Steiner Shallow-Light Trees Are Exponentially Lighter than Spanning Ones

Abstract

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 $(\alpha,\beta)$-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 $\alpha$ (resp., $\beta$) 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, 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 $\epsilon > 0$ 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 logarithmic lightness, i.e., $\beta = O(\log \frac{1}{\epsilon})$. This establishes an exponential separation between spanning SLTs and Steiner ones. In the regime $\epsilon = 0$ our construction provides a 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.