Abstract
The paper proposes algorithms for the iterative construction of optimal coverings of nonconvex flat figures using sets of circles. These algorithms are based on the procedures of dividing the figure into zones of influence of points that serve as the centers of the initial coverings and finding the Chebyshev centers of these zones. To generate the initial array of points, we use stochastic procedures based on the synthesis of optimal hexagonal grids and random vectors.
The work was supported by the Decree no. 211 of the Government of the Russian Federation, contract no. 02.A03.21.0006.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Kazakov, A.L., Lebedev, P.D.: Algorithms for constructing optimal \(n\)-networks in metric spaces. Autom. Remote Control 78(7), 1290–1301 (2017)
Melissen, H.: Densest packings of eleven congruent circles in a circle. Geom. Dedicata. 50(1), 15–25 (1994)
Heppes, A., Melissen, H.: Covering a rectangle with equal circles. Period. Math. Hung. 34(1–2), 65–81 (1997)
Kazakov, A.L., Lempert, A.A., Bukharov, D.S.: On segmenting logistical zones for servicing continuously developed consumers. Autom. Remote Control 74(6), 968–977 (2013)
Desyatov, V.G.: Proektirovanie Sistem Ob’ectov Obshchestvennogo Kompleksa Promyshlennykh Predpriyatiy (Systems Design for Public Service Objects of Industrial Plants). MARKhI, Moscow (1989)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1988)
Dem’yanov, V.F., Vasil’ev, L.V.: Nondifferentiable Optimization. Nauka, Moscow (1981); Springer, New York (1985)
Hausdorff, F.: Set Theory. Chelsea Publishing Co., New York (1962); Komkniga, Moscow (2006)
Garkavi, A.L.: On the existence of an optimal network and best diameter of a set in a Banach space. Usp. Mat. Nauk 15(2), 210–211 (1960). (in Russian)
Garkavi, A.L.: The best possible net and the best possible cross-section of a set in a normed space. Am. Math. Soc. Transl. II. Ser. 39, 111–132 (1964); transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 26(1), 87–106 (1962)
Mestetskiy, L.M.: Continuous Morphology of Binary Images. Figures, skeletons, Circular. Fizmatlit, Moscow (2009). (in Russian)
Brusov, V.S., Piyavskii, S.A.: A computational algorithm for optimally covering a plane region. USSR Comput. Math. Math. Phys. 11(2), 17–27 (1971)
Lebedev, P.D., Uspenskii, A.A., Ushakov, V.N.: Algorithms of the best approximations of the flat sets by the union of circles. Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki (4), 88–99 (2013). (in Russian)
Ushakov, V.N., Lakhtin, A.S., Lebedev, P.D.: Optimization of the Hausdorff distance between sets in Euclidean space. Proc. Steklov Inst. Math. 291(1), S222–S238 (2015)
Ushakov, V.N., Lebedev, P.D.: Algorithms for the construction of an optimal cover for sets in three-simensional Euclidean space. Proc. Steklov Inst. Math. 293(1), S225–S237 (2016)
Ushakov, V.N., Lebedev, P.D.: Algorithms of optimal set covering on the planar \(\mathbf{R}^2\). Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 26(2), 258–270 (2016). (in Russian)
Piyavskii, S.A.: On optimization of networks. Izv. Akad. Nauk SSSR Tekh. Kibern. (1), 68–80 (1968). (in Russian)
Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin (1953); Fizmatlit, Moscow (1958)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Lebedev, P., Ushakov, V. (2019). Iterative Methods for Constructing Approximations to Optimal Coverings of Nonconvex Polygons. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-33394-2_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-33393-5
Online ISBN: 978-3-030-33394-2
eBook Packages: Computer ScienceComputer Science (R0)