Algorithms for Constructing Optimal Covering of Planar Figures with Disks Sets of Linearly Different Radii

• Published in: Izvestiâ Irkutskogo gosudarstvennogo universiteta. Seriâ "Matematika" (Online) Vol. 46; no. 1; pp. 35 - 50
• Main Authors: Lebedev, P. D., Stoychin, K. L.
• Format: Journal Article
• Language: English
• Published: Irkutsk State University 01-01-2023
• Subjects: chebyshev center; dirichlet zone; disks coverage; domain of dominance; minimization
• ISSN: 1997-7670; 2541-8785
• DOI: 10.26516/1997-7670.2023.46.35

The problem of optimal covering of plane figures with sets of a fixed number of different circles is considered. We suppose that each circle has a radius equal to the sum of the parameter common to all and its individual number. The main aim of the paper is to develop algorithms that allow the construction of a covering with a minimum common parameter. It is proved that the problem can be reduced to minimizing a function of several variables depending on the coordinates of the centers of the circles. The zones of influence of points serving as the centers of circles for a fixed set of individual numbers have been studied. Iterative algorithm for solving the problem is proposed using the concepts of the Chebyshev center and a generalization of the Dirichlet zone. The possibilities of applying the results of the article to the construction of sensor networks are shown.