Abstract
We show that any k-fold covering using translates of an arbitrary convex polygon can be decomposed into Ω(k) covers. Such a decomposition can be computed using an efficient (polynomial-time) algorithm.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Orden, D., Ramos, P.: Decomposition of multiple coverings into more parts. In: SODA’09: Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 302–310. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Buchsbaum, A.L., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: SODA’07: Proceedings of the Eighteenth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 1056–1063. Society for Industrial and Applied Mathematics, Philadelphia (2007)
Mani, P., Pach, J.: Decomposition problems for multiple coverings with unit balls. Manuscript (1986)
Pach, J.: Covering the plane with convex polygons. Discrete Comput. Geom. 1, 73–81 (1986)
Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. Comput. Geom. 42(2), 127–133 (2009)
Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT. Lecture Notes in Computer Science, vol. 4381, pp. 135–148. Springer, Berlin (2005)
Pálvölgyi, D.: Indecomposable coverings with concave polygons. Discrete Comput. Geom. 44(3), 577–588 (2010). doi:10.1007/s00454-009-9194-y
Pálvölgyi, D., Tóth, G.: Convex polygons are cover-decomposable. Discrete Comput. Geom. 43(3), 483–496 (2010)
Pandit, S., Pemmaraju, S.V., Varadarajan, K.R.: Approximation algorithms for domatic partitions of unit disk graphs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J.D.P. (eds.) APPROX-RANDOM. Lecture Notes in Computer Science, vol. 5687, pp. 312–325. Springer, Berlin (2009)
Pemmaraju, S.V., Pirwani, I.A.: Energy conservation via domatic partitions. In: MobiHoc’06: Proceedings of the 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 143–154. ACM, New York (2006)
Tardos, G., Tóth, G.: Multiple coverings of the plane with triangles. Discrete Comput. Geom. 38(2), 443–450 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of the results in this article appears in Gibson and Varadarajan, Decomposing Coverings and the Planar Sensor Cover Problem, Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), 2009.
Rights and permissions
About this article
Cite this article
Gibson, M., Varadarajan, K. Optimally Decomposing Coverings with Translates of a Convex Polygon. Discrete Comput Geom 46, 313–333 (2011). https://doi.org/10.1007/s00454-011-9353-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-011-9353-9