“Tri, Tri Again”: Finding Triangles and Small Subgraphs in a Distributed Setting

Abstract

Let G = (V,E) be an n-vertex graph and M _d a d-vertex graph, for some constant d. Is M _d 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 $\mathcal{O}(\log n)$ bits. A simple deterministic algorithm that requires $\mathcal{O}(n^{(d-2)/d}/\log n)$ communication rounds is presented. For the special case that M _d is a triangle, we present a probabilistic algorithm that requires an expected $\mathcal{O}(n^{1/3}/(t^ {2/3}+1))$ rounds of communication, where t is the number of triangles in the graph, and $\mathcal{O}(\min\{n^{1/3}\log^{2/3}n/(t^ {2/3}+1),n^{1/3}\})$ with high probability.

We also present deterministic algorithms that are specially suited for sparse graphs. In graphs of maximum degree Δ, we can test for arbitrary subgraphs of diameter D in $\mathcal{O}(\Delta^{D+1}/n)$ rounds. For triangles, we devise an algorithm featuring a round complexity of $\mathcal{O}((A^2\log_{2+n/A^2} n)/n)$, where A denotes the arboricity of G.