Abstract
We consider the problem of finding, for a given n point set P in the plane and an integer k ≤ n, the smallest circle enclosing at least k points of P. We present a randomized algorithm that computes in O( nk ) expected time such a circle, improving over all previously known algorithms. Since this problem is believed to require Ω(nk) time, we present a linear time δ-approximation algorithm that outputs a circle that contains at least k points of P, and of radius less than (1 + δ)r opt (P,K), where r opt (P,K) is the radius of the minimal disk containing at least k points of P. The expected running time of this approximation algorithm is \(O (n + n \cdot {\rm min}(\frac{1}{k\delta^3}{\rm log^2}\frac{1}{\delta},k))\).
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Har-Peled, S., Mazumdar, S. (2003). Fast Algorithms for Computing the Smallest k-Enclosing Disc. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_27
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DOI: https://doi.org/10.1007/978-3-540-39658-1_27
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