Euclidean Steiner shallow-light trees

A spanning tree that simultaneously approximates a shortestpath tree and a minimum spanning tree is called a shallow- light tree (shortly, SLT). More specifically, an (α, β)-SLT of a weighted undirected graph G = (V,E,w) with respect to a designated vertex rt ∈ V is a spanning tree of G with: (1) root-stretch α - it preserves all distances between rt and the other vertices up to a factor of α. (2) lightness β - it has weight at most β times the weight of a minimum spanning tree MST(G) of G. Tight tradeoffs between the parameters of SLTs were established by Awerbuch et al. in PODC'90 and by Khuller et al. in SODA'93. They showed that for any ∈ > 0, any graph admits a (1 + ∈,O(1/∈ ))-SLT with respect to any root vertex, and complemented this result with a matching lower bound. Khuller et al. asked if the upper bound β = O(1/∈ ) on the lightness of SLTs can be improved in constant-dimensional Euclidean spaces. In FOCS'11 Elkin and this author gave a negative answer to this question, showing a lower bound of β = Ω(1/∈ ) that applies to 2-dimensional Euclidean spaces. In this paper we show that Steiner points lead to a quadratic improvement in Euclidean SLTs, by presenting a construction of Euclidean Steiner (1 + ∈,O(√ 1/∈ ))-SLTs. While the lightness bound β = O(√ 1/∈ ) of our construction applies to Euclidean spaces of any constant dimension, there is a matching lower bound of β = Ω(√ 1/∈ ) even in 2-dimensional Euclidean spaces. The runtime of our construction, and thus the number of Steiner points used, are bounded by O(n).