P.D. Lebedev, A.L. Kazakov. Construction of optimal covers by disks of different radii for convex planar sets ... P. 137-148

We consider the problem of constructing an optimal cover of a planar set $M$ by the union of a given number of disks. In the general case, the radii of the disks are assumed to be different; each radius is the product of some positive factor specific for each disk and a parameter $r$, which is common for all elements of the cover. The optimality criterion is the minimum of $r$ under the condition that $M$ is a subset of the union of the disks. For a set of points $S$, we write the value of $r$ that defines the minimum radius of the disks centered at the points of $S$ and implementing a cover of $M$. Expressions are found that analytically describe the impact zones (the so-called generalized Dirichlet zones) of the points of $S$, which differ significantly from the expressions for the case of congruent circles. A procedure for the iterative correction of coordinates of $S$ based on finding Chebyshev centers of impact zones of points is proposed. It is shown that the procedure does not degrade the properties of the cover, while its parameters can be changed in the process of starting the software complex. Numerical experiments on the construction of optimal covers by families of disks were carried out with different coefficients defining the radii of the disks. Various convex polygons were taken as the set $M$, and the results were visualized.

Keywords: optimal cover, generalized Dirichlet zone, Chebyshev center, iterative algorithm, minimization

Received April 2, 2019

Revised May 14, 2019

Accepted May 20, 2019

Funding Agency:Theorems 1 and 2 were proved by Lebedev with support from the Russian Foundation for Basic Research (project nos. 18-01-00221 and 18-31-00018 mol_a) and from the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University). The computational experiment was carried out by Kazakov with support from the Russian Foundation for Basic Research (project no. 18-07-00604).

Pavel Dmitrievich Lebedev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia,
e-mail: pleb@yandex.ru

Aleksandr Leonidovich Kazakov, Dr. Phys.-Math. Sci., Matrosov Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences, Irkutsk, 664033 Russia,
e-mail: kazakov@icc.ru

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Cite this article as: P.D.Lebedev, A.L.Kazakov. Construction of optimal covers by disks of different radii for convex planar sets, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 137–148.