Let $\mathcal{D}$ be a set of n pairwise disjoint disks in the plane. Consider the metric space in which the distance between any two disks D and $D^{\prime }$ in $\mathcal{D}$ is the length of the shortest line segment that connects D and $D^{\prime }$. For any real number $\epsilon >0$, we show how to obtain a $(1+\epsilon )$-spanner for this metric space that has at most $(2\pi /\epsilon )\cdot n$ edges. The previously best known result is by Bose et al. (2011) [1]. Their $(1+\epsilon )$-spanner is a variant of the Yao graph and has at most $(8\pi /\epsilon )\cdot n$ edges. Our new spanner is also a variant of the Yao graph.

中文翻译：

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磁盘扳手的改进结构

让 $\mathcal{d}$是平面中n个成对的不相交磁盘的集合。考虑任意两个磁盘D和D之间的距离的度量空间$d^{\prime }$ 在 $\mathcal{d}$是连接D和的最短线段的长度$d^{\prime }$。对于任何实数$\epsilon >0$，我们展示了如何获取 $（\mathrm{1个}+\epsilon ）$最多具有此度量标准空间的-spanner $（2\pi /\epsilon ）\cdot ñ$边缘。先前最著名的结果是Bose等人的论文。（2011）[1]。其$（\mathrm{1个}+\epsilon ）$-spanner是Yao图的一种变体，最多具有 $（8\pi /\epsilon ）\cdot ñ$边缘。我们的新扳手也是Yao图形的一种变体。