Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

For a pair of parameters α, β ≥ 1, a spanning tree T of a weighted undirected n-vertex graph G = (V,E,w) is called an (α, β)-shallow-light tree (shortly, (α, β)-SLT) of G with respect to a designated vertex rt ∈ V if (1) it approximates all distances from rt to the other vertices up to a factor of α, and (2) its weight is at most β times the weight of the minimum spanning tree MST(G) of G. The parameter α (resp., β) is called the root-distortion (resp., lightness) of the tree T. Shallow-light trees (SLTs) constitute a fundamental graph structure, with numerous theoretical and practical applications. In particular, they were used for constructing spanners in network design, for VLSI-circuit design, for various data gathering and dissemination tasks in wireless and sensor networks, in overlay networks, and in the message-passing model of distributed computing. Tight tradeoffs between the parameters of SLTs were established by Awerbuch, Baratz, and Peleg [Proceedings of the 9th Annual ACM Symposium on Principles of Distributed Computing (PODC), 1990, pp. 177-187, Efficient Broadcast and Light-Weight Spanners, manuscript, 1991] and Khuller, Raghavachari, and Young [Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 1993, pp. 243-250]. They showed that for any ε > 0 there always exist (1 + ε, O(Formula presented))-SLTs and that the upper bound β = O(Formula presented) on the lightness of SLTs cannot be improved. In this paper we show that using Steiner points one can build SLTs with logarithmic lightness, i.e., β = O(log Formula presented). This establishes an exponential separation between spanning SLTs and Steiner ones. In the regime ε = 0 our construction provides a shortest-path tree with weight at most O(log n)· w(MST(G)). Moreover, we prove matching lower bounds that show that all our results are tight up to constant factors.