Fully Dynamic Algorithms for Graph Spanners via Low-Diameter Router Decomposition

TL;DR

The paper tackles maintaining graph spanners in fully dynamic graphs under adaptive adversaries, a long-standing open problem in which sublogarithmic stretch, sublinear update time, and subquadratic spanner size have not been simultaneously achieved. It introduces a deterministic approach based on a low-diameter router decomposition, a proper partition of the graph into edge-disjoint clusters each serving as a router, enabling short routing paths and controlled congestion within clusters. The main technical contribution is a suite of tools—the Nice Router construction with pruning, embedding of routers, router witnesses, and decremental low-diameter clustering—that together yield a dynamic spanner with stretch $poly(k)\cdot 2^{O(1/\delta^6)}$, size $O(n^{1+O(1/k)})$, worst-case update time $n^{O(\delta)}$, and recourse $n^{O(1/k)}$. The framework extends to fault-tolerant and low-congestion spanners and to connectivity certificates, providing near-linear-time guarantees against adaptive adversaries and broad applicability to dynamic network design and routing algorithms.

Abstract

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.

Figures (1)

• Figure 1: An illustration for the proof of Lemma \ref{['thm:resilient-router']}. Shown is an $i$-unsafe pair $(x_{\ell},y_{\ell})$ with their corresponding trees $T_{i}(x_{\ell})$ and $T_{i}(y_{\ell})$. The routing instance for iteration $i$ is defined by sending $\lambda$ flow units between matched leaf vertices in $L_{i}(x_{\ell})$ and $L_{i}(y_{\ell})$. The solid edges correspond to $\lambda$ units of demand between the matched pair.

Theorems & Definitions (70)

• Theorem 1.1: Dynamic Deterministic Spanners, Informal
• Theorem 1.2: Dynamic Router Decomposition, Informal
• Theorem 1.3
• Definition 1: $\varphi$-expander
• Definition 2: $\overline{w}$-restricted demand
• Definition 3: Routing of a demand
• Definition 4: $(\overline{w},d,\eta)$-Router.
• Definition 5: Well-Connected Graph
• Claim 1
• Theorem 2.1: Corollary 5.3 from chuzhoy2023distanced