Constant-time Connectivity and 2-Edge Connectivity Querying in Dynamic Graphs

TL;DR

This paper introduces a new dynamic graph connectivity algorithm that achieves constant-time queries and improved update efficiency by combining spanning trees and disjoint-set structures, also extending to 2-edge connectivity.

Contribution

The paper presents a novel spanning-tree-based method that maintains connectivity and 2-edge connectivity in fully dynamic graphs with constant query time and enhanced update performance.

Findings

Achieves constant-time connectivity queries in dynamic graphs.

Significantly improves theoretical running time for edge updates.

Demonstrates superior performance on large real datasets.

Abstract

Connectivity query processing is a fundamental problem in graph processing. Given an undirected graph and two query vertices, the problem aims to identify whether they are connected via a path. Given frequent edge updates in real graph applications, in this paper, we study connectivity query processing in fully dynamic graphs, where edges are frequently inserted or deleted. A recent solution, called D-tree, maintains a spanning tree for each connected component and applies several heuristics to reduce the depth of the tree. To improve efficiency, we propose a new spanning-tree-based solution by maintaining a disjoint-set tree simultaneously. By combining the advantages of two trees, we achieve the constant query time complexity and also significantly improve the theoretical running time in both edge insertion and edge deletion. In addition, we extend our connectivity maintenance algorithms to maintain 2-edge connectivity. Our performance studies on real large datasets show considerable improvement of our algorithms.