Colorful triangle counting and a MapReduce implementation

Abstract

In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, let G be a graph with n vertices, t triangles and let Δ be the maximum number of triangles an edge of G is contained in. Our randomized algorithm colors the vertices of G with N=1/p colors uniformly at random, counts monochromatic triangles, i.e., triangles whose vertices have the same color, and scales that count appropriately. We show that if p⩾max(Δlognt,lognt) then for any constant ϵ>0 our unbiased estimate T is concentrated around its expectation, i.e., Pr[|T−E[T]|⩾ϵE[T]]=o(1). Finally, our algorithm is amenable to being parallelized. We present a simple MapReduce implementation of our algorithm.