On the Algorithm for Equal Balls Packing into a Multi-connected Set

Abstract

The paper is devoted to the problem of densest packing a given number of equal balls into multi-connected containers. The objective is to find the maximum radius associated with balls. We consider the problem both in three-dimensional Euclidean and one class of non-Euclidean spaces. In this study, the distance between points means the minimum time required to overcome the path between them. The algorithm based on the optical-geometric approach and billiard simulation combination is suggested and implemented. The idea is the following: we initially construct sufficiently small balls and enlarge them step by step as long as all the balls keep inside without overlaps. Computational experiments show the applicability and validity of the method.

Packing spheres with quasi-containment conditions

“…Correspondingly, the sphere packing problem is approximated by a binary optimization problem to select grid nodes for the centers of the spheres [21,22]. Many approaches like [3,5,9,[11][12][13]19] combine several techniques. Eckard Specht's website [33] provides instances of benchmark problems with known (approximate) solutions and their objective values, see also [23].…”