Optimal Bounds for Spanners and Tree Covers in Doubling Metrics

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2) a $(1+\epsilon)$-tree cover with $\tilde{O}(n \cdot \epsilon^{-d+1})$ 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 $(1+\epsilon)$-spanner has $\tilde{\Omega}(n \cdot \epsilon^{-d+1})$ edges and lightness $\tilde{\Omega}(\epsilon^{-d})$. 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 $\tilde{O}(n \cdot \epsilon^{-d})$ edges, and it could be transformed into a spanner of maximum degree $\tilde{O}(\epsilon^{-d})$ and lightness $\tilde{O}(n \cdot \epsilon^{-(d+1)})$ by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a $(1+\epsilon)$-tree cover with $\tilde{O}(\epsilon^{-d})$ 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 $(1+\epsilon)$-tree covers with $\tilde{O}(n \cdot \epsilon^{-d})$ trees, with maximum degree $O(1)$ in each tree. This construction is optimal with respect to the number of trees and maximum degree.