Distributed Triangle Detection via Expander Decomposition

Abstract: We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $\tilde{O}(n<sup>{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $\tilde{O}(n<sup>{2/3})$ and $\tilde{O}(n<sup>{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $\tilde{O}(n<sup>{1-\delta})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_m\cup E_s\cup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $\Omega(n<sup>\delta)$ and conductance $\Omega(1/\text{poly} \log(n))$. As a consequence the mixing time of a random walk within the component is $O(\text{poly} \log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n<sup>{\delta}$. (c) $|E_r| \leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.