Sparsification—a technique for speeding up dynamic graph algorithms
Journal of the ACM (JACM)
Managing RBAC states with transitive relations
ASIACCS '07 Proceedings of the 2nd ACM symposium on Information, computer and communications security
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Let $P$ be a property of undirected graphs. We consider the following problem: given a graph $G$ that has property $P$, find a minimal spanning subgraph of $G$ with property $P$. We describe general algorithms for this problem and prove their correctness under fairly weak assumptions about $P$. We establish that the worst-case running time of these algorithms is $\Theta(m+n \log n)$ for 2-edge-connectivity and biconnectivity where $n$ and $m$ denote the number of vertices and edges, respectively, in the input graph. By refining the basic algorithms we obtain the first linear time algorithms for computing a minimal 2-edge-connected spanning subgraph and for computing a minimal biconnected spanning subgraph. We also devise general algorithms for computing a minimal spanning subgraph in directed graphs. These algorithms allow us to simplify an earlier algorithm of Gibbons, Karp, Ramachandran, Soroker, and Tarjan for computing a minimal strongly connected spanning subgraph. We also provide the first tight analysis of the latter algorithm, showing that its worst-case time complexity is $\Theta(m+n \log n).$