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 (respectively, 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 Awer buch et al. [5], [6] and Khuller et al. [33]. They showed that for any ε gt; 0 there always exist (1+ ε, O(1/ε))-SLTs, and that the upper bound β = O(1/ε) 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 1/ε). This establishes an exponential separation between spanning SLTs and Steiner ones. One particularly remarkable point on our tradeoff curve is ε =0. In this regime 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. Finally, on our way to these results we settle (up to constant factors) a number of open questions that were raised by Khuller et al. [33] in SODA'93.