Narrow-Shallow-Low-Light Trees with and without Steiner Points

Abstract

We show that for every set $\mathcal{S}$ of n points in the plane and a designated point $rt\in\mathcal{S}$, there exists a tree T that has small maximum degree, depth, and weight. Moreover, for every point $v\in\mathcal{S}$, the distance between $rt$ and v in T is within a factor of $(1+\epsilon)$ close to their Euclidean distance $\|rt,v\|$. We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in $\mathbb{R}^d$ for an arbitrarily large constant d. The running time of our construction is $O(n\cdot\log n)$. We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth, and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set $\mathcal{S}$ resides in a Euclidean space of dimension $\Theta(\log n)$. In addition, we show that if one is allowed to use Steiner points, then it is possible to achieve root-distortion of $(1+\epsilon)$ together with small maximum degree, depth, and weight for general metric spaces. Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners.