Simple Algorithms for Fully Dynamic Edge Connectivity

2025-08-11

AI Summary

This paper studies the fully dynamic edge-connectivity maintenance problem: given a simple graph $G$ undergoing edge insertions and deletions, maintain its edge connectivity $lambda_G$ in real time. We propose two randomized dynamic algorithms. The first achieves $ ilde{O}(n)$ worst-case update time, matching the current state-of-the-art bound. The second yields a breakthrough for highly connected graphs—when $lambda_G = omega(sqrt{n})$, it reduces update time to $ ilde{O}(n/lambda_G)$ and query time to $ ilde{O}(n^2/lambda_G^2)$, achieving the first sublinear (i.e., $o(n)$) worst-case update and query times for high-connectivity graphs. Our algorithms integrate randomized sampling, dynamic forest maintenance, and a succinct potential-function analysis—significantly improving upon prior approaches whose polynomial dependence on $lambda_G$ incurred prohibitive overhead.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $λ_G$. We present two simple randomized algorithms solving this problem. The first algorithm maintains the edge connectivity in worst-case update time $ ilde{O}(n)$ per edge update, matching the known bound but with simpler analysis. Our second algorithm achieves worst-case update time $ ilde{O}(n/λ_G)$ and worst-case query time $ ilde{O}(n^2/λ_G^2)$, which is the first algorithm with worst-case update and query time $o(n)$ for large edge connectivity, namely, $λ_G = ω(sqrt{n})$.

Problem

• Maintain edge connectivity in dynamic graphs
• Achieve efficient update time for connectivity
• Provide fast query time for large connectivity

Innovation

• Simple randomized algorithms for dynamic edge connectivity
• Worst-case update time matching known bounds
• First algorithm with sublinear update and query time