Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time
نویسندگان
چکیده
We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time O(n). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an O(m/ log n)-time algorithm for 2edge strongly connected components, and thus improve over the O(mn) running time also when m = O(n). Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of O(n log n) for edges and O(n) for vertices.
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