Steiner Shallow-Light Trees are Exponentially Lighter than Spanning Ones

For a pair of parameters $\alpha,\beta \ge 1$, a spanning tree $T$ of a weighted undirected $n$-vertex graph $G = (V,E,w)$ is called an \emph{$(\alpha,\beta)$-shallow-light tree} (shortly, $(\alpha,\beta)$-SLT)of $G$ with respect to a designated vertex $rt \in V$ if (1) it approximates all distances from $rt$ to the other vertices up to a factor of $\alpha$, and(2) its weight is at most $\beta$ times the weight of the minimum spanning tree $MST(G)$ of $G$. The parameter $\alpha$ (respectively, $\beta$) is called the \emph{root-distortion}(resp., \emph{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.\ \cite{ABP90, ABP91} and Khuller et al.\ \cite{KRY93}. They showed that for any $\epsilon >, 0$there always exist $(1+\epsilon, O(\frac{1}{\epsilon}))$-SLTs, and that the upper bound $\beta = O(\frac{1}{\epsilon})$on the lightness of SLTs cannot be improved. In this paper we show that using Steiner points one can build SLTs with \emph{logarithmic lightness}, i.e., $\beta = O(\log \frac{1}{\epsilon})$. This establishes an \emph{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 \emph{shortest-path tree} with weight at most $O(\log n) \cdot w(MST(G))$. 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.\ \cite{KRY93} in SODA'93.