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

Abstract

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.