2026-07-25T12:08:01Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21102 2024-09-23T09:13:16Z ddc:004 open_access

String 2-Covers with No Length Restrictions Boneh, Itai Golan, Shay Shur, Arseny Quasi-periodicity String cover Range query Range stabbing A λ-cover of a string S is a set of strings {C_i}₁^λ such that every index in S is contained in an occurrence of at least one string C_i. The existence of a 1-cover defines a well-known class of quasi-periodic strings. Quasi-periodicity can be decided in linear time, and all 1-covers of a string can be reported in linear time as well. Since in general it is NP-complete to decide whether a string has a λ-cover, the natural next step is the development of efficient algorithms for 2-covers. Radoszewski and Straszyński [ESA 2020] analysed the particular case where the strings in a 2-cover must be of the same length. They provided an algorithm that reports all such 2-covers of S in time near-linear in |S| and in the size of the output. In this work, we consider 2-covers in full generality. Since every length-n string has Ω(n²) trivial 2-covers (every prefix and suffix of total length at least n constitute such a 2-cover), we state the reporting problem as follows: given a string S and a number m, report all 2-covers {C₁,C₂} of S with length |C₁|+|C₂| upper bounded by m. We present an Õ(n + output) time algorithm solving this problem, with output being the size of the output. This algorithm admits a simpler modification that finds a 2-cover of minimum length. We also provide an Õ(n) time construction of a 2-cover oracle which, given two substrings C₁,C₂ of S, reports in poly-logarithmic time whether {C₁,C₂} is a 2-cover of S. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Itai Boneh and Shay Golan and Arseny Shur 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.2024.31 urn:nbn:de:0030-drops-211029 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.31 eng https://creativecommons.org/licenses/by/4.0/legalcode