Dynamic Light Spanners in Doubling Metrics

TL;DR

This paper presents a new efficient dynamic algorithm for maintaining light-weight $(1+ ext{epsilon})$-spanners in metric spaces with constant doubling dimension, with updates in polylogarithmic time.

Contribution

It introduces the first efficient dynamic algorithm for maintaining light spanners in low-dimensional metric spaces, with polylogarithmic update time.

Findings

Maintains a $(1+ ext{epsilon})$-spanner with weight within a constant factor of the MST.

Supports insertions and deletions in polylogarithmic time.

Works in metric spaces of constant doubling dimension.

Abstract

A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, \delta)$ 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 $\delta(u,v)$. 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 $\operatorname{poly}(\log \Phi)$ time, where $\Phi$ 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.