Faster Randomized Worst-Case Update Time for Dynamic Subgraph Connectivity

TL;DR

This paper introduces a randomized data structure for dynamic subgraph connectivity that improves worst-case update times in general undirected graphs, making it faster than previous deterministic methods.

Contribution

It presents a novel randomized data structure achieving faster worst-case update times for dynamic subgraph connectivity problems.

Findings

Achieves ((m^{3/4})) worst-case update time

Improves upon previous deterministic worst-case update bounds

Provides efficient connectivity queries in dynamic networks

Abstract

Real-world networks are prone to breakdowns. Typically in the underlying graph $G$, besides the insertion or deletion of edges, the set of active vertices changes overtime. A vertex might work actively, or it might fail, and gets isolated temporarily. The active vertices are grouped as a set $S$. $S$ is subjected to updates, i.e., a failed vertex restarts, or an active vertex fails, and gets deleted from $S$. Dynamic subgraph connectivity answers the queries on connectivity between any two active vertices in the subgraph of $G$ induced by $S$. The problem is solved by a dynamic data structure, which supports the updates and answers the connectivity queries. In the general undirected graph, the best results for it include $\widetilde{O}(m^{2/3})$ deterministic amortized update time, $\widetilde{O}(m^{4/5})$ and $\widetilde{O}(\sqrt{mn})$ deterministic worst-case update time. In the paper, we propose a randomized data structure, which has $\widetilde{O}(m^{3/4})$ worst-case update time.