2026-07-21T12:47:42Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21079 2024-09-23T09:13:16Z ddc:004 open_access

Interval Selection in Sliding Windows Alexandru, Cezar-Mihail Konrad, Christian Sliding window algorithms Streaming algorithms Interval selection We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the L most recent intervals, for some integer L. We give the following results: 1) In the unit-length intervals case, we give a 2-approximation sliding window algorithm with space Õ(|OPT|), and we show that any sliding window algorithm that computes a (2-ε)-approximation requires space Ω(L), for any ε > 0. 2) In the arbitrary-length case, we give a (11/3+ε)-approximation sliding window algorithm with space Õ(|OPT|), for any constant ε > 0, which constitutes our main result. We also show that space Ω(L) is needed for algorithms that compute a (2.5-ε)-approximation, for any ε > 0. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass 2-approximation streaming algorithm by Cabello and Pérez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a (4+ε)-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Cezar-Mihail Alexandru and Christian Konrad 2024 Is Part Of LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2024.8 urn:nbn:de:0030-drops-210795 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.8 eng https://creativecommons.org/licenses/by/4.0/legalcode