Connectivity oracles for failure prone graphs
Proceedings of the forty-second ACM symposium on Theory of computing
New data structures for subgraph connectivity
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Connectivity oracles for planar graphs
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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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 $S \subseteq V$ to support insertions/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 a combination of several simple ideas. Our method requires $\widetilde O(|E|^{4\omega/(3\omega+3)})=O(|E|^{0.94})$ amortized update time, and $\widetilde O(|E|^{1/3})$ query time, after $\widetilde O(|E|^{(5\omega+1)/(3\omega+3)})$ preprocessing time, where &ohgr; is the matrix multiplication exponent and $\widetilde O$ hides polylogarithmic factors.