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

TL;DR

This paper resolves key questions about spanners in planar domains, showing a sharp contrast between stretch $2$ and stretch $2+\varepsilon$: any $2$-spanner must have at least $\Omega(n\log n)$ edges, while a $2+\varepsilon$-spanner can be built with $O(n)$ edges. The authors develop a novel non-Steiner tree-cover framework for trees and leverage it to construct sparse distance-preserving structures in planar domains, culminating in efficient Steiner and non-Steiner spanners for planar graphs, polygonal domains, and polyhedral terrains. Central to the approach is a threshold phenomenon around stretch $2$, plus a sequence of reductions via net-trees, additive spanners, sparse covers, and recursive shortest-path separators, enabling near-linear spanners with rigorous guarantees. The work also connects these constructions to reliable spanners and locality-sensitive orderings, yielding new lower bounds that separate planar and tree metrics from Euclidean settings and highlighting the practical impact for routing in planar domains and related geometric networks.

Abstract

We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant $ε\in (0,1)$, one could construct a $(2+ε)$-spanner with $O(n\log(n))$ edges (SICOMP 2019), and there is a lower bound of $Ω(n^2)$ 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 $ε\in (0,1)$, $O(n)$ 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).

Figures (11)

• Figure 1: (a) A polygon with holes, blue terminals, and a shortest path between terminals $x$ and $y$. (b) A polyhedral terrain and a shortest path between two points $x$ and $y$.
• Figure 2: (a) $\varepsilon$-chops; (b) a jump chop; (c) a collection of chops $\mathbb{C}$ for $k = 3$ that is partitioned into 3 buckets $\mathbb{B}_0,\mathbb{B}_1,\mathbb{B}_2$ in (d). Square vertices are terminals.
• Figure 3: (a) The second bucket $\mathbb{B}_2$; (b) and (c) a tree constructed from $\mathbb{B}_2$; (d) all the $\varepsilon$-chops close enough to a jump chop are segments on the same edge; and (e) $z = \mathcal{LCA}(x,y)$ belongs to a chop at some level $i+sk$.
• Figure 4: (a) illustration for \ref{['clm:reg-chop-decrease']}; (b) illustration for \ref{['lm:dist-chops-modk']}; (c) illustration for \ref{['lm:dist-x-r_isk']}.
• Figure 5: Construct two trees $T_{i,0}$ and $T_{i,1}$ from $T_{i}$ by skipping $3$ levels; here $p=3$.

Theorems & Definitions (72)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Lemma 1.5
• Theorem 1.6: Theorem 1.2 in ChangCLMST23
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1
• Claim 2.2
• proof