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## Archive of Issues

Russia Yekaterinburg
Year
2016
Volume
26
Issue
2
Pages
258-270
 Section Mathematics Title Algorithms of optimal set covering on the planar $\mathbb{R}^2$ Author(-s) Ushakov V.N.a, Lebedev P.D.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa Abstract The problem of optimal covering of planar convex sets with a union of a given number $n$ of equal disks is studied. Criterion of optimality is a minimization of disks' radius, which gives an opportunity to reduce the optimization problem to a construction of the best Chebyshev $n$-net of a convex set. Numerical methods based on dividing the set into Dirichlet zones and finding characteristic points are suggested and proved in the present paper. One of the main elements of the methods is a Chebyshev center calculation for a compact convex set. Stochastic algorithms for generating an initial position of the $n$-net points are presented. Modeling of some examples is computed and visualization of the constructed covering is realized. Keywords disk covering, best Chebyshev net, Chebyshev center, Dirichlet zone, characteristic points, closed curve UDC 514.174.3 MSC 05B40 DOI 10.20537/vm160212 Received 15 April 2016 Language Russian Citation Ushakov V.N., Lebedev P.D. Algorithms of optimal set covering on the planar $\mathbb{R}^2$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 258-270. References Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Мoscow: Nauka, 1974, 456 p. Ushakov V.N., Lebedev P.D., Matviychuk A.R., Malev A.G. Differential games with fixed terminal time and estimation of the instability degree of sets in these games, Proceedings of the Steklov Institute of Mathematics, 2012, vol. 277, issue 1, pp. 266-277. Mestetskii L.M. 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