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 cylinders of radius at most w*. We describe an algorithm that, given P and an ɛ > 0, computes k cylinders of radius at most (1 + ɛ)w* that cover P. The running time of the algorithm is O(n log n), with the constant of proportionality depending on k, d, and ɛ. We first show that there exists a small “certificate” Q ⊆ P, whose size does not depend on n, such that for any k-cylinders that cover Q, an expansion of these cylinders by a factor of (1+ɛ) covers P. We then use a well-known scheme based on sampling and iterated re-weighting for computing the cylinders.
Research by the first author is supported by NSF under grants CCR-00-86013 ITR- 333-1050, EIA-98-70724, EIA-01-31905, and CCR-97-32787, and by a grant from the U.S.-Israel Binational Science Foundation.
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Agarwal, P.K., Procopiuc, C.M., Varadarajan, K.R. (2002). Approximation Algorithms for k-Line Center. In: Möhring, R., Raman, R. (eds) Algorithms — ESA 2002. ESA 2002. Lecture Notes in Computer Science, vol 2461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45749-6_9
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