Local Construction of Spanners in the 3-D Space

In this paper we present local distributed algorithms for constructing spanners in wireless sensor networks modeled as unit ball graphs (shortly UBGs) and quasi-unit ball graphs (shortly quasi-UBGs), in the 3-dimensional Euclidean space. Our first contribution is a local distributed algorithm that, given a UBG U and a parameter alpha < pi/3, constructs a sparse spanner of U with stretch factor 1/(1 - 2 sin (alpha/2)), improving the previous upper bound of 1/(1 - alpha) by Althofer et al. which is applicable only when alpha < 1/(1 + 2 root-2) < pi/3. The second contribution of this paper is in presenting the first local distributed algorithm for the construction of bounded-degree lightweight spanners of UBGs and quasi-UBGs. The simulation results we obtained show that, empirically, the weight of the spanners, the stretch factor and locality of the algorithms, are much better than the theoretical upper bounds proved in this paper.