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

Abstract

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight, and a spanning tree T ′ with weight w(T ′ ) = O(k) · w(M ST (M )) and unweighted diameter O(k · n 1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.We prove that these tradeoffs between unweighted diameter and weight are tight up to constant factors in the entire range of parameters. Moreover, our lower bounds apply to a basic 1-dimensional Euclidean space.Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem 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) · w(M ST (M )). 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.Finally, our upper bounds imply improved approximation algorithms for the minimum h-hop spanning tree and bounded diameter minimum spanning tree problems for metric spaces.

Euclidean Steiner Shallow-Light Trees

“…In addition to small root-stretch and lightness, these trees have small depth. Similarly to the SLTs of [5,6,29], the shallow-low-light trees of [19,20] exhibit an inverse-linear tradeoff of 1 + versus Ω( 1 ) between the root-stretch and lightness.…”

“…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].…”

“…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.…”