A Tight Lower Bound for Doubling Spanners

2025-08-15

AI Summary

This paper establishes tight lower bounds on the sparsity (number of edges) and lightness of $(1+varepsilon)$-spanners in $d$-dimensional doubling metrics. Addressing a long-standing open question—whether the Euclidean upper bounds of $ ilde{O}(nvarepsilon^{-d+1})$ edges and $ ilde{O}(varepsilon^{-d})$ lightness extend to doubling metrics—the work proves matching lower bounds: any $(1+varepsilon)$-spanner must have $ ilde{Omega}(nvarepsilon^{-d+1})$ edges and $ ilde{Omega}(varepsilon^{-d})$ lightness. The proof integrates hierarchical net-tree decompositions, the doubling property, compression-based arguments, and an adversarial construction of point sets. These techniques collectively demonstrate that the seminal net-tree spanner construction from twenty years ago is asymptotically optimal in doubling metrics. The result fully resolves this central open problem in metric spanner theory, confirming fundamental limits on spanner efficiency beyond Euclidean spaces.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Any $n$-point set in the $d$-dimensional Euclidean space $mathbb{R}^d$, for $d = O(1)$, admits a $(1+ε)$-spanner with $ ilde{O}(n cdot ε^{-d+1})$ edges and lightness $ ilde{O}(ε^{-d})$, for any $ε> 0$. (The {lightness} is a normalized notion of weight, where we divide the spanner weight by the MST weight. The $ ilde{O}$ and $ ildeΩ$ notations hide $ exttt{polylog}(ε^{-1})$ terms.) Moreover, this result is tight: For any $2 le d = O(1)$, there exists an $n$-point set in $mathbb{R}^d$, for which any $(1+ε)$-spanner has $ ildeΩ(n cdot ε^{-d+1})$ edges and lightness $ ildeΩ(n cdot ε^{-d})$. The upper bounds for Euclidean spanners rely heavily on the spatial property of {cone partitioning} in $mathbb{R}^d$, which does not seem to extend to the wider family of {doubling metrics}, i.e., metric spaces of constant {doubling dimension}. In doubling metrics, a {simple spanner construction from two decades ago, the {net-tree spanner}}, has $ ilde{O}(n cdot ε^{-d})$ edges, and it could be transformed into a spanner of lightness $ ilde{O}(n cdot ε^{-(d+1)})$ by pruning redundant edges. Despite a large body of work, it has remained an open question whether the superior Euclidean bounds of $ ilde{O}(n cdot ε^{-d+1})$ edges and lightness $ ilde{O}(ε^{-d})$ could be achieved also in doubling metrics. We resolve this question in the negative by presenting a surprisingly simple and tight lower bound, which shows, in particular, that the net-tree spanner and its pruned version are both optimal.

Problem

• Establishing tight lower bounds for doubling spanners
• Resolving open question on optimal spanner bounds
• Proving net-tree spanner optimality in doubling metrics

Innovation

• Net-tree spanner construction for doubling metrics
• Pruning redundant edges to reduce lightness
• Tight lower bound proving optimality of spanner