Technology Category
Machine Learning: Graph-based Machine LearningData Mining & Knowledge Management: Graph Mining, Social Network Analysis & CommunityReasoning under Uncertainty: Stochastic Optimization
Simple Algorithms for Fully Dynamic Edge Connectivity
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Maintain edge connectivity in dynamic graphs
Achieve efficient update time for connectivity
Provide fast query time for large connectivity
Innovation
Methods, ideas, or system contributions that make the work stand out.
Simple randomized algorithms for dynamic edge connectivity
Worst-case update time matching known bounds
First algorithm with sublinear update and query time
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