The aligned K-center problem

Handle URI:
http://hdl.handle.net/10754/561744
Title:
The aligned K-center problem
Authors:
Braß, Peter; Knauer, Christian; Na, Hyeonsuk; Shin, Chansu; Vigneron, Antoine E. ( 0000-0003-3586-3431 )
Abstract:
In this paper we study several instances of the aligned k-center problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line ℓ such that their union covers S and the maximum radius of the disks is minimized. This problem is a constrained version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, or a polygon. We first consider the simplest version of the problem where the line ℓ is given in advance; we can solve this problem in time O(n log2 n). In the case where only the direction of ℓ is fixed, we give an O(n2 log 2 n)-time algorithm. When ℓ is an arbitrary line, we give a randomized algorithm with expected running time O(n4 log2 n). Then we present (1+ε)-approximation algorithms for these three problems. When we denote T(k, ε) = (k/ε2+(k/ε) log k) log(1/ε), these algorithms run in O(n log k + T(k, ε)) time, O(n log k + T(k, ε)/ε) time, and O(n log k + T(k, ε)/ε2) time, respectively. For k = O(n1/3/log n), we also give randomized algorithms with expected running times O(n + (k/ε2) log(1/ε)), O(n+(k/ε3) log(1/ε)), and O(n + (k/ε4) log(1/ε)), respectively. © 2011 World Scientific Publishing Company.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer Science Program; Visual Computing Center (VCC); Geometric Algorithms Group
Publisher:
World Scientific Pub Co Pte Lt
Journal:
International Journal of Computational Geometry & Applications
Issue Date:
Apr-2011
DOI:
10.1142/S0218195911003597
Type:
Article
ISSN:
02181959
Sponsors:
Author for correspondence; Supported by Korean Research Foundation Grant (KRF-2007-531-D00018).Supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0016416), and the HUFS Research Fund.
Appears in Collections:
Articles; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorBraß, Peteren
dc.contributor.authorKnauer, Christianen
dc.contributor.authorNa, Hyeonsuken
dc.contributor.authorShin, Chansuen
dc.contributor.authorVigneron, Antoine E.en
dc.date.accessioned2015-08-03T09:03:38Zen
dc.date.available2015-08-03T09:03:38Zen
dc.date.issued2011-04en
dc.identifier.issn02181959en
dc.identifier.doi10.1142/S0218195911003597en
dc.identifier.urihttp://hdl.handle.net/10754/561744en
dc.description.abstractIn this paper we study several instances of the aligned k-center problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line ℓ such that their union covers S and the maximum radius of the disks is minimized. This problem is a constrained version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, or a polygon. We first consider the simplest version of the problem where the line ℓ is given in advance; we can solve this problem in time O(n log2 n). In the case where only the direction of ℓ is fixed, we give an O(n2 log 2 n)-time algorithm. When ℓ is an arbitrary line, we give a randomized algorithm with expected running time O(n4 log2 n). Then we present (1+ε)-approximation algorithms for these three problems. When we denote T(k, ε) = (k/ε2+(k/ε) log k) log(1/ε), these algorithms run in O(n log k + T(k, ε)) time, O(n log k + T(k, ε)/ε) time, and O(n log k + T(k, ε)/ε2) time, respectively. For k = O(n1/3/log n), we also give randomized algorithms with expected running times O(n + (k/ε2) log(1/ε)), O(n+(k/ε3) log(1/ε)), and O(n + (k/ε4) log(1/ε)), respectively. © 2011 World Scientific Publishing Company.en
dc.description.sponsorshipAuthor for correspondence; Supported by Korean Research Foundation Grant (KRF-2007-531-D00018).Supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0016416), and the HUFS Research Fund.en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectapproximate algorithmen
dc.subjectThe k-center problemen
dc.titleThe aligned K-center problemen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentComputer Science Programen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.contributor.departmentGeometric Algorithms Groupen
dc.identifier.journalInternational Journal of Computational Geometry & Applicationsen
dc.contributor.institutionDepartment of Computer Science, City College, NY, United Statesen
dc.contributor.institutionInstitut für Informatik, Universität Bayreuth, Germanyen
dc.contributor.institutionSchool of Computing, Soongsil University, Seoul, South Koreaen
dc.contributor.institutionSchool of Electronics and Information Engineering, Hankuk University of Foreign Studies, South Koreaen
kaust.authorVigneron, Antoine E.en
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