The k-centre problem for classes of cyclic words

Duncan Adamson*, Argyrios Deligkas, Vladimir V. Gusev, Igor Potapov

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)/n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)/N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique.

Original languageEnglish
Title of host publicationSOFSEM 2023 - Theory and practice of computer science
Subtitle of host publication48th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2023, Nový Smokovec, Slovakia, January 15–18, 2023, Proceedings
EditorsLeszek Gąsieniec
Place of PublicationCham
PublisherSpringer
Pages385-400
Number of pages16
ISBN (Electronic)9783031231018
ISBN (Print)9783031231001
DOIs
Publication statusPublished - 1 Jan 2023
Event48th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2023 - Nový Smokovec, Slovakia
Duration: 15 Jan 202318 Jan 2023

Publication series

NameLecture notes in computer science
Volume13878
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference48th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2023
Country/TerritorySlovakia
CityNový Smokovec
Period15/01/2318/01/23

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