Shallow-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 · n^1/k) · w(MST(M)), and a spanning tree T′ with weight w(T′) = O(k) · w(MST(M)) and unweighted diameter O(k · n^1/k). Moreover, there is a designated point rt such that for every other point v, both dist_T(rt, v) and dist_T(rt,v) are at most (1 + ε) · dist_M(rt,v), for an arbitrarily small constant ε > O. We prove that the above tradeoffs are tight up to constant factors in the entire range of parameters. Furthermore, our lower bounds apply to a basic one-dimensional Euclidean space. Finally, our lower bounds for the particular case of unweighted diameter O(log n) settle a long-standing open problem in Computational Geometry.