Approximately Counting Triangles in Sublinear Time.

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 < epsilon < 1, the algorithm provides an estimate (t) over cap such that, with high constant probability, (1 - epsilon) . t < <(t)over cap> < (1 + epsilon) . t, where t is the number s12triangles in the graph G. The expected query complexity of the algorithm is ((n)(t1/3) + min{m, (m3/2)(t) }) . poly(log n, 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 (n/(t)1/3 + m(3/2)/t) . poly(log n,1/epsilon) We also prove that Omega(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 the dependence on poly(log n, 1/epsilon).