Abstract
In wireless communication networks, optimal use of the directional antenna is very important. The directional antenna coverage (DAC) problem is to cover all clients with the smallest number of directional antennas. In this paper, we consider the variable-size rectangle covering (VSRC) problem, which is a transformation of the DAC problem. There are n points above the base line y = 0 of the two-dimensional plane. The target is to cover all these points by minimum number of rectangles, such that the dimension of each rectangle is not fixed but the area is at most 1, and the bottom edge of each rectangle is on the base line y = 0. In some applications, the number of rectangles covering any position in the two-dimensional plane is bounded, so we also consider the variation when each position in the plane is covered by no more than two rectangles. We give two polynomial time algorithms for finding the optimal covering. Further, we propose two 2-approximation algorithms that use less running time (O(nlogn) and O(n)).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bao, L., Garia-Luna-Aceves, J.J.: Transmission Scheduling in Ad Hoc Networks with Directional Antennas. In: Proceedings of the 8th International Conference on Mobile computing and networking, pp. 48–58 (2002)
Berman, P., Jeong, J., Kasiviswanathan, S., Urgaonkar, B.: Packing to Angles and Sectors. In: Gibbons, P.B., Scheideler, C. (eds.) SPAA 2007, pp. 171–180 (2007)
Charikar, M., Chekuri, C., Feder, T., Motwani, R.: Incremental Clustering and Dynamic Information Retriveal. SIAM J. Comput. 33(6), 1417–1440 (2004)
Chan, T.M., Zarrabi-Zadeh, H.: A randomize Algorithm for Online Unit Clustering. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 121–131. Springer, Heidelberg (2007)
Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal Packing and Covering in the Plane are NP-Complete. Information Processing Letters 12(3), 133–137 (1981)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide o the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Hochbaum, D.S., Mass, W.: Approximation Scheme for Covering and Packing Problems in Image Processing and VLSI. JACM 32, 130–136 (1985)
Karmarkar, N., Karp, R.M.: An Efficient Approximation Scheme for the One-Dimensional Bin-Packing Problem. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, pp. 312–320 (1982)
Roy, S., Hu, Y., Peroulis, D., Li, X.Y.: Minimum-Energy Broadcast Using Practical Directional Antennas in All-Wireless Networks. In: Proc. of the 25th INFORCOM, pp. 1–12 (2006)
Seiden, S.: On the Online Bin Packing Problem. JACM 49(5), 640–671 (2002)
Spyropoulos, A., Raghavendra, C.: Energy Efficient Communication in Ad Hac Networks Using Directional Antennas. In: Proc. of the 21st INFORCOM, pp. 220–228 (2002)
Voipio, V., Vainikainen, P.: Narrowbeam Cylindrical Antenna Array with Sparse Antenna Spacing. In: Proc. of the 48th IEEE Vehicular Technology Conference, vol. 1, pp. 465–469 (1998)
Yi, S., Pei, Y., Kalyanaraman, S.: On the Capacity Improvement of Ad Hoc Wireless Networks Using Directional Antennas. In: Proceedings of the 9th International Conference on Mobile computing and networking, pp. 108–116 (2003)
Zarrabi-Zadeh, H., Chan, T.M.: An Improved Algorithm for Online Unit Clustering. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 383–393. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chin, F.Y.L., Ting, HF., Zhang, Y. (2009). Variable-Size Rectangle Covering. In: Du, DZ., Hu, X., Pardalos, P.M. (eds) Combinatorial Optimization and Applications. COCOA 2009. Lecture Notes in Computer Science, vol 5573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02026-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-02026-1_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02025-4
Online ISBN: 978-3-642-02026-1
eBook Packages: Computer ScienceComputer Science (R0)