Dynamic Connectivity, Minimum Spanning Tree, and 2-Edge Connectivity with Polylogarithmic Worst-Case Update Time

2026-09-30

AI Summary

This study addresses the inefficiency of worst-case update times for dynamic graph connectivity, minimum spanning tree, and 2-edge connectivity problems. To overcome these limitations, this work proposes randomized algorithms that withstand adaptive adversaries by relying solely on the randomness inherent in static expander decompositions, while further establishing a deterministic reduction pathway. Consequently, the proposed approach reduces the worst-case update time for these dynamic graph problems to polylogarithmic bounds, breaking previous subpolynomial barriers. By operating with high probability against adaptive adversaries, the developed algorithms significantly improve upon the best-known existing time complexity results.

Abstract

We give fully dynamic algorithms for maintaining connectivity, minimum spanning tree, and $2$-edge connectivity of a graph with worst-case polylogarithmic update time. Our algorithms are randomized and succeed with high probability against an adaptive adversary. For the minimum spanning tree and $2$-edge connectivity problems, this improves over the subpolynomial update time bounds obtained by Nanongkai, Saranurak, and Wulff-Nilsen [FOCS'17], Jin and Sun [FOCS'21], and Jin, Sun, and Thorup [SODA'24], respectively. The only randomized component of our algorithms is the computation of static expander decompositions, and a deterministic algorithm for said problem would directly imply deterministic algorithms for all three problems. This reduction is novel even for the connectivity problem.

Problem

• Dynamic Connectivity
• Minimum Spanning Tree
• 2-Edge Connectivity
• Fully Dynamic Graphs
• Worst-Case Update Time

Innovation

• Dynamic Connectivity
• Minimum Spanning Tree
• 2-Edge Connectivity
• Polylogarithmic Update Time
• Expander Decomposition