It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+ε)$-spanner with maximum degree $\tilde{O}(ε^{-d+1})$ and with lightness $\tilde{O}(ε^{-d})$; 2) a $(1+ε)$-tree cover with $\tilde{O}(n \cdot ε^{-d+1})$ trees and maximum degree of $O(1)$ in each tree. Moreover, all the parameters in these constructions are optimal: there exists an $n$-point set in $\mathbb{R}^d$, for which any $(1+ε)$-spanner has $\tildeΩ(n \cdot ε^{-d+1})$ edges and lightness $\tildeΩ(ε^{-d})$. The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in $\mathbb{R}^d$, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has $\tilde{O}(n \cdot ε^{-d})$ edges, and it could be transformed into a spanner of maximum degree $\tilde{O}(ε^{-d})$ and lightness $\tilde{O}(n \cdot ε^{-(d+1)})$ by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a $(1+ε)$-tree cover with $\tilde{O}(ε^{-d})$ trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) a surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters, 2) a new construction of $(1+ε)$-tree covers with $\tilde{O}(n \cdot ε^{-d})$ trees, with maximum degree $O(1)$ in each tree. This construction is optimal with respect to the number of trees and maximum degree.

翻译：已知对于$d$维欧几里得空间$\mathbb{R}^d$（其中$d = O(1)$）中的任意$n$点集，存在：1) 一个$(1+ε)$-单跨器，其最大度数为$\tilde{O}(ε^{-d+1})$，轻度为$\tilde{O}(ε^{-d})$；2) 一个$(1+ε)$-树覆盖，包含$\tilde{O}(n \cdot ε^{-d+1})$棵树，且每棵树的最大度数为$O(1)$。此外，这些构造中的所有参数都是最优的：存在一个$\mathbb{R}^d$中的$n$点集，使得任何$(1+ε)$-单跨器都有$\tildeΩ(n \cdot ε^{-d+1})$条边和$\tildeΩ(ε^{-d})$的轻度。欧几里得单跨器的上界严重依赖于$\mathbb{R}^d$中锥体划分的空间性质，而这似乎无法推广到更广泛的加倍度量族（即常加倍维数的度量空间）。在加倍度量中，二十年前的一种简单单跨器构造——网树单跨器，具有$\tilde{O}(n \cdot ε^{-d})$条边，并且可以通过剪枝冗余边转化为最大度数为$\tilde{O}(ε^{-d})$、轻度为$\tilde{O}(n \cdot ε^{-(d+1)})$的单跨器。此外，对网树单跨器的仔细改进可得到包含$\tilde{O}(ε^{-d})$棵树的$(1+ε)$-树覆盖。尽管已有大量研究，但在更广泛的加倍度量族中获得单跨器和树覆盖的紧界问题仍然难以解决。我们通过以下结果解决了这一问题：1) 一个出人意料简单且紧的下界，表明网树单跨器及其剪枝版本在所有相关参数上都是最优的；2) 一种新的$(1+ε)$-树覆盖构造，包含$\tilde{O}(n \cdot ε^{-d})$棵树，且每棵树的最大度数为$O(1)$。这一构造在树的数量和最大度数方面是最优的。