## Two algorithms for the minimum enclosing ball problem

 dc.citation.epage 1391 en_US dc.citation.issueNumber 3 en_US dc.citation.spage 1368 en_US dc.citation.volumeNumber 19 en_US dc.contributor.author Yıldırım, E. A. en_US dc.date.accessioned 2016-02-08T10:06:24Z dc.date.available 2016-02-08T10:06:24Z dc.date.issued 2008 en_US dc.department Department of Industrial Engineering en_US dc.description.abstract Given A := {a1.....am} ⊂ ℝn and ε > 0, we propose and analyze two algorithms for the problem of computing a (1 + ε)-approximation to the radius of the minimum enclosing ball of A. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of the minimum enclosing ball problem. We establish that this algorithm converges in O(1/ε) iterations with an overall complexity bound of O(mn/ε) arithmetic operations. In addition, the algorithm returns a "core set" of size O(1/ε), which is independent of both m and n. The latter algorithm is obtained by incorporating "away" steps into the former one at each iteration and achieves the same asymptotic complexity bound as the first one. While the asymptotic bound on the size of the core set returned by the second algorithm also remains the same as the first one, the latter algorithm has the potential to compute even smaller core sets in practice, since, in contrast to the former one, it allows "dropping" points from the working core set at each iteration. Our analysis reveals that the leading terms in the asymptotic complexity analysis are reasonably small. In contrast to the first algorithm, we also establish that the second algorithm asymptotically exhibits linear convergence, which provides further insight into our computational results, indicating that the latter algorithm indeed terminates faster with smaller core sets in comparison with the first one. We also discuss how our algorithms can be extended to compute an approximation to the minimum enclosing ball of more general input sets without sacrificing the iteration complexity and the bound on the core set size. In particular, we establish the existence of a core set of size O(1/ε) for a much wider class of input sets. We adopt the real number model of computation in our analysis. en_US dc.identifier.doi 10.1137/070690419 en_US dc.identifier.eissn 1095-7189 dc.identifier.issn 1052-6234 dc.identifier.uri http://hdl.handle.net/11693/22911 dc.language.iso English en_US dc.publisher Society for Industrial and Applied Mathematics en_US dc.relation.isversionof http://dx.doi.org/10.1137/070690419 en_US dc.source.title SIAM Journal on Optimization en_US dc.subject Minimum enclosing balls en_US dc.subject Core sets en_US dc.subject Approximation algorithms en_US dc.title Two algorithms for the minimum enclosing ball problem en_US dc.type Article en_US
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