Abstract
(MATH) Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph $G=(V,E)$ and a subset of vertices $S\on V$, to support insertions and deletions in~$S$ and connectivity queries (are two vertices connected\@?) in the subgraph induced by~$S$. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, $\OO (|E|^{4\w/(3\w+3)})=O(|E|^{0.94})$ amortized update time, and $\OO(|E|^{1/3})$ query time, where $\w$ is the matrix multiplication exponent and $\OO$ hides polylogarithmic factors.
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