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    The aligned K-center problem

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    Type
    Article
    Authors
    Braß, Peter
    Knauer, Christian
    Na, Hyeonsuk
    Shin, Chansu
    Vigneron, Antoine E. cc
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Computer Science Program
    Visual Computing Center (VCC)
    Geometric Algorithms Group
    Date
    2011-11-20
    Online Publication Date
    2011-11-20
    Print Publication Date
    2011-04
    Permanent link to this record
    http://hdl.handle.net/10754/561744
    
    Metadata
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    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.
    Citation
    BRASS, P., KNAUER, C., NA, H.-S., SHIN, C.-S., & VIGNERON, A. (2011). THE ALIGNED K-CENTER PROBLEM. International Journal of Computational Geometry & Applications, 21(02), 157–178. doi:10.1142/s0218195911003597
    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.
    Publisher
    World Scientific Pub Co Pte Lt
    Journal
    International Journal of Computational Geometry & Applications
    DOI
    10.1142/S0218195911003597
    ae974a485f413a2113503eed53cd6c53
    10.1142/S0218195911003597
    Scopus Count
    Collections
    Articles; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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