Maintaining Light Spanners via Minimal Updates
Hadi Khodabandeh, David Eppstein
TL;DR
The paper addresses maintaining a $(1+\varepsilon)$-spanner for a dynamic point set in $\mathbb{R}^d$ with constant degree and lightness while minimizing recourse. It introduces a hierarchical clustering framework, bucketing, and a potential-function analysis to build a sparse spanner and then a light spanner on top, achieving amortized $O(1)$ edge updates after insertions and $O(\log \Delta)$ after deletions, with constants depending on $\varepsilon$ and $d$. The key contributions are the fully-dynamic algorithm, the two-tier invariants ensuring $(1+\varepsilon)$ stretch and constant lightness, and the amortized recourse bounds proven via potential decrease per maintenance step. The results have practical implications for dynamic network design and geometric data structures where small, bounded updates are more costly than computation, and they open avenues for extending the framework to semi-dynamic or online settings and exploring running-time optimizations.
Abstract
We study the problem of maintaining a lightweight bounded-degree $(1+\varepsilon)$-spanner of a dynamic point set in a $d$-dimensional Euclidean space, where $\varepsilon>0$ and $d$ are arbitrary constants. In our fully-dynamic setting, points are allowed to be inserted as well as deleted, and our objective is to maintain a $(1+\varepsilon)$-spanner that has constant bounds on its maximum degree and its lightness (the ratio of its weight to that of the minimum spanning tree), while minimizing the recourse, which is the number of edges added or removed by each point insertion or deletion. We present a fully-dynamic algorithm that handles point insertion with amortized constant recourse and point deletion with amortized $O(\logΔ)$ recourse, where $Δ$ is the aspect ratio of the point set.