Finding Triangles and Small Subgraphs in a Distributed Setting

Let G = (V,E) be an n-vertex graph and Md a d-vertex graph, for some constant d. Is Md a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n(d−2)/d/ log n) communication rounds is presented. For the special case that Md is a triangle, we present a probabilistic algorithm that requires an expected O(n/(t + 1)) rounds of communication, where t is the number of triangles in the graph, and O(min{n log n/(t + 1), n}) with high probability. We also present deterministic algorithms specially suited for sparse graphs. In any graph of maximum degree ∆, we can test for arbitrary subgraphs of diameter D in O(∆/n) rounds. For triangles, we devise an arboricity-based algorithm, featuring a round complexity of O((A)/n + log n) ⊆ O(|E|/n + log n), where A denotes the arboricity of G.