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, if $p \geq \max{(\frac{\Delta \log{n}}{t}, \frac{\log{n}}{\sqrt{t}})}$, where $n$, $t$, $\Delta$ denote the number of vertices in $G$, the number of triangles in $G$, the maximum number of triangles an edge of $G$ is contained, then for any constant $\epsilon>0$ our unbiased estimate $T$ is concentrated around its expectation, i.e., $ \Prob{|T - \Mean{T}| \geq \epsilon \Mean{T}} = o(1)$. Finally, we present a \textsc{MapReduce} implementation of our algorithm.