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 $O(\log\Delta)$ recourse, where $\Delta$ is the aspect ratio of the point set.

翻译：我们研究在 $d$ 维欧几里得空间中动态点集上维持轻量级有界度 $(1+\varepsilon)$-稀疏子图的问题，其中 $\varepsilon>0$ 和 $d$ 为任意常数。在全动态场景下，点可被插入或删除，目标是在维持一个最大度和轻量度（子图权重与最小生成树权重之比）均为常数的 $(1+\varepsilon)$-稀疏子图的同时，最小化每次点插入或删除所导致的边增减次数（即更新代价）。我们提出一种全动态算法：点插入的平摊更新代价为常数，点删除的平摊更新代价为 $O(\log\Delta)$，其中 $\Delta$ 为点集的长宽比。