Steiner shallow-light trees are exponentially lighter than spanning ones

Michael Elkin, Shay Solomon

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)996-1025
Number of pages30
JournalSIAM Journal on Computing
Issue number4
StatePublished - 2015
Externally publishedYes


  • Shallow-light tree
  • Shortest-path tree
  • Steiner point
  • Steiner tree minimum spanning tree


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