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.

翻译：解开2006年的开放问题,我们证明,在捆绑的双倍维度中,存在对单位球形图的轻量度约束度比量,并且我们在LOCAL计算模型中设计了一个简单的$mathcal{O}(log ⁇ n)美元四轮分布算法,给LOCAL计算模型中的单位球形1G$G$和正常数$epsilon < 1美元,发现一个1美元(log\delta)一美元,在最大度和亮度上都有恒定的界限,只使用2点的周边信息。这立即改进了最佳的先前亮度约束,Damian、Pandit和Pemmaraju的算法,在$\mathcal{O}(log ⁇ n)一回合中运行,在LOCOL模型中有一个单位球形的正数,但是在光度上找到一个(log\Delta)一美元,在最大度和亮度的深度上确定了我们最短的距离与最短的距离的比率, 在单位精度边缘上,我们开始的CEEEEEST的计算, 在C的计算中,我们将一个最接近的计算中进行一个最精确的计算。