Shallow, Low, and Light Trees, and Tight Lower Bounds for Euclidean Spanners

We show that for every n-point metric space M there exists a spanning tree T with unweighted diameter O(log n) and weight ω(T ) = O(log n) · ω(MST (M)). Moreover, there is a designated point rt such that for every point v, distT (rt, v) ≤ (1 + ǫ) · distM (rt, v), for an arbitrarily small constant ǫ > 0. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is tight up to constant factors in the entire range of parameters. These results enable us to settle a long-standing open question 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) ·ω(MST (M)). Ten years later in SODA’05 Agarwal et al. showed that this result is tight up to a factor of O(log logn). We close this gap and show that the result of Arya et al. is tight up to constant factors. Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel. E-mail: {dinitz,elkinm,shayso}@cs.bgu.ac.il Partially supported by the Lynn and William Frankel Center for Computer Sciences. This research has been supported by the Israeli Academy of Science, grant 483/06.