Abstract
Given a set P of n points in ℝd and an integer k ≥ 1, let w* denote the minimum value so that P can be covered by k congruent cylinders of radius w*. We describe a randomized algorithm that, given P and an ε > 0, computes k cylinders of radius (1 + ε) w* that cover P. The expected running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ε. We first show that there exists a small ”certificate” Q ⫅ P, whose size does not depend on n, such that for any k congruent cylinders that cover Q, an expansion of these cylinders by a factor of (1 + ε) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.
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Agarwal, P., Procopiuc, C. & Varadarajan, K. Approximation Algorithms for a k-Line Center. Algorithmica 42, 221–230 (2005). https://doi.org/10.1007/s00453-005-1166-x
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DOI: https://doi.org/10.1007/s00453-005-1166-x