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

2026-01-28

AI Summary

This work addresses the open problem of efficiently maintaining a low-stretch, small-size, and rapidly updatable graph spanner in fully dynamic graphs under an adaptive adversary. The paper presents the first deterministic fully dynamic algorithm for this setting, whose core innovation is the introduction of a “low-diameter router decomposition”—a genuine decomposition technique that partitions the graph into edge-disjoint clusters with limited vertex overlap, ensuring internally closed paths within each cluster and enabling low-congestion multi-commodity flow routing. This approach achieves, for the first time, polynomial stretch, near-linear size ($O(n^{1+O(1/k)})$), and sublinear worst-case update time ($n^{O(\delta)}$) for any $512 \leq k \leq (\log n)^{1/49}$ and $1/k \leq \delta \leq 1/400$. The framework further extends to fault-tolerant and low-congestion spanner constructions.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $1/k\leq \delta \leq 1/400$, maintains a spanner $H$ of a fully dynamic graph with stretch $poly(k)\cdot 2^{O(1/\delta^6)}$ and size $|E(H)|\leq O(n^{1+O(1/k)})$, with worst-case update time $n^{O(\delta)}$ 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.

Problem

• dynamic spanners
• graph algorithms
• adaptive adversary
• stretch
• update time

Innovation

• low-diameter router decomposition
• fully dynamic spanners
• deterministic dynamic algorithms
• proper clustering
• sublinear update time