Fully Dynamic Exact Edge Connectivity in Sublinear Time

TL;DR

This paper introduces two new fully dynamic algorithms for maintaining exact edge connectivity in graphs with efficient worst-case and amortized update times, answering an open question in the field.

Contribution

It presents the first fully dynamic algorithms for exact edge connectivity with sublinear update times, improving upon previous methods that were approximate or restricted.

Findings

Achieved worst-case update time of O(n)

Achieved amortized update time of O(m^{1-1/31})

Answered an open question by Thorup about dynamic edge connectivity

Abstract

Given a simple $n$-vertex, $m$-edge graph $G$ undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of $G$ in $\tilde{O}(n)$ worst-case update time and $\tilde{O}(m^{1-1/31})$ amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms either assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results provide an affirmative answer to an open question posed by Thorup [Combinatorica'07].