Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

TL;DR

The paper introduces the tau-lanky criterion to identify geometric graphs with sublinear separators and proves it yields tight separators for greedy spanners in Euclidean and related metric spaces. It shows greedy $(1+\epsilon)$-spanners are $O(1)$-lanky, enabling separators of size $O(k^{1-1/d})$ for $k$-vertex subgraphs in $\mathbb{R}^d$, and extends the framework to unit-ball graphs and point sets with low fractal dimension. The authors derive both probabilistic and deterministic separator constructions, including a linear-time derandomization, and apply the approach to doubling metrics, resolving longstanding open questions (e.g., Abam–Har-Peled) via the CGMZ spanner, which they prove is $\tau$-lanky with constant max degree. Collectively, the results yield sublinear separators across multiple geometric graph families and provide algorithmic implications, such as PTAS feasibility on graphs with polynomial expansion. The work advances a simple, general criterion for sublinear separators in Euclidean and doubling settings and broadens the applicability of greedy spanners in practical algorithmic contexts.

Abstract

The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh showed that the greedy spanner in $\mathbb{R}^2$ admits a sublinear separator in a strong sense: any subgraph of $k$ vertices of the greedy spanner in $\mathbb{R}^2$ has a separator of size $O(\sqrt{k})$. Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in $\mathbb{R}^d$ for any constant $d\geq 3$ as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh by showing that any subgraph of $k$ vertices of the greedy spanner in $\mathbb{R}^d$ has a separator of size $O(k^{1-1/d})$. We introduce a new technique that gives a simple characterization for any geometric graph to have a sublinear separator that we dub $τ$-lanky: a geometric graph is $τ$-lanky if any ball of radius $r$ cuts at most $τ$ edges of length at least $r$ in the graph. We show that any $τ$-lanky geometric graph of $n$ vertices in $\mathbb{R}^d$ has a separator of size $O(τn^{1-1/d})$. We then derive our main result by showing that the greedy spanner is $O(1)$-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in $\mathbb{R}^d$. Our technique naturally extends to doubling metrics. We use the $τ$-lanky characterization to show that there exists a $(1+ε)$-spanner for doubling metrics of dimension $d$ with a constant maximum degree and a separator of size $O(n^{1-\frac{1}{d}})$; this result resolves an open problem posed by Abam and Har-Peled a decade ago.

Figures (4)

• Figure 1: (a) Points $y$ and $q$ of $Y$ are in the same cone bounded by two red dashed lines at apex $x$ and (b) the triangle $xqy$ with $\angle yxq = \gamma$.
• Figure 2: A cone containing two points in $V(M_{x, D})$. We later prove that this case cannot happen.
• Figure 3: An edge $(v \rightarrow w) \in E(\overrightarrow{G_1})$ incident to a vertex $w\in \mathbf{B}(p,c\cdot r)$ is rerouted to some point $u\in \mathbf{B}(p,c\cdot r)$. That is, $(v \rightarrow u) \in E(\overrightarrow{G_2})$.
• Figure 4: $(a)$: The positions of $u_1, u_2, v_1, v_2$ and $w$ in the net tree and $(b)$ the edges $(v_1 \rightarrow w)$ and $(v_2 \rightarrow w)$ are rerouted to $u_1$ and $u_2$, respectively. We show in \ref{['lm:reroutedInBallLargeEdges']} that $u_2$ must be outside $\mathbf{B}(p, c \cdot r)$.

Theorems & Definitions (86)

• Definition 1: $\tau$-Lanky
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2: $c$-Separated Pair
• Definition 3: Doubling Metric
• Lemma 1
• Definition 4: Fractal Dimension
• Lemma 2: Lemma 2.4 HPM06