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 em 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<ε<1, the algorithm provides an estimate widehatt such that with high constant probability, (1-ε)· t< widehatt<(1+ε)· t, where t is the number of triangles in the graph G. The expected query complexity of the algorithm is !left(fracnt1/3 + minleftm, fracm3/2trightright)· rm poly(log n, 1/ε), where n is the number of vertices in the graph and m is the number of edges, and the expected running time is !left(fracnt1/3 + fracm3/2tright)· rm poly(log n, 1/ε). We also prove that Omega!left(fracnt1/3 + minleftm, fracm3/2trightright) 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/ε).