{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T17:39:06Z","timestamp":1790185146150,"version":"4.1.0"},"reference-count":53,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100011643","name":"Blavatnik Family Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100011643","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["671\/13"],"award-info":[{"award-number":["671\/13"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sublinear-time algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries, and neighbor queries. We show that for any given approximation parameter $0&lt;\\epsilon&lt;1$, the algorithm provides an estimate $\\widehat{t}$ such that, with high constant probability, $(1-\\epsilon)\\cdot t&lt; \\widehat{t}&lt;(1+\\epsilon)\\cdot t$, where $t$ is the number of triangles in the graph $G$. The expected query complexity of the algorithm is $(\\frac{n}{t^{1\/3}} + \\min\\{m, \\frac{m^{3\/2}}{t}\\})\\cdot {poly}(\\log n, \\frac{1}{\\epsilon})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges. The expected running time of the algorithm is $(\\frac{n}{t^{1\/3}} + \\frac{m^{3\/2}}{t})\\cdot {poly}(\\log n, \\frac{1}{\\epsilon})$. We also prove that $\\Omega(\\frac{n}{t^{1\/3}} + \\min\\{m, \\frac{m^{3\/2}}{t}\\})$ queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to the dependence on ${poly}(\\log n, \\frac{1}{\\epsilon})$.<\/jats:p>","DOI":"10.1137\/15m1054389","type":"journal-article","created":{"date-parts":[[2017,10,10]],"date-time":"2017-10-10T16:21:21Z","timestamp":1507652481000},"page":"1603-1646","source":"Crossref","is-referenced-by-count":48,"title":["Approximately Counting Triangles in Sublinear Time"],"prefix":"10.1137","volume":"46","author":[{"given":"Talya","family":"Eden","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Amit","family":"Levi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dana","family":"Ron","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Seshadhri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,10,10]]},"reference":[{"key":"atypb1","first-page":"1446","author":"Ahmed N. 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Avron,\n                      Counting triangles in large graphs using randomized matrix trace estimation\n                      , in Proceedings of the Second Workshop on Large-Scale Data Mining: Theory and Applications, vol. 10, 2010."},{"key":"atypb6","first-page":"623","author":"Bar-Yossef Z.","year":"2002","journal-title":"SIAM"},{"key":"atypb7","first-page":"16","author":"Becchetti L.","year":"2008","journal-title":"ACM"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.056119"},{"key":"atypb9","first-page":"223","author":"Bj\u00f6rklund A.","year":"2014","journal-title":"Springer"},{"key":"atypb10","first-page":"253","author":"Buriol L. S.","year":"2006","journal-title":"ACM"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1086\/421787"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702403244"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"H. Chernoff,\n                      A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations\n                      , Ann. Math. Statist. (1952), pp. 493-507.","DOI":"10.1214\/aoms\/1177729330"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/0214017"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"S. Chu and J. Cheng,\n                      Triangle listing in massive networks\n                      , ACM Trans. Knowl. Discov. Data (TKDD), 6 (2012), 17.","DOI":"10.1145\/2382577.2382581"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/MCSE.2009.120"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1086\/228943"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539703435297"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/060672121"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.032093399"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539704447304"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-001-0078-7"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20203"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/100783066"},{"key":"atypb25","first-page":"22","author":"Hassidim A.","year":"2009","journal-title":"IEEE"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1086\/224954"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/0207033"},{"key":"atypb28","first-page":"589","author":"Jha M.","year":"2013","journal-title":"ACM"},{"key":"atypb29","first-page":"710","author":"Jowhari H.","year":"2005","journal-title":"Springer"},{"key":"atypb30","first-page":"598","author":"Kane D. 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Upfal,\n                      Probability and Computing-Randomized Algorithms and Probabilistic Analysis\n                      , Cambridge University Press, UK, 2005.","DOI":"10.1017\/CBO9780511813603"},{"key":"atypb37","first-page":"327","author":"Nguyen H. 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