On Metric Clustering to Minimize the Sum of Radii

Authors:
Matt Gibson;Gaurav Kanade;Erik Krohn;Imran A. Pirwani;Kasturi Varadarajan
Affiliations:
University of Iowa, Department of Computer Science, 52242-1419, Iowa City, IA, USA;University of Iowa, Department of Computer Science, 52242-1419, Iowa City, IA, USA;University of Iowa, Department of Computer Science, 52242-1419, Iowa City, IA, USA;University of Alberta, Department of Computing Science, T6G 2E8, Edmonton, Alberta, Canada;University of Iowa, Department of Computer Science, 52242-1419, Iowa City, IA, USA
Venue:
Algorithmica - Special Issue: Scandinavian Workshop on Algorithm Theory; Guest Editor: Joachim Gudmundsson
Year:
2010

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Abstract

Given an n-point metric (P,d) and an integer k0, we consider the problem of covering P by k balls so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in n O(log n⋅log Δ) time and returns with high probability the optimal solution. Here, Δ is the ratio between the maximum and minimum interpoint distances in the metric space. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and in metrics of constant doubling dimension.