Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time
We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs with m edges and n vertices only rather simple O(mn)-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time \(O(n^2)\). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an \(O(m^2 /\log n)\)-time algorithm for 2-edge 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 \(O(n^2 \log n)\) for k-edge-connectivity and \(O(n^3)\) for k-vertex-connectivity.
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