A $t$-spanner of an undirected $n$-vertex graph $G$ is a sparse subgraph $H$ of $G$ that preserves all pairwise distances between its vertices to within multiplicative factor $t$, also called the \emph{stretch}. We investigate the problem of maintaining spanners in the fully dynamic setting with an adaptive adversary. Despite a long line of research, this problem is still poorly understood: no algorithm achieving a sublogarithmic stretch, a sublinear in $n$ update time, and a strongly subquadratic in $n$ spanner size is currently known. One of our main results is a deterministic algorithm, that, for any $512 \leq k \leq (\log n)^{1/49}$ and $1/k\leq δ\leq 1/400$, maintains a spanner $H$ of a fully dynamic graph with stretch $poly(k)\cdot 2^{O(1/δ^6)}$ and size $|E(H)|\leq O(n^{1+O(1/k)})$, with worst-case update time $n^{O(δ)}$ and recourse $n^{O(1/k)}$. Our algorithm relies on a new technical tool that we develop, called low-diameter router decomposition. We design a deterministic algorithm that maintains a decomposition of a fully dynamic graph into edge-disjoint clusters with bounded vertex overlap, where each cluster $C$ is a bounded-diameter router, meaning that any reasonable multicommodity demand over the vertices of $C$ can be routed along short paths and with low congestion. A similar graph decomposition notion was introduced by [Haeupler et al., STOC 2022] and strengthened by [Haeupler et al., FOCS 2024]. However, in contrast to these and other prior works, the decomposition that our algorithm maintains is proper, ensuring that the routing paths between the pairs of vertices of each cluster $C$ are contained inside $C$, rather than in the entire graph $G$. We show additional applications of our router decomposition, including dynamic algorithms for fault-tolerant spanners and low-congestion spanners.

翻译：在无向$n$顶点图$G$中，$t$-生成树是$G$的一个稀疏子图$H$，它将其顶点间所有成对距离保持至多$t$倍乘因子内，该因子亦称为\emph{伸展度}。我们研究了在自适应敌手模型下的全动态环境中维护生成树的问题。尽管已有长期研究，该问题仍未被充分理解：目前尚不存在能够同时实现次对数伸展度、$n$的次线性更新时间以及$n$的强次二次方生成树规模的算法。我们的主要成果之一是一个确定性算法，对于任意满足$512 \leq k \leq (\log n)^{1/49}$和$1/k\leq δ\leq 1/400$的参数，该算法维护一个全动态图的生成树$H$，其伸展度为$poly(k)\cdot 2^{O(1/δ^6)}$，规模$|E(H)|\leq O(n^{1+O(1/k)})$，最坏情况更新时间为$n^{O(δ)}$，调整代价为$n^{O(1/k)}$。我们的算法依赖于我们开发的一种称为低直径路由器分解的新技术工具。我们设计了一个确定性算法，用于维护将全动态图分解为边不相交簇的结构，这些簇具有有界的顶点重叠，其中每个簇$C$是一个有界直径路由器，这意味着在$C$的顶点上任何合理的多商品需求都可以通过短路径且低拥塞地进行路由。类似的图分解概念由[Haeupler等人，STOC 2022]提出，并由[Haeupler等人，FOCS 2024]加强。然而，与这些及其他先前工作不同的是，我们的算法维护的分解是严格的，确保每个簇$C$的顶点对之间的路由路径包含在$C$内部，而非整个图$G$中。我们展示了路由器分解的更多应用，包括容错生成树和低拥塞生成树的动态算法。