Dynamic Light Spanners in Doubling Metrics

2026-03-24

AI Summary

This work proposes the first algorithm capable of dynamically maintaining a $(1+\varepsilon)$-spanner for a point set undergoing insertions and deletions in a doubling metric space of constant dimension. By leveraging the doubling property of the metric space, dynamic graph data structures, and geometric divide-and-conquer techniques, the algorithm achieves an update time of $\mathrm{poly}(\log \Phi)$ per operation—where $\Phi$ denotes the aspect ratio—while ensuring that the total weight of the spanner remains within a constant factor of the minimum spanning tree weight. This result overcomes a long-standing limitation: despite extensive study in static settings, even in low-dimensional Euclidean spaces, no efficient dynamic construction was previously known for lightweight spanners. The proposed method thus substantially advances both the theoretical foundations and practical applicability of dynamic geometric spanners.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Problem

• dynamic spanner
• doubling metrics
• lightweight spanner
• point set
• insertions and deletions

Innovation

• dynamic spanner
• doubling metric
• lightweight spanner
• polylogarithmic update time
• minimum spanning tree