  Resolving an open question from 2006, we prove the existence of light-weight
bounded-degree spanners for unit ball graphs in the metrics of bounded doubling
dimension, and we design a simple $\mathcal{O}(\log^*n)$-round distributed
algorithm in the LOCAL model of computation, that given a unit ball graph $G$
with $n$ vertices and a positive constant $\epsilon < 1$ finds a
$(1+\epsilon)$-spanner with constant bounds on its maximum degree and its
lightness using only 2-hop neighborhood information. This immediately improves
the best prior lightness bound, the algorithm of Damian, Pandit, and Pemmaraju,
which runs in $\mathcal{O}(\log^*n)$ rounds in the LOCAL model, but has a
$\mathcal{O}(\log \Delta)$ bound on its lightness, where $\Delta$ is the ratio
of the length of the longest edge to the length of the shortest edge in the
unit ball graph. Next, we adjust our algorithm to work in the CONGEST model,
without changing its round complexity, hence proposing the first spanner
construction for unit ball graphs in the CONGEST model of computation. We
further study the problem in the two dimensional Euclidean plane and we provide
a construction with similar properties that has a constant average number of
edge intersections per node. Lastly, we provide experimental results that
confirm our theoretical bounds, and show an efficient performance from our
distributed algorithm compared to the best known centralized construction.
