Fast and Longest Rollercoasters

Abstract

AbstractFor$$k\ge 3$$k≥3, ak-rollercoasteris a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at leastk; 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 ak-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 ofndistinct real numbers contains a rollercoaster of length at least$$\lceil n/2\rceil $$⌈n/2⌉for$$n>7$$n>7, and that a longest rollercoaster contained in such a sequence can be computed in$$O(n\log n)$$O(nlogn)-time (or faster, in$$O(n \log \log n)$$O(nloglogn)time, when the input sequence is a permutation of$$\{1,\ldots ,n\}$${1,…,n}). They have also shown that every sequence of$$n\geqslant (k-1)^2+1$$n⩾(k-1)2+1distinct real numbers contains ak-rollercoaster of length at least$$\frac{n}{2(k-1)}-\frac{3k}{2}$$n2(k-1)-3k2, and gave an$$O(nk\log n)$$O(nklogn)-time (respectively,$$O(n k\log \log n)$$O(nkloglogn)-time) algorithm computing a longestk-rollercoaster in a sequence of lengthn(respectively, a permutation of$$\{1,\ldots ,n\}$${1,…,n}). In this paper, we give an$$O(nk^2)$$O(nk2)-time algorithm computing the length of a longestk-rollercoaster contained in a sequence ofndistinct real numbers; hence, for constantk, our algorithm computes the length of a longestk-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respectivek-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 longestk-rollercoaster in$$O(n \log ^2 n)$$O(nlog2n)-time, that is, subquadratic even for large values of$$k\le n$$k≤n. Again, the rollercoaster can be easily retrieved. Finally, we show an$$\Omega (n \log k)$$Ω(nlogk)lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longestk-rollercoaster.