Fast and Longest Rollercoasters

Abstract

For kge 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 real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (in: ICALP, volume 107 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 18:1–18:15, 2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least lceil n/2rceil for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(nlog n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,ldots ,n}). They have also shown that every sequence of ngeqslant (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(nklog n)-time (respectively, O(n klog log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,ldots ,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. (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 kle 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.