Approximately Counting Triangles in Sublinear Time

Talya Eden, Amit Levi, Dana Ron, C. Seshadhri

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 sub linear-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 <epsilon<1, the algorithm provides an estimate hat{t} such that with high constant probability, (1-epsilon) t<hat{t}<(1+epsilon)t, where t is the number of triangles in the graph G. The expected query complexity of the algorithm is O(n/t{1/3} + min {m, m {3/2}/t}) poly(log n, 1/epsilon), where n is the number of vertices in the graph and m is the number of edges, and the expected running time is (n/t{1/3} + m {3/2}/t) poly(log n, 1/epsilon). We also prove that Ω(n/t {1/3} + min {m, m {3/2}/t}) queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to polylogarithmic factors in n (and the dependence on 1/epsilon).

Original languageEnglish
Title of host publicationProceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015
PublisherIEEE Computer Society
Number of pages20
ISBN (Electronic)9781467381918
StatePublished - 11 Dec 2015
Externally publishedYes
Event56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States
Duration: 17 Oct 201520 Oct 2015

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428


Conference56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© 2015 IEEE.


  • Sublinear Approximation Algorithm
  • Triangles Counting

ASJC Scopus subject areas

  • General Computer Science


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