An optimal algorithm for 3-edge-connectivity is presented. The algorithm performs only one pass over the given graph to determine a set of cut-pairs whose removal leads to the 3-edge-connected components. An additional pass determines all the 3-edge-connected components of the given graph. The algorithm is simple, easy to implement and runs in linear time and space. Experimental results show that it outperforms all the previously known linear-time algorithms for 3-edge-connectivity in determining if a given graph is 3-edge-connected and in determining cut-pairs. Its performance is also among the best in determining the 3-edge-connected components.
In this paper, we present efficient parallel algorithms for the following graph problems: finding the lowest common ancestors for vertex pairs of a directed tree; finding all fundamental cycles, a directed spanning forest, all bridges, all bridge-connected components, all separation vertices, all biconnected components, and testing the biconnectivity of an undirected graph. All these algorithms achieve the O(lg n) time bound, with the first two algorithms using n[n/lg n] processors and the remaining algorithms using n[n/lg n] processors. In all cases, our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n. Moreover, our algorithms are optimal with respect to the time-processor product for dense graphs, with the exception of the first two algorithms.The machine model we use is the PRAM which is a SIMD model allowing simultaneous reads but not simultaneous writes to the same memory location.
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