Steiner Shallow-Light Trees are Exponentially Better than Spanning Ones

The power of Steiner points was studied in a number of different settings in the context of metric embeddings. Perhaps most notably in the context of probabilistic tree embeddings Bartal and Fakcharoenphol et al. [8, 9, 21] used Steiner points to devise near-optimal constructions of such embeddings. However, Konjevod et al. [24] and Gupta [22] demonstrated that Steiner points do not help in this context. Specifically, they showed that any probabilistic tree embedding with distortion D that employs Steiner points can be converted into a probabilistic tree embedding of distortion O(D) that uses only points of the original metric. Steiner points were also studied in the context of graph spanners [3], in the context of Euclidean spanners [27, 19, 28], and in the context of distance preservers [10]. In all these contexts it is known [3, 27, 19, 28, 10] that Steiner points cannot be used to significantly improve inherent tradeoffs between the involved parameters. The situation is similar in the context of low-light trees. Specifically, it is known [17, 19] that essentially the same tradeoff between unweighted diameter and weight that applies to spanning low-light trees applies to Steiner low-light trees too. These results may lead to a far-reaching conclusion that Steiner points do not help in metric embeddings in general. In this paper we show that this is not the case, and demonstrate that Steiner points do help dramatically in the context of shallow-light trees. Shallow-light trees [6, 7, 23, 12, 13, 14] combine small weight with small distortion with respect to a designated root vertex rt (henceforth, root-distortion). Awerbuch et al. [7] and Khuller et al. [23] showed that for any positive real parameter ǫ > 0, one can simultaneously achieve root-distortion (1 + ǫ) and weight O( 1 ǫ ) times the weight of the minimum spanning tree w(MST ). Moreover, this tradeoff is tight up to constant factors. In this paper we show that by using Steiner points one can simultaneously achieve root-distortion (1+ ǫ) and weight O(log ( 1 ǫ ) ) ·w(MST ). In particular, one can also construct a Steiner tree with weight O(log n) ·w(MST ) that preserves all distances between rt and other vertices. Furthermore, we show that up to constant factors this tradeoff is tight. These results imply that there is an exponential separation between shallow-light trees that use Steiner points and shallow-light trees that do not use them. Finally, on the way to these results we also address a number of open questions that were posed by Khuller et al. [23]. Specifically, we show that the lower bound on the tradeoff between root-distortion (1 + ǫ) and weight Ω( ǫ ) · w(MST ) of spanning shallow-light trees (1) applies even to 2-dimensional Euclidean metrics rather than to general metrics; (2) applies even if we replace (worst-case) rootdistortion by average root-distortion; (3) applies even if the designated root vertex is selected at will. Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel. E-mail: {elkinm,shayso}@cs.bgu.ac.il Both authors are partially supported by the Lynn and William Frankel Center for Computer Sciences. This research has been supported by the BSF grant No. 2008430. This research has been supported by the Clore Fellowship grant No. 81265410.