{"oa_version":"Preprint","extern":"1","status":"public","publisher":"Springer Nature","abstract":[{"lang":"eng","text":"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(m n)-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/log𝑛)-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(𝑛2log𝑛) for k-edge-connectivity and 𝑂(𝑛3) for k-vertex-connectivity."}],"year":"2015","publication_identifier":{"issn":["0302-9743"],"isbn":["9783662476710"]},"month":"07","date_updated":"2023-02-10T09:21:47Z","title":"Finding 2-edge and 2-vertex strongly connected components in quadratic time","article_processing_charge":"No","scopus_import":"1","date_created":"2022-08-11T09:38:34Z","citation":{"apa":"Henzinger, M. H., Krinninger, S., &#38; Loitzenbauer, V. (2015). Finding 2-edge and 2-vertex strongly connected components in quadratic time. In <i>2nd International Colloquium on Automata, Languages and Programming</i> (Vol. 9134, pp. 713–724). Kyoto, Japan: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-662-47672-7_58\">https://doi.org/10.1007/978-3-662-47672-7_58</a>","chicago":"Henzinger, Monika H, Sebastian Krinninger, and Veronika Loitzenbauer. “Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time.” In <i>2nd International Colloquium on Automata, Languages and Programming</i>, 9134:713–24. Springer Nature, 2015. <a href=\"https://doi.org/10.1007/978-3-662-47672-7_58\">https://doi.org/10.1007/978-3-662-47672-7_58</a>.","mla":"Henzinger, Monika H., et al. “Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time.” <i>2nd International Colloquium on Automata, Languages and Programming</i>, vol. 9134, Springer Nature, 2015, pp. 713–24, doi:<a href=\"https://doi.org/10.1007/978-3-662-47672-7_58\">10.1007/978-3-662-47672-7_58</a>.","ama":"Henzinger MH, Krinninger S, Loitzenbauer V. Finding 2-edge and 2-vertex strongly connected components in quadratic time. In: <i>2nd International Colloquium on Automata, Languages and Programming</i>. Vol 9134. Springer Nature; 2015:713-724. doi:<a href=\"https://doi.org/10.1007/978-3-662-47672-7_58\">10.1007/978-3-662-47672-7_58</a>","ieee":"M. H. Henzinger, S. Krinninger, and V. Loitzenbauer, “Finding 2-edge and 2-vertex strongly connected components in quadratic time,” in <i>2nd International Colloquium on Automata, Languages and Programming</i>, Kyoto, Japan, 2015, vol. 9134, pp. 713–724.","short":"M.H. Henzinger, S. Krinninger, V. Loitzenbauer, in:, 2nd International Colloquium on Automata, Languages and Programming, Springer Nature, 2015, pp. 713–724.","ista":"Henzinger MH, Krinninger S, Loitzenbauer V. 2015. Finding 2-edge and 2-vertex strongly connected components in quadratic time. 2nd International Colloquium on Automata, Languages and Programming. ICALP: International Colloquium on Automata, Languages, and Programming, LNCS, vol. 9134, 713–724."},"type":"conference","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"713 - 724","intvolume":"      9134","language":[{"iso":"eng"}],"quality_controlled":"1","oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1412.6466","open_access":"1"}],"author":[{"first_name":"Monika H","id":"540c9bbd-f2de-11ec-812d-d04a5be85630","orcid":"0000-0002-5008-6530","last_name":"Henzinger","full_name":"Henzinger, Monika H"},{"full_name":"Krinninger, Sebastian","last_name":"Krinninger","first_name":"Sebastian"},{"full_name":"Loitzenbauer, Veronika","last_name":"Loitzenbauer","first_name":"Veronika"}],"doi":"10.1007/978-3-662-47672-7_58","alternative_title":["LNCS"],"_id":"11787","volume":9134,"external_id":{"arxiv":["1412.6466"]},"date_published":"2015-07-06T00:00:00Z","day":"06","publication_status":"published","publication":"2nd International Colloquium on Automata, Languages and Programming","conference":{"end_date":"2015-07-10","name":"ICALP: International Colloquium on Automata, Languages, and Programming","start_date":"2015-07-06","location":"Kyoto, Japan"}}