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 ldr n^1/k) ldr w(MST(M)), and a spanning tree T´ with weight w(T´) = O(k) ldr w(MST(M)) and unweighted diameter O(k ldr 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 + epsiv) ldr dist_M(rt,v), for an arbitrarily small constant epsiv > 0. 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.