Finding 2-edge and 2-vertex strongly connected components in quadratic time

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20155101705918
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(1) University of Vienna, Vienna, Austria | 1600年 / ERATO Kawarabayashi Large Graph Project; MEXT Grant-in-Aid for Scientific Research on Innovative Areas: Exploring the Limits of Computation; Research Institute for Mathematical Sciences; Tateisi Science and Technology Foundation卷 / Springer Verlag期
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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(n2). For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an O(m2/ 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(n2 log n) for k-edge-connectivity and O(n3) for k-vertex-connectivity. © Springer-Verlag Berlin Heidelberg 2015;
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