New Data Structures for Subgraph Connectivity

We study the “subgraph connectivity” problem for undirected graphs with sublinear vertex update time. In this problem, we can make vertices active or inactive in a graph G, and answer the connectivity between two vertices in the subgraph of G induced by the active vertices. In this paper, we solve two open problems in subgraph connectivity. We give the first subgraph connectivity structure with worst-case sublinear time bounds for both updates and queries. Our worst-case subgraph connectivity structure supports Õ(m) update time, Õ(m) query time and occupies Õ(m) space, where m is the number of edges in the whole graph G. In the second part of our paper, we describe another dynamic subgraph connectivity structure with amortized Õ(m) update time, Õ(m) query time and linear space, which improves the structure introduced by [Chan, Pǎtraşcu, Roditty, FOCS’08] that takes Õ(m) space.