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2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (2015)
Berkeley, CA, USA
Oct. 17, 2015 to Oct. 20, 2015
ISSN: 0272-5428
ISBN: 978-1-4673-8191-8
pp: 614-633
ABSTRACT
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 &amp;#x03A9;(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).
INDEX TERMS
Approximation algorithms, Approximation methods, Algorithm design and analysis, Complexity theory, TV, Yttrium, Computer science
CITATION

T. Eden, A. Levi, D. Ron and C. Seshadhri, "Approximately Counting Triangles in Sublinear Time," 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, CA, USA, 2015, pp. 614-633.
doi:10.1109/FOCS.2015.44
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