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arXiv · 1412.6466

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

Monika Henzinger·Sebastian Krinninger·Veronika Loitzenbauer
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Abstract

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 $O(m n)$-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 $O(m n)$ 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 $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices.

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Monika Henzinger, Sebastian Krinninger, Veronika Loitzenbauer. 2014-12-19. Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time. https://doi.org/10.1007/978-3-662-47672-7_58

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