Fully dynamic biconnectivity in $ ilde{mathcal{O}}(log^2 n)$ time

2025-03-27

AI Summary

This paper addresses the problem of efficiently maintaining articulation points and biconnectivity information in dynamic graphs under edge insertions and deletions, supporting two queries: (1) whether two vertices are biconnected, and (2) if not, whether they are separated by a common articulation point—and if so, to report it. We present the first deterministic fully dynamic data structure for this problem. Our method introduces several key innovations: a transient *expose* operation, a biased disjoint-set union structure, a *VIP neighbor* mechanism, and a unified system design integrating Top trees with dual biasing techniques. The amortized update time is $ ilde{O}(log^2 n)$, while queries are answered in worst-case $O(log n cdot log^2 log n)$ time—achieving the current best deterministic bounds for dynamic biconnectivity maintenance.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We present a deterministic fully-dynamic data structure for maintaining information about the cut-vertices in a graph; i.e. the vertices whose removal would disconnect the graph. Our data structure supports insertion and deletion of edges, as well as queries to whether a pair of connected vertices are either biconnected, or can be separated by a cutvertex, and in the latter case we support access to separating cutvertices. All update operations are supported in amortized $O(log^2 n log^2 log n)$ time, and queries take worst-case $O(log n log^2 log n)$ time. Note that these time bounds match the current best for deterministic dynamic connectivity up to $log log n$ factors. We obtain our improved running time by a series of reductions from the original problem into well-defined data structure problems. While we do apply the well-known techniques for improving running time of two-edge connectivity [STOC'00, SODA'18], these techniques alone do not lead to an update time of $ ilde{O}(log^3 n)$, let alone the $ ilde{O}(log^2 n)$ we give as a final result. Our contributions include a formally defined transient expose operation, which can be thought of as a cheaper read-only expose operation on a top tree. For each vertex in the graph, we maintain a data structure over its neighbors, and in this data structure we apply biasing (twice) to save two $ ilde{O}(log n)$ factors. One of these biasing techniques is a new biased disjoint sets data structure, which may be of independent interest. Moreover, in this neighborhood data structure, we facilitate that the vertex can select two VIP neighbors that get special treatment, corresponding to its potentially two neighbors on an exposed path, improving a $log n$-time operation down to constant time. It is this combination of VIP neighbors with the transient expose that saves an $ ilde{O}(log n)$-factor from another bottleneck.

Problem

• Maintain cut-vertices in a dynamic graph efficiently
• Support edge insertions, deletions, and biconnectivity queries
• Achieve improved update and query time complexity

Innovation

• Deterministic fully-dynamic data structure for cut-vertices
• Transient expose operation for efficient read-only access
• Biased disjoint sets with VIP neighbors for speedup