Spanners in Planar Domains via Steiner Spanners and non-Steiner Tree Covers^*

We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant ε ∈ (0, 1), one could construct a (2 + ε)-spanner with O(n log(n)) edges (SICOMP 2019), and there is a lower bound of Ω(n²) edges for any (2 − ε)-spanner (SoCG 2015). The main open question is whether a linear number of edges suffices and the stretch can be reduced to 2. We resolve this problem by showing that for stretch 2, one needs Ω(n log n) edges, and for stretch 2 + ε for any fixed ε ∈ (0, 1), O(n) edges are sufficient. Our lower bound is the first super-linear lower bound for stretch 2. En route to achieve our result, we introduce the problem of constructing non-Steiner tree covers for metrics, which is a natural variant of the well-known Steiner point removal problem for trees (SODA 2001). Given a tree and a set of terminals in the tree, our goal is to construct a collection of a small number of dominating trees such that for every two points, at least one tree in the collection preserves their distance within a small stretch factor. Here, we identify an unexpected threshold phenomenon around 2 where a sharp transition from n trees to Θ(log n) trees and then to O(1) trees happens. Specifically, (i) for stretch 2 − ε, one needs Ω(n) trees; (ii) for stretch 2, Θ(log n) tree is necessary and sufficient; and (iii) for stretch 2 + ε, a constant number of trees suffice. Furthermore, our lower bound technique for the non-Steiner tree covers of stretch 2 has further applications in proving lower bounds for two related constructions in tree metrics: reliable spanners and locality-sensitive orderings. Our lower bound for locality-sensitive orderings matches the best upper bound (STOC 2022). Finally, we study (1 + ε)-spanners in planar domains using Steiner points. In planar domains, Steiner points are necessary to obtain a stretch arbitrarily close to 1. Here, we construct a (1 + ε)-spanner with an almost linear dependency on ε in the number of edges; the precise bound is O((n/ε) · log(ε⁻¹α(n)) · log ε⁻¹) edges, where α(n) is the inverse Ackermann function. Our result generalizes to graphs of bounded genus. For n points in a polyhedral metric, we construct a Steiner (1 + ε)-spanner with O((n/ε) · log(ε⁻¹α(n)) · log ε⁻¹) edges.