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 a tree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distance between rt and v in T is within a factor of (1+ ) 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 Rd , 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 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 S resides in a Euclidean space of dimension Θ(log n). On the bright side, we show that ...

• Type: Conference
• Year: 2009
• Where: ESA
• Authors: Michael Elkin, Shay Solomon
• Keywords: Algorithms, ESA 2009, Euclidean Distance Rt, General Metric Spaces, Small Maximum Degree