A $(1+\varepsilon)\textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated $\textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $\textit{maximum degree}$ of a point in the tree cover may be as large as $\Omega(\log \Phi)$, where $\Phi$ is the aspect ratio of the input point set. In this work we present a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$ trees, which is optimal (up to the $\log(1/\varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $\textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+\varepsilon)$-stretch $\textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n \log n)$ term; this improves over the running time underlying the Dumbbell Theorem.

翻译：度量空间的一个$(1+\varepsilon)$-伸缩树覆盖是一组树的集合，其中任意两点在某一棵树中均存在一条$(1+\varepsilon)$-伸缩路径。著名的“哑铃定理”[Arya等人，STOC'95]指出：$d$维欧几里得空间中任意$n$个点的集合，均存在一个具有$O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$棵树的$(1+\varepsilon)$-伸缩树覆盖，其中$O_d$记号抑制了仅依赖于维度$d$的项。该构造的运行时间为$O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$。由于同一个点可能出现在树的多个层次中，树覆盖中点的最大度数可能高达$\Omega(\log \Phi)$，其中$\Phi$为输入点集的纵横比。本文提出一个具有$O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$棵树的$(1+\varepsilon)$-伸缩树覆盖，该结果（在$\log(1/\varepsilon)$因子范围内）是最优的。此外，对于任意$d$，各棵树中点的最大度数均为绝对常数。作为直接推论，我们在低维欧几里得空间中得到一个最优路由方案。我们还提出一个具有$O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$棵树的$(1+\varepsilon)$-伸缩Steiner树覆盖（可能使用Steiner点），该结果同样是最优的。忽略附加的$O_d(n \log n)$项，我们两种构造的运行时间与各自树覆盖中的边数成线性关系；这一结果改进了哑铃定理的运行时间下界。