Low-light trees, and tight lower bounds for Euclidean spanners

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T)=O(k{dot operator}n^1/k){dot operator}w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k){dot operator}w(MST(M)) and unweighted diameter O(k{dot operator}n^1/k). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic one-dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n){dot operator}w(MST(M)). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the minimumh-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.