Counting Triangles in Sub-linear Time

We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in two models: exact counting algorithms, which require reading the entire graph, and streaming algorithms, where the edges are given in a stream and the memory is limited. 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. The problem of counting triangles in sublinear time was previously considered in the work of Gonen et al. (SIAM Journal on Discrete Mathematics, 2011), whose main focus was on counting the number of stars (of a given size) in sublinear time. They gave a linear lower bound on the running time of any algorithm for approximating the number of triangles in a graph when the algorithm is allowed degree and neighbor queries (and when the number of edges and number of triangles are linear in the number of vertices). Following this lower bound they ask if sublinear-time algorithms can be designed when the algorithm is also allowed vertex-pair queries. We answer this question affirmatively by presenting an algorithm that, for any given approximation parameter 0 < < 1, provides an estimation ∆̂ such that with high constant probability, (1 − )∆(G) < ∆̂ < (1 + )∆(G), where ∆(G) is the number of triangles in the graph G. The query complexity of the algorithm is O ( n ∆(G)1/3 + min { m, m 3/2 ∆(G) }) ·poly(log n, 1/ ), where n is the number of vertices in the graph and m is the number of edges. For values of ∆ and m such that ∆ ≥ √ m the running time is of the same order as the query complexity. We also prove lower bounds that show this algorithm is tight up to polylogarithmic factors in n and the dependence on 1/ .