Improved Approximation Results for the Minimum Energy Broadcasting Problem

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.