{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T13:33:46Z","timestamp":1775828026077,"version":"3.50.1"},"reference-count":9,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":6068,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In a random <jats:italic>n<\/jats:italic>\u2010vertex digraph, each arc is present with probability <jats:italic>p<\/jats:italic>, independently of the presence or absence of other arcs. We investigate the structure of the strong components of a random digraph and present an algorithm for the construction of the transitive closure of a random digraph. We show that, when <jats:italic>n<\/jats:italic> is large and <jats:italic>np<\/jats:italic> is equal to a constant <jats:italic>c<\/jats:italic> greater than 1, it is very likely that all but one of the strong components are very small, and that the unique large strong component contains about \u0398<jats:italic><jats:sub>2<\/jats:sub>n<\/jats:italic> vertices, where \u0398 is the unique root in [0, 1] of the equation 1 \u2212 <jats:italic>x<\/jats:italic> \u2212 <jats:italic>e<\/jats:italic><jats:sup>\u2212ex<\/jats:sup> = 0. Nearly all the vertices outside the large strong component line in strong components of size 1. Provided that the expected degree of a vertex is bounded away from 1, our transitive closure algorithm runs in expected time <jats:italic>O(n).<\/jats:italic> for all choices of <jats:italic>n<\/jats:italic> and <jats:italic>p<\/jats:italic>, the expected execution time of the algorithm is <jats:italic>O(w(n)<\/jats:italic> (<jats:italic>n<\/jats:italic> log <jats:italic>n<\/jats:italic>)<jats:sup>4\/3<\/jats:sup>), where <jats:italic>w(n)<\/jats:italic> is an arbitrary nondecreasing unbounded function. To circumvent the fact that the size of the transitive closure may be \u03a9(<jats:italic>n<jats:sup>2<\/jats:sup>)<\/jats:italic> the algorithm presents the transitive closure in the compact form <jats:italic>(A \u00d7 B)<\/jats:italic>U <jats:italic>C<\/jats:italic>, where <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic> are sets of vertices, and <jats:italic>C<\/jats:italic> is a set of arcs.<\/jats:p>","DOI":"10.1002\/rsa.3240010106","type":"journal-article","created":{"date-parts":[[2007,5,31]],"date-time":"2007-05-31T05:08:48Z","timestamp":1180588128000},"page":"73-93","source":"Crossref","is-referenced-by-count":134,"title":["The transitive closure of a random digraph"],"prefix":"10.1002","volume":"1","author":[{"given":"Richard M.","family":"Karp","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"205","article-title":"On the strong connectedness of directed random graphs","volume":"1","author":"Pal\u00e1sti I.","year":"1966","journal-title":"Studia Sci. 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Hungar."},{"key":"e_1_2_1_3_2","unstructured":"T.Uczak \u201cThe phase transition in the evolution of random graphs \u201d manuscript (1988)."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579431"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1137\/0207011"},{"key":"e_1_2_1_6_2","volume-title":"Random Graphs","author":"Boll\u00f3bas B.","year":"1985"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1137\/1115007"},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","unstructured":"P.Raghavan \u201cProbabilistic Construction of deterministic algorithms: approximating packing integer programs \u201dProceedings of the 27th Annual IEEE Symp. on Foundations of Computer Science 1986 pp.10\u201318.","DOI":"10.1109\/SFCS.1986.45"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-51866-9"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65371-1"}],"container-title":["Random Structures &amp; Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Frsa.3240010106","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/rsa.3240010106","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T09:41:44Z","timestamp":1697967704000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/rsa.3240010106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["10.1002\/rsa.3240010106"],"URL":"https:\/\/doi.org\/10.1002\/rsa.3240010106","archive":["Portico"],"relation":{},"ISSN":["1042-9832","1098-2418"],"issn-type":[{"value":"1042-9832","type":"print"},{"value":"1098-2418","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}