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A linear time deterministic algorithm to find a small subset that approximates the centroid. (English) Zbl 1184.68575

Summary: Given a set of points \(S\) in any dimension, we describe a deterministic algorithm for finding a \(T \subset S, |T| = O(1/\varepsilon)\) such that the centroid of \(T\) approximates the centroid of \(S\) within a factor \(1+\varepsilon \) for any fixed \(\varepsilon >0\). We achieve this in linear time by an efficient derandomization of the algorithm in [M. Inaba, N. Katoh and H. Imai, in: Proceedings of the Tenth Annual Symposium on Computational Geometry, 332–339 (1991)].

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68W05 Nonnumerical algorithms
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References:

[1] A. Kumar, Y. Sabharwal, S. Sen, A simple linear time \((1 + \operatorname{\\(\#\#\#\#\\)})k\); A. Kumar, Y. Sabharwal, S. Sen, A simple linear time \((1 + \operatorname{\\(\#\#\#\#\\)})k\)
[2] M. Inaba, N. Katoh, H. Imai, Applications of weighted Voronoi diagrams and randomization to variance-based \(k\); M. Inaba, N. Katoh, H. Imai, Applications of weighted Voronoi diagrams and randomization to variance-based \(k\)
[3] Feigenbaum, J.; Kannan, S.; McGregor; Suri, S.; Zhang, J., On graph problems in a semi-streaming problem, Theoretical Computer Science, 348, 2 (2005) · Zbl 1081.68069
[4] Motwani, R.; Raghavan, P., Randomized Algorithms (2000), Cambridge University Press
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