Geodesic
On some problems of covering the plane with circles
Diskretnyj analiz i issledovanie operacij, Tome 21 (2014) no. 1, pp. 84-102.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study regular circle plane coverings in which the plane is split into regular polygons (tiles) and all the tiles are covered identically. The density of a regular covering can be calculated by dividing total area of circles covering a tile by the tile's area. We focus on regular coverings containing circles of four, five and six different radii. We prove optimality of several known coverings in their classes, find tight lower bounds for densities depending on radii of circles in a covering, and propose new coverings which are optimal in their classes under some additional constraints on radii. Ill. 14, bibliogr. 15.
Keywords: plane covering, density, regular covering, wireless sensor network. 
@article{DA_2014_21_1_a6,
     author = {I. I. Takhonov},
     title = {On some problems of covering the plane with circles},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {84--102},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2014_21_1_a6/}
}
TY  - JOUR
AU  - I. I. Takhonov
TI  - On some problems of covering the plane with circles
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2014
SP  - 84
EP  - 102
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DA_2014_21_1_a6/
LA  - ru
ID  - DA_2014_21_1_a6
ER  - 
%0 Journal Article
%A I. I. Takhonov
%T On some problems of covering the plane with circles
%J Diskretnyj analiz i issledovanie operacij
%D 2014
%P 84-102
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DA_2014_21_1_a6/
%G ru
%F DA_2014_21_1_a6
I. I. Takhonov. On some problems of covering the plane with circles. Diskretnyj analiz i issledovanie operacij, Tome 21 (2014) no. 1, pp. 84-102. http://geodesic.mathdoc.fr/item/DA_2014_21_1_a6/

[1] Aldyn-ool T. A., Erzin A. I., Zalyubovskii V. V., “Pokrytie ploskoi oblasti sluchaino raspredelennymi sensorami”, Vestn. NGU. Ser. Matematika, mekhanika, informatika, 10:4 (2010), 7–25 | Zbl

[2] Astrakov S. N., Erzin A. I., Zalyubovskii V. V., “Sensornye seti i pokrytie ploskosti krugami”, Diskret. analiz i issled. operatsii, 16:3 (2009), 3–19 | MR | Zbl

[3] Astrakov S. N., Erzin A. I., “Postroenie effektivnykh modelei pokrytiya pri monitoringe protyazhënnykh ob'ektov”, Vychisl. tekhnologii, 17:1 (2010), 26–34

[4] Erzin A. I., Plotnikov R. V., “Maksimizatsiya vremeni funktsionirovaniya pokrytii v usloviyakh ogranichennosti resursov ikh elementov”, Diskret. analiz i issled. operatsii, 18:6 (2011), 17–32 | MR | Zbl

[5] Erzin A. I., Plotnikov R. V., Shamardin Yu. V., “On some polynomially solvable cases and approximate algorithms in the optimal communication tree construction problem”, J. Appl. Industr. Math., 7:2 (2013), 142–152 | DOI | MR

[6] Tot L. F., Raspolozheniya na ploskosti, na sfere i v prostranstve, Fizmatgiz, M., 1958, 364 pp.

[7] Cardei M., Wu J., Lu M., “Improving netwotk lifetime using sensors with adjustible sensing ranges”, Int. J. Sensor Networks, 1:1–2 (2006), 41–49 | DOI

[8] Erzin A., Astrakov S., “Min-density stripe covering and applications in sensor networks”, Lect. Notes Comput. Sci., 6784, Springer-Verl., Berlin–Heidelberg, 2011, 152–162 | DOI | MR

[9] Erzin A., Plotnikov R., “Wireless sensor network's lifetime maximization problem in case of given set of covers”, Lect. Notes Comput. Sci., 6786, Springer-Verl., Berlin–Heidelberg, 2011, 44–57 | DOI

[10] Kershner R., “The number of circles covering a set”, Amer. J. Math., 61:3 (1939), 665–671 | DOI | MR | Zbl

[11] Nguen N. D., Zalyubovsky V., Ha M. Th., Le T. D., Choo H., “Energy-efficient models for coverage problem in sensor networks with adjustable ranges”, Ad Hoc Sensor Wireless Networks, 16 (2012), 1–28

[12] Toth F. G., “Covering the plane with two kinds of circles”, Discrete Comput. Geometry, 13:3 (1995), 445–457 | DOI | MR | Zbl

[13] Wu J., Dai F., “Virtual backbone construction in MANETs using adjustable transmission ranges”, IEEE Trans. Mobile Comput., 5:9 (2006), 1188–1200 | DOI

[14] Wu J., Yang S., “Energy-efficient node scheduling models in sensor networks with adjustible ranges”, Int. J. Foundat. Comput. Sci., 16:1 (2005), 3–17 | DOI | MR | Zbl

[15] Zalyubovskiy V., Erzin A., Astrakov S., Choo H., “Energy-efficient area coverage by sensors with adjustable ranges”, Sensors, 9 (2009), 2446–2460 | DOI