. We present algorithms allowing us to determine the longest plateau-$k$-roller\-coasters appearing as a subsequence in any given word $w$ of length $n$ over an alphabet of size $\sigma$ in $O(n \sigma k)$ time, to count the number of plateau-$k$-rollercoasters in $w$ of maximum length in $O(n \sigma k)$ time, and to output all of them with $O(n)$ delay after $O(n \sigma k)$ preprocessing. Furthermore, we present an algorithm to determine the longest common plateau-$k$-rollercoaster within a set of words in $O(N k \sigma)$ where $N$ is the product of all word lengths within the set.