For a pair of parameters alpha, beta >= 1, a spanning tree T of a weighted undirected n-vertex graph G = (V, E, w) is called an (alpha, beta)-shallow-light tree (shortly, (alpha, beta)-SLT) of G with respect to a designated vertex rt is an element of V if (1) it approximates all distances from rt to the other vertices up to a factor of a, and (2) its weight is at most beta times the weight of the minimum spanning tree MST(G) of G. The parameter a (respectively, beta) 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 et al. [5], [6] and Khuller et al. [33]. They showed that for any epsilon > 0 there always exist (1 + epsilon, O(1/epsilon))-SLTs, and that the upper bound beta = O(1/epsilon) 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., beta = O(log 1/epsilon). This establishes an exponential separation between spanning SLTs and Steiner ones. One particularly remarkable point on our tradeoff curve is epsilon = 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.
