Abstract
In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d≥2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, for any fixed instance, as a limit for α going to infinity, the ratio tends to the lower bound of Clementi et al. (Proceedings of the 18th annual symposium on theoretical aspects of computer science (STACS), pp. 121–131, 2001), Wan et al. (Wirel. Netw. 8(6):607–617, 2002) given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.45-approximation ratio for the 2-dimensional case, thus significantly decreasing the previously known 12 upper bound (Wan et al. in Wirel. Netw. 8(6):607–617, 2002) (actually corrected to 12.15 in Klasing et al. (Proceedings of the 3rd IFIP-TC6 international networking conference, pp. 866–877, 2004)). Finally, we extend our analysis to any number of dimensions d≥2 and any α≥d, obtaining a general approximation ratio of 3d−1, again independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d.
Similar content being viewed by others
References
Calinescu, G., Kapoor, S., Olshevsky, A., Zelikovsky, A.: Network lifetime and power assignment in ad-hoc wireless networks. In: Proceedings of the 11th Annual European Symposium on Algorithms (ESA). Lecture Notes in Computer Science, vol. 2832, pp. 114–126. Springer, New York (2003)
Calinescu, G., Mandoiu, I., Zelikovsky, A.: Symmetric connectivity with minimum power consumption in radio networks. In: Proceedings of the 2nd IFIP International Conference on Theoretical Computer Science, pp. 119–130. Kluwer, Dordrecht (2002)
Caragiannis, I., Kaklamanis, C., Kanellopoulos, P.: New results for energy-efficient broadcasting in wireless networks. In: Proceedings of the 13th Annual International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 2518, pp. 332–343. Springer, New York (2002)
Caragiannis, I., Kaklamanis, C., Kanellopoulos, P.: Energy-efficient wireless network design. In: Proceedings of the 14th Annual International Symposium on Algorithms and Computation (ISAAC). Lecture Notes in Computer Science, vol. 2906, pp. 585–594. Springer, New York (2003)
Cartigny, J., Simplot, D., Stojmenovic, I.: Localized minimum-energy broadcasting in ad-hoc networks. In: Proceedings of the 22nd Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), vol. 3, pp. 2210–2217. IEEE Computer Society, Los Alamitos (2003)
Chen, W., Huang, N.: The strongly connecting problem on multihop packet radio networks. IEEE Trans. Commun. 37(3), 293–295 (1989)
Clementi, A., Crescenzi, P., Penna, P., Rossi, G., Vocca, P.: On the complexity of computing minimum energy consumption broadcast subgraphs. In: Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 2010, pp. 121–131. Springer, New York (2001)
Clementi, A., Huiban, G., Penna, P., Rossi, G., Verhoeven, Y.C.: On the approximation ratio of the MST-based heuristic for the energy-efficient broadcast problem in static ad-hoc radio networks. In: Proceedings of the 3rd IEEE IPDPS Workshop on Wireless, Mobile and Ad Hoc Networks (WMAN), p. 222. IEEE Computer Society, Los Alamitos (2003)
Conway, J.H., Sloane, N.J.A.: The kissing number problem. Bounds on kissing numbers. In: Sphere Packings, Lattices, and Groups, 3rd edn. Springer, New York (1998), Chaps. 2.1 and 13
Dong, Q., Banerjee, S., Adler, M., Misra, A.: Minimum energy reliable paths using unreliable wireless links. In: Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), pp. 449–459. ACM Press, New York (2005)
Doshi, S., Bhandare, S., Brown, T.X.: An on-demand minimum energy routing protocol for a wireless ad hoc network. SIGMOBILE Mob. Comput. Commun. Rev. 6(3), 50–66 (2002)
Edmonds, J.: Optimal branchings. J. Res. Nat. Bur. Stand. B 71, 233–240 (1967)
Flammini, M., Klasing, R., Navarra, A., Perennes, S.: Improved approximation results for the minimum energy broadcasting problem. In: Proceedings of ACM Joint Workshop on Foundations of Mobile Computing (DIALM-POMC), pp. 85–91. ACM Press, New York (2004)
Frieze, A.M., McDiarmid, C.J.H.: On random minimum length spanning trees. Combinatorica 9, 363–374 (1989)
Gilbert, E., Pollak, H.: Steiner minimal trees. SIAM J. Appl. Math. 16, 1–29 (1968)
Hac, A.: Wireless Sensor Network Designs. Wiley, New York (2003)
Kirousis, L., Kranakis, E., Krizanc, D., Pelc, A.: Power consumption in packet radio networks. Theor. Comput. Sci. 243(1–2), 289–305 (2000)
Klasing, R., Navarra, A., Papadopoulos, A., Perennes, S.: Adaptive broadcast consumption (ABC), a new heuristic and new bounds for the minimum energy broadcast routing problem. In: Proceedings of the 3rd IFIP-TC6 International Networking Conference. Lecture Notes in Computer Science, vol. 3042, pp. 866–877. Springer, New York (2004)
Li, X., Wan, P.: Constructing minimum energy mobile wireless networks. SIGMOBILE Mob. Comput. Commun. Rev. 5(4), 55–67 (2001)
Liang, W.: Constructing minimum-energy broadcast trees in wireless ad hoc networks. In: Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc), pp. 112–122. ACM Press, New York (2002)
Penna, P., Ventre, C.: Energy-efficient broadcasting in ad-hoc networks: combining MSTs with shortest-path trees. In: Proceedings of the 1st ACM International Workshop on Performance Evaluation of Wireless Ad Hoc, Sensor, and Ubiquitous Networks (PE-WASUN), pp. 61–68. ACM Press, New York (2004)
Rappaport, T.: Wireless Communications: Principles and Practice. Prentice-Hall, Englewood Cliffs (1996)
Wan, P.J., Calinescu, G., Li, X., Frieder, O.: Minimum energy broadcast routing in static ad hoc wireless networks. Wirel. Netw. 8(6), 607–617 (2002)
Wieselthier, J.E., Nguyen, G.D., Ephremides, A.: On the construction of energy-efficient broadcast and multicast trees in wireless networks. In: Proceedings of the 19th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pp. 585–594. IEEE Computer Society, Los Alamitos (2000)
Zagalj, M., Hubaux, J., Enz, C.: Minimum-energy broadcast in all-wireless networks: NP-completeness and distribution issues. In: Proceedings of the 8th Annual International Conference on Mobile Computing and Networking (MobiCom), pp. 172–182. ACM Press, New York (2002)
Zhao, F., Guibas, L.: Wireless Sensor Networks: An Information Processing Approach. Kaufmann, Los Altos (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research was partially funded by the European project COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL).
Preliminary version of this paper appeared in Flammini et al. (Proceedings of ACM joint workshop on foundations of mobile computing (DIALM-POMC), pp. 85–91, 2004).
Rights and permissions
About this article
Cite this article
Flammini, M., Klasing, R., Navarra, A. et al. Improved Approximation Results for the Minimum Energy Broadcasting Problem. Algorithmica 49, 318–336 (2007). https://doi.org/10.1007/s00453-007-9077-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9077-7