Dynamic Light Spanners in Doubling Metrics

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 $\epsilon >0$, we maintain a $(1+\epsilon )$-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 $\mathrm{poly}(\log \mathrm{\Phi })$ time, where $\mathrm{\Phi }$ 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.