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

We show that for every set S of n points in the plane and a designated point rt ∈ S, there exists atree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distancebetween rt and v in T is within a factor of (1+2) close to their Euclidean distance ‖rt, v‖. We call thesetrees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (upto constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to pointsets in R, for an arbitrarily large constant d. The running time of our construction is O(n · log n).We also study this problem in general metric spaces, and show that NSLLTs with small maximumdegree, 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 Sresides in a Euclidean space of dimension Θ(log n).On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieveroot-distortion (1+2) 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 Euclideanspanning trees and spanners.