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		<doi>10.1109/SIBGRAPI.2006.20</doi>
		<citationkey>MartinetzMadaMota:2006:FaEaCo</citationkey>
		<title>Fast and Easy Computation of Approximate Smallest Enclosing Balls</title>
		<format>On-line</format>
		<year>2006</year>
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		<author>Martinetz, Thomas,</author>
		<author>Madany Mamlouk, Amir,</author>
		<author>Mota, Cicero,</author>
		<affiliation>Institute for Neuro- and Bioinformatics, University of L uebeck</affiliation>
		<affiliation>Institute for Neuro- and Bioinformatics, University of L uebeck</affiliation>
		<affiliation>Departamento de Matemática, Universidade Federal do Amazonas</affiliation>
		<editor>Oliveira Neto, Manuel Menezes de,</editor>
		<editor>Carceroni, Rodrigo Lima,</editor>
		<e-mailaddress>cicmota@gmail.com</e-mailaddress>
		<conferencename>Brazilian Symposium on Computer Graphics and Image Processing, 19 (SIBGRAPI)</conferencename>
		<conferencelocation>Manaus, AM, Brazil</conferencelocation>
		<date>8-11 Oct. 2006</date>
		<publisher>IEEE Computer Society</publisher>
		<publisheraddress>Los Alamitos</publisheraddress>
		<booktitle>Proceedings</booktitle>
		<tertiarytype>Full Paper</tertiarytype>
		<transferableflag>1</transferableflag>
		<versiontype>finaldraft</versiontype>
		<keywords>computational geometry, smallest enclosing ball, pattern recognition.</keywords>
		<abstract>The incremental Badoiu-Clarkson algorithm finds the smallest ball enclosing n point in d dimensions with at least O(1/t^0.5) precision, after t iteration steps. The extremely simple incremental step of the algorithm makes it very attractive both for theoreticians and practitioners. A simplified proof for this convergence is given. This proof allows to show that the precision increases, in fact, even as O(u/t) with the number of iteration steps. Computer experiments, but not yet a proof, suggest that the u, which depends only on the data instance, is actually bounded by min{(2d)^0.5,(2n)^0.5}. If it holds, then the algorithm finds the smallest enclosing ball with epsilon precision in at most O(nd (d')^0.5 }/epsilon) time, with d=min{d,n}.</abstract>
		<language>en</language>
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