Simple Algorithms for Fully Dynamic Edge Connectivity

TL;DR

This paper introduces two simple randomized algorithms for fully dynamic edge connectivity, achieving efficient worst-case update and query times, especially for graphs with high edge connectivity.

Contribution

The paper presents two new randomized algorithms for dynamic edge connectivity with improved worst-case times and simpler analysis compared to prior work.

Findings

First algorithm maintains edge connectivity with worst-case time $ ilde{O}(n)$.

Second algorithm achieves worst-case update time $ ilde{O}(n/\lambda_G)$ and query time $ ilde{O}(n^2/\lambda_G^2)$.

Effective for graphs with high edge connectivity $ ext{}\lambda_G = ext{ } ext{} ext{o}(n)$.

Abstract

In the fully dynamic edge connectivity problem, the input is a simple graph $G$ undergoing edge insertions and deletions, and the goal is to maintain its edge connectivity, denoted $\lambda_G$. We present two simple randomized algorithms solving this problem. The first algorithm maintains the edge connectivity in worst-case update time $\tilde{O}(n)$ per edge update, matching the known bound but with simpler analysis. Our second algorithm achieves worst-case update time $\tilde{O}(n/\lambda_G)$ and worst-case query time $\tilde{O}(n^2/\lambda_G^2)$, which is the first algorithm with worst-case update and query time $o(n)$ for large edge connectivity, namely, $\lambda_G = \omega(\sqrt{n})$.