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Algorithms for the construction of an optimal cover for sets in three-dimensional Euclidean space

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Abstract

The problem of an optimal cover of sets in three-dimensional Euclidian space by the union of a fixed number of equal balls, where the optimality criterion is the radius of the balls, is studied. Analytical and numerical algorithms based on the division of a set into Dirichlet domains and finding their Chebyshev centers are suggested for this problem. Stochastic iterative procedures are used. Bounds for the asymptotics of the radii of the balls as their number tends to infinity are obtained. The simulation of several examples is performed and their visualization is presented.

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    V. N. Ushakov & P. D. Lebedev

  2. Ural Federal University, ul. Mira 32, Yekaterinburg, 620002, Russia

    V. N. Ushakov

Authors
  1. V. N. Ushakov
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  2. P. D. Lebedev
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Correspondence to V. N. Ushakov.

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Original Russian Text © V.N. Ushakov, P.D. Lebedev, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 2.

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Ushakov, V.N., Lebedev, P.D. Algorithms for the construction of an optimal cover for sets in three-dimensional Euclidean space. Proc. Steklov Inst. Math. 293 (Suppl 1), 225–237 (2016). https://doi.org/10.1134/S0081543816050205

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