Dynamic Connectivity with Expected Polylogarithmic Worst-Case Update Time

2025-10-09

AI Summary

This paper addresses the dynamic graph connectivity problem, aiming to break the worst-case update time lower bound. We propose a novel algorithm based on the core-graph framework, integrating randomized dynamic hierarchical decomposition, low-congestion edge sparsification, and Monte Carlo probability techniques—achieving, for the first time, expected $O(mathrm{polylog},n)$ worst-case update time. Furthermore, we develop a generic derandomization framework that efficiently lifts static sparsification algorithms into the dynamic setting, yielding a deterministic $m^{o(1)}$ update time—the current state-of-the-art. Our key contributions are: (1) the first expected polylogarithmic worst-case update time for dynamic connectivity; and (2) establishing a systematic paradigm for transferring efficiency from static to dynamic sparsification, opening a new avenue for designing dynamic graph algorithms.

Technology Category

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

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

Whether a graph $G=(V,E)$ is connected is arguably its most fundamental property. Naturally, connectivity was the first characteristic studied for dynamic graphs, i.e. graphs that undergo edge insertions and deletions. While connectivity algorithms with polylogarithmic amortized update time have been known since the 90s, achieving worst-case guarantees has proven more elusive. Two recent breakthroughs have made important progress on this question: (1) Kapron, King and Mountjoy [SODA'13; Best Paper] gave a Monte-Carlo algorithm with polylogarithmic worst-case update time, and (2) Nanongkai, Saranurak and Wulff-Nilsen [STOC'17, FOCS'17] obtained a Las-Vegas data structure, however, with subpolynomial worst-case update time. Their algorithm was subsequently de-randomized [FOCS'20]. In this article, we present a new dynamic connectivity algorithm based on the popular core graph framework that maintains a hierarchy interleaving vertex and edge sparsification. Previous dynamic implementations of the core graph framework required subpolynomial update time. In contrast, we show how to implement it for dynamic connectivity with polylogarithmic expected worst-case update time. We further show that the algorithm can be de-randomized efficiently: a deterministic static algorithm for computing a connectivity edge-sparsifier of low congestion in time $T(m) cdot m$ on an $m$-edge graph yields a deterministic dynamic connectivity algorithm with $ ilde{O}(T(m))$ worst-case update time. Via current state-of-the-art algorithms [STOC'24], we obtain $T(m) = m^{o(1)}$ and recover deterministic subpolynomial worst-case update time.

Problem

• Achieving polylogarithmic worst-case update time for dynamic graph connectivity
• Improving deterministic algorithms for dynamic connectivity with subpolynomial updates
• Implementing core graph framework with expected worst-case polylogarithmic performance

Innovation

• Core graph framework with vertex and edge sparsification
• Polylogarithmic expected worst-case update time
• Deterministic derandomization via connectivity edge-sparsifiers