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

TL;DR

This paper proves tight bounds for Euclidean spanners, showing that low-diameter, low-weight spanning trees with near-optimal distance properties exist for any metric space, resolving a long-standing open problem.

Contribution

It establishes tight bounds for Euclidean spanners, demonstrating the existence of shallow, low-weight trees with near-optimal distances, and closes the gap from previous approximate results.

Findings

Existence of spanning trees with unweighted diameter O(log n) and weight O(log n) times MST weight.

Extension of results to a tradeoff between unweighted diameter and weight, tight up to constant factors.

Resolution of a long-standing open question in Computational Geometry.

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.