TAILIEUCHUNG - On reserve and double covering problems for the sets with non euclidean metrics
The article is devoted to Circle covering problem for a bounded set in a two- dimensional metric space with a given amount of circles. Here we focus on a more complex problem of constructing reserve and multiply coverings. Besides that, we consider the case where covering set is a multiply-connected domain. The numerical algorithms based on fundamental physical principles, established by Fermat and Huygens, is suggested and implemented. It allows us to solve the problems for the cases of non-convex sets and non-Euclidean metrics. | Yugoslav Journal of Operations Research xx (xxx), Number nn, zzz–zzz DOI: ON RESERVE AND DOUBLE COVERING PROBLEMS FOR THE SETS WITH NON-EUCLIDEAN METRICS Anna LEMPERT Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Science, Lermontova 134, Irkutsk, Russia lempert@ Alexander KAZAKOV Matrosov Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Science, Lermontova 134, Irkutsk, Russia kazakov@ Quang Mung LE Irkutsk National Research Technical University, Lermontova 83, Irkutsk, Russia quangmungle2010@ Received: November 2017 / Accepted: February 2018 Abstract: The article is devoted to Circle covering problem for a bounded set in a twodimensional metric space with a given amount of circles. Here we focus on a more complex problem of constructing reserve and multiply coverings. Besides that, we consider the case where covering set is a multiply-connected domain. The numerical algorithms based on fundamental physical principles, established by Fermat and Huygens, is suggested and implemented. It allows us to solve the problems for the cases of non-convex sets and non-Euclidean metrics. Preliminary results of numerical experiments are presented and discussed. Calculations show the applicability of the proposed approach. Keywords: Covering Problem, Fermat Principle, Huygens Principle, Wave Front, NonEuclidean Metric, Reserve Covering, Double Covering, Computational Experiment. MSC: 65K10, 90B06. 2 A. Kazakov, A. Lempert, . Le / On Reserve and Double Covering Problems . 1. INTRODUCTION Circle covering is the dual problem of circle packing problem. The covering problem means to locate congruent geometric objects in a metric space so that its given area lies entirely within their union. Usually, scientists deal with the circles covering problem (CCP) in special cases, when the covered area is a square, a circle, a rectangle
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