Fast and Longest Rollercoasters

TL;DR

This paper introduces efficient algorithms for finding the longest k-rollercoaster subsequence in a sequence of distinct numbers, improving previous methods and establishing lower bounds for comparison-based approaches.

Contribution

It presents an $O(nk^2)$-time algorithm for the longest k-rollercoaster, achieving optimal linear time for constant k, and an $O(n \\log^2 n)$-time algorithm for large k, along with lower bounds.

Findings

Longest rollercoaster can be computed in linear time for constant k.

New algorithms are faster than previous $O(nk \\log n)$-time methods.

Established an $\\Omega(n \\log k)$ lower bound for comparison-based algorithms.

Abstract

For $k\geq 3$, 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 numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a $k$-rollercoaster. Biedl et al. [ICALP 2018] have shown that each sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, and that a longest rollercoaster contained in such a sequence can be computed in $O(n\log n)$-time. They have also shown that every sequence of $n\geq (k-1)^2+1$ distinct real numbers contains a $k$-rollercoaster of length at least $\frac{n}{2(k-1)}-\frac{3k}{2}$, and gave an $O(nk\log n)$-time algorithm computing a longest $k$-rollercoaster in a sequence of length $n$. 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. [ICALP 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 $k\leq n$. 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.