We give a randomized ARBITRARY CRCW PRAM algorithm for computing a spanning forest of an \(n\)-vertex, \(m\)-edge graph. Let \(d\) be the maximum component diameter. With good probability, the algorithm runs in \(O(\log d+\log\log n)\) time using \(O(m+n)\) processors and outputs a rooted spanning forest of original edges with maximum depth \(2^{O(\log d+\log\log n)}\). For finding bridges, our algorithm identifies every bridge edge in \(O(\log d+\log\log n)\) time. For biconnectivity, it outputs all blocks in \(O(\log d+\log d'+\log\log n)\) time, where \(d'\) is the bi-diameter. All these problems are fundamental in graph algorithms with numerous applications. Our results improve the best previous time bounds parameterized by the number of vertices and diameter. Finally, we prove deterministic \(Ω(\log n/\log\log n)\) lower bounds for outputting a DFS Sequence, a weighted spanning forest, and an ear decomposition.

翻译：暂无翻译