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

Abstract: We show that for every $n$-point metric space $M$ there exists a spanning tree $T$ with unweighted diameter $O(\log n)$ and weight $\omega(T) = O(\log n) \cdot \omega(MST(M))$. Moreover, there is a designated point $rt$ such that for every point $v$, $dist_T(rt,v) \le (1+\epsilon) \cdot dist_M(rt,v)$, for an arbitrarily small constant $\epsilon > 0$. We extend this result, and provide a tradeoff between unweighted diameter and weight, and prove that this tradeoff is \emph{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 $O(\log n) \cdot \omega(MST(M))$. Ten years later 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.