Fully Dynamic Edge Connectivity in $\tilde{O}(n^{12/13})$ Time

2026-07-12

AI Summary

This work addresses the problem of efficiently maintaining edge connectivity in dynamic graphs under edge insertions and deletions. The authors propose novel randomized and deterministic algorithms that leverage innovative dynamic graph data structures and connectivity maintenance techniques. Their key contribution is the first algorithm achieving $o(n)$ worst-case update and query time for arbitrary edge connectivity values. Specifically, the randomized algorithm attains $\tilde{O}(n^{12/13})$ worst-case time per operation. The deterministic variant achieves either $n^{1+o(1)}$ worst-case or $\tilde{O}(n)$ amortized time on simple graphs, and $\tilde{O}(n^{3/2})$ time on unweighted multigraphs, significantly improving upon prior results.

Abstract

In the (fully) dynamic edge connectivity problem, the goal is to maintain the edge connectivity $λ_G$ of an $n$-vertex graph $G$ that undergoes edge insertions and deletions. Our main result is a randomized algorithm for maintaining edge connectivity in dynamic simple graphs using worst-case update and query time $\tilde{O}(n^{12/13})$, for all values of $λ_G$. This is the first algorithm that has $o(n)$ update and query time, as all existing algorithms achieve this only when $λ_G$ is below $n^{1/11}$ or above $n^{1/2}$ (up to polylogarithmic factors). We then use the tools developed for this purpose to design two additional algorithms. The first one is a deterministic algorithm for the exact same task, that uses $n^{1+o(1)}$ worst-case update and query time or $\tilde{O}(n)$ amortized update and query time; this gives a polynomial improvement over existing deterministic algorithms. The second one is a deterministic algorithm for the same task but in dynamic unweighted multigraphs, that uses $\tilde{O}(n^{3/2})$ worst-case update and query time.

Problem

• dynamic graph
• edge connectivity
• graph algorithms
• data structures
• connectivity maintenance

Innovation

• dynamic graph algorithms
• edge connectivity
• sublinear time
• randomized algorithm
• deterministic algorithm