Local Construction of Spanners in the 3D 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 3D euclidean space. Our first contribution is a local distributed algorithm that, given a UBG U and a parameter α<;π/3, constructs a sparse spanner of U with stretch factor 1/(1-2 sin(α/2)), improving the previous upper bound of 1/(1 - α ) by Althöfer et al. which is applicable only when α<;1/(1+2√2) <;π/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 and the stretch factor of the spanners, and the locality of the algorithms, are much better than the theoretical upper bounds proved in this paper.