Faster Randomized Worst-Case Update Time for Dynamic Subgraph Connectivity

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. The set 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, we propose a randomized data structure, which has \(\widetilde{O}(m^{3/4})\) worst-case update time. The former best results for it include \(\widetilde{O}(m^{2/3})\) deterministic amortized update time by Chan, Pǎtrascu and Roditty [4], \(\widetilde{O}(m^{4/5})\) by Duan [8] and \(\widetilde{O}(\sqrt{mn})\) by Baswana, Chaudhury, Choudhary and Khan [2] deterministic worst-case update time.