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

Abstract

Abstract. 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 R d , 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 if one is allowed to use Steiner points then it is possible to achieve root-distortion (1 + ) 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.

Euclidean Steiner Shallow-Light Trees

“…Moreover, SLTs find applications in routing [3,42,28,46] and in network and VLSI-circuit design [15,16,17,41]. In addition, SLTs are embedded within various related structures, such as light approximate routing trees [46], shallow-low-light trees [19,20], light spanners [6,40], and others [41,36,35].…”

“…Yet another example occurs in the context of low-light trees, which combine small lightness with small depth [19,20]. It is known that Steiner points do not help in this context either: Any Euclidean Steiner tree T can be converted into a spanning tree with the same (up to constants) lightness and depth as those of T [20].…”

“…It is known that Steiner points do not help in this context either: Any Euclidean Steiner tree T can be converted into a spanning tree with the same (up to constants) lightness and depth as those of T [20]. See Section 1.5 for more examples of this "incompetence" of Steiner points.…”

“…The low-light trees of [19,20] mentioned in Section 1.3 apply in fact to arbitrary metrics, and also there Steiner points do not help beyond constants. We refer to [21] for more examples where Steiner points do not help much.…”

“…Another closely related notion is shallow-low-light trees [19,20]. In addition to small root-stretch and lightness, these trees have small depth.…”