Faster Randomized Worst-Case Update Time for Dynamic Subgraph   Connectivity
  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. 
Ran Duan, Le Zhang
cs.DS