Abstract
Barrier coverage, as one of the most important applications of wireless sensor network (WSNs), is to provide coverage for the boundary of a target region. We study the barrier coverage problem by using a set of n sensors with adjustable coverage radii deployed along a line interval or circle. Our goal is to determine a range assignment \(\mathbf {R}=({r_{1}},{r_{2}}, \ldots , {r_{n}})\) of sensors such that the line interval or circle is fully covered and its total cost \(C(\mathbf {R})=\sum _{i=1}^n {r_{i}}^\alpha \) is minimized. For the line interval case, we formulate the barrier coverage problem of line-based offsets deployment, and present two approximation algorithms to solve it. One is an approximation algorithm of ratio 4 / 3 runs in \(O(n^{2})\) time, while the other is a fully polynomial time approximation scheme (FPTAS) of computational complexity \(O(\frac{n^{2}}{\epsilon })\). For the circle case, we optimally solve it when \(\alpha = 1\) and present a \(2(\frac{\pi }{2})^\alpha \)-approximation algorithm when \(\alpha > 1\). Besides, we propose an integer linear programming (ILP) to minimize the total cost of the barrier coverage problem such that each point of the line interval is covered by at least k sensors.
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References
Andrews AM, Wang H (2017) Minimizing the aggregate movements for interval coverage. Algorithmica 78(1):47–85
Arora A, Dutta P, Bapat S, Kulathumani V, Zhang H, Naik V, Mittal V, Cao H, Demirbas M, Gouda M et al (2004) A line in the sand: a wireless sensor network for target detection, classification, and tracking. Comput Netw 46(5):605–634
Bar-Noy A, Baumer B (2011) Maximizing network lifetime on the line with adjustable sensing ranges. In: Proceedings of the 7th international workshop on algorithms for sensor systems, wireless ad hoc networks, and autonomous mobile entities. Springer, pp 28–41
Bar-Noy A, Rawitz D, Terlecky P (2017) Maximizing barrier coverage lifetime with mobile sensors. SIAM J Discrete Math 31(1):573–596
Carmi P, Katz M, Lev-Tov N (2007) Covering points by unit disks of fixed location. In: Proceedings of 18th international symposium algorithms and computation. Springer, pp 644–655
Chen D, Gu Y, Li J, Wang H (2013) Algorithms on minimizing the maximum sensor movement for barrier coverage of a linear domain. Discrete Comput Geom 50:374–408
Chen DZ, Tan X, Wang H, Wu G (2015) Optimal point movement for covering circular regions. Algorithmica 72(2):379–399
Czyzowicz J, Kranakis E, Krizanc D, Lambadaris I, Narayanan L, Opatrny J, Stacho L, Urrutia J, Yazdani M (2010) On minimizing the sum of sensor movements for barrier coverage of a line segment. In: Proceedings of the 9th international conference on ad-hoc, mobile and wireless networks, pp 29–42
de Rezende P, Miyazawa F, Sasaki A (2013) A ptas for the disk cover problem of geometric objects. Oper Res Lett 41(5):552–555
Dobrev S, Kranakis E, Krizanc D, Lafond M, Maňuch J, Narayanan L, Opatrny J, Shende S, Stacho L (2017) Weak coverage of a rectangular barrier. In: International conference on algorithms and complexity. Springer, pp 196–208
Fan H, Lee VC, Li M, Zhang X, Zhao Y (2014a) Barrier coverage using sensors with offsets. In: International conference on wireless algorithms, systems, and applications. Springer, pp 389–400
Fan H, Li M, Sun X, Wan PJ, Zhao Y (2014b) Barrier coverage by sensors with adjustable ranges. ACM Trans Sensor Netw 11(1):14
Huang Y, Gao X, Zhang Z, Wu W (2009) A better constant-factor approximation for weighted dominating set in unit disk graph. J Comb Optim 18(2):179–194
Kumar S, Lai T, Arora A (2005) Barrier coverage with wireless sensors. In: Proceedings of the 11th international conference on mobile computing and networking. ACM, pp 284–298
Kumar S, Lai T, Arora A (2007) Barrier coverage with wireless sensors. Wirel Netw 13(6):817–834
Pananjady A, Bagaria VK, Vaze R (2017) Optimally approximating the coverage lifetime of wireless sensor networks. IEEE/ACM Trans Netw 25(1):98–111
Saipulla A, Westphal C, Liu B, Wang J (2009) Barrier coverage of line-based deployed wireless sensor networks. In: Proceedings of the 28th IEEE international conference on computer communications. IEEE, pp 127–135
Wan PJ, Yi CW (2006) Coverage by randomly deployed wireless sensor networks. IEEE/ACM Trans Netw 14(SI):2658–2669
Wan PJ, Xu X, Wang Z (2011) Wireless coverage with disparate ranges. In: Proceedings of the 12th ACM international symposium on mobile ad hoc networking and computing. ACM, pp 1–8
Wan PJ, Chen D, Dai G, Wang Z, Yao F (2012) Maximizing capacity with power control under physical interference model in duplex mode. In: Proceedings of the 31st IEEE international conference on computer communications. IEEE, pp 415–423
Wang B, Xu H, Liu W, Liang H (2013) A novel node placement for long belt coverage in wireless networks. IEEE Trans Comput 62(12):2341–2353
Wang H, Zhang X (2015) Minimizing the maximum moving cost of interval coverage. In: Proceedings of the 26th international symposium on algorithms and computation, pp 188–198
Xie K, Ning X, Wang X, He S, Ning Z, Liu X, Wen J, Qin Z (2017) An efficient privacy-preserving compressive data gathering scheme in wsns. Inf Sci 390:82–94
Zou F, Wang Y, Xu XH, Li X, Du H, Wan PJ, Wu W (2011) New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs. Theor Comput Sci 412(3):198–208
Acknowledgements
The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. UGC/FDS11/E04/15) and National Natural Science Foundation of China (No. 61772154).
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Zhang, X., Fan, H., Lee, V.C.S. et al. Minimizing the total cost of barrier coverage in a linear domain. J Comb Optim 36, 434–457 (2018). https://doi.org/10.1007/s10878-018-0306-6
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DOI: https://doi.org/10.1007/s10878-018-0306-6