## Abstract

We study the problem of assigning transmission ranges to radio stations in the plane such that any pair of stations can communicate within a bounded number of hops *h* and the cost of the network is minimized. We consider two settings of the problem: collinear station locations and arbitrary locations. For the case of collinear stations, we introduce the pioneer polynomial-time exact algorithm for any \(\alpha \ge 1\) and constant *h*, and thus conclude that the 1D version of the problem, where *h* is a constant, is in \(\mathcal {P}\). Furthermore, we provide a 3 / 2-approximation algorithm for the case where *h* is an arbitrary number and \(\alpha =1\), improving the previously best known approximation ratio of 2. For the case of stations placed arbitrarily in the plane and \(\alpha =1\), we first present a \((1.5+ \varepsilon )\)-approximation algorithm for a case where a deviation of one hop (\(h+1\) hops in total) is acceptable. Then, we show a \((3+\varepsilon )\)-approximation algorithm that complies with the exact hop bound. To achieve that, we introduce the following two auxiliary problems, which are of independent interest. The first is the *bounded-hop multi-sink range* problem, for which we present a PTAS which can be applied to compute a \((1+\varepsilon )\)-approximation for the bounded diameter minimum spanning tree, for any \(\varepsilon >0\). The second auxiliary problem is the *bounded-hop dual-sink pruning* problem, for which we show a polynomial-time algorithm. To conclude, we consider the initial bounded-hop all-to-all range assignment problem and show that a combined application of the aforementioned problems yields the \((3+\varepsilon )\)-approximation ratio for this problem, which improves the previously best known approximation ratio of \(4(9^{h-2})/(\root h \of {2}-1)\).

## Keywords

Computational geometry Approximation algorithms Wireless networks## References

- 1.Alt, H., Arkin, E.M., Brönnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proceedings of the 22nd ACM Symposium on Computational Geometry, SOCG’06, pp. 449–458 (2006)Google Scholar
- 2.Ambühl, C., Clementi, A.E.F., Penna, P., Rossi, G., Silvestri, R.: On the approximability of the range assignment problem on radio networks in presence of selfish agents. Theor. Comput. Sci.
**343**(1–2), 27–41 (2005)MathSciNetCrossRefMATHGoogle Scholar - 3.Arora, S., Raghavan, P., Rao, S.: Approximation schemes for euclidean
*k*-medians and related problems. In: Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, STOC’98, pp. 106–113 (1998)Google Scholar - 4.Călinescu, G., Kapoor, S., Sarwat, M.: Bounded-hops power assignment in ad-hoc wireless networks. In: Proceedings of the IEEE Wireless Communications and Networking Conference, WCNC’04, pp. 1494–1499 (2004)Google Scholar
- 5.Chambers, E.W., Fekete, S.P., Hoffmann, H.F., Marinakis, D., Mitchell, J.S.B., Srinivasan, V., Stege, U., Whitesides, S.: Connecting a set of circles with minimum sum of radii. In: Proceedings of the 12th International Symposium on Algorithms and Data Structures, WADS’11, pp. 183–194 (2011)Google Scholar
- 6.Chen, J., Salim, M.B., Matsumoto, M.: Modeling the energy performance of object tracking in wireless sensor network using dual-sink. In: Proceedings of the 16th Asia-Pacific Conference on Communications, APCC’10, pp. 204–209 (2010)Google Scholar
- 7.Clementi, A.E.F., Di Ianni, M., Silvestri, R.: The minimum broadcast range assignment problem on linear multi-hop wireless networks. Theor. Comput. Sci.
**1–3**(299), 751–761 (2003)MathSciNetCrossRefMATHGoogle Scholar - 8.Clementi, A.E.F., Penna, P., Ferreira, A., Perennes, S., Silvestri, R.: The minimum range assignment problem on linear radio networks. Algorithmica
**35**(2), 95–110 (2003)MathSciNetCrossRefMATHGoogle Scholar - 9.Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., Bedford (1990)MATHGoogle Scholar
- 10.Gouvy, N., Elhafsi, E.H., Mitton, N., Zorbas, D.: Energy efficient multi-flow routing in mobile sensor networks. In: IEEE Wireless Communications and Networking Conference, WCNC’13, pp. 1968–1973 (2013)Google Scholar
- 11.Gruber, M., Raidl, G.R.: Solving the euclidean bounded diameter minimum spanning tree problem by clustering-based (meta-)heuristics. In: Revised Selected Papers of the 12th International Conference on Computer Aided Systems Theory, EUROCAST’09, pp. 665–672 (2009)Google Scholar
- 12.Kantor, E., Peleg, D.: Approximate hierarchical facility location and applications to the bounded depth steiner tree and range assignment problems. J. Discrete Algorithms
**7**(3), 341–362 (2009)MathSciNetCrossRefMATHGoogle Scholar - 13.Laue, S., Matijevic, D.: Approximating k-hop minimum spanning trees in Euclidean metrics. Inf. Process. Lett.
**107**(3–4), 96–101 (2008)MathSciNetCrossRefMATHGoogle Scholar - 14.Lev-Tov, N., Peleg, D.: Exact algorithms and approximation schemes for base station placement problems. In: Proceedings of the 8th Scandinavian Workshop on Algorithm Theory, Turku, SWAT’02, pp. 90–99 (2002)Google Scholar
- 15.Lev-Tov, N., Peleg, D.: Polynomial time approximation schemes for base station coverage with minimum total radii. Comput. Netw.
**47**(4), 489–501 (2005)CrossRefMATHGoogle Scholar - 16.Mitton, N., Simplot-Ryl, D., Zheng, J.: Guaranteed delivery in k-anycast routing in multi-sinkwireless networks. Trans. Mobile Commun. Appl.
**3**, e1 (2013)Google Scholar - 17.Pahlavan, K.: Wireless information networks. Wiley, New York (2005)CrossRefGoogle Scholar
- 18.Pellenz, M.E., Jamhour, E., Penna, M.C., Souza, R.D., de Oliveira Brante, G.G.: A power assignment method for multi-sink WSN with outage probability constraints. In: Proceedings of the 28th IEEE International Conference on Advanced Information Networking and Applications, AINA’14, pp. 533–540 (2014)Google Scholar
- 19.Pouryazdanpanah, M.K., Anjomshoa, M., Salehi, A.S., Afroozeh, A., Moshfegh, M.G.: DS-VBF: dual sink vector-based routing protocol for underwater wireless sensor network. In: Proceedings of the 5th IEEE Control and System Graduate Research Colloquium, ICSGRC’14, pp. 227–232 (2014)Google Scholar
- 20.Sen, A., Das Gupta, M., De D.: Energy efficient layered cluster based hierarchical routing protocol with dual sink. In: Proceedings of the 5th International Conference on Computers and Devices for Communication, CODEC 2012, pp. 1–4 (2012)Google Scholar
- 21.Wang, X., Wang, J., Lu, K., Xu, Y.: GKAR: a novel geographic \({K}\)-anycast routing for wireless sensor networks. IEEE Trans. Parallel Distrib. Syst.
**24**(5), 916–925 (2013)CrossRefGoogle Scholar - 22.Wu, X., Chen, G.: Dual-sink: using mobile and static sinks for lifetime improvement in wireless sensor networks. In: Proceedings of the 16th IEEE International Conference on Computer Communications and Networks, ICCCN’07, pp. 1297–1302 (2007)Google Scholar