Tight Bounds for Distributed Graph Computations

Motivated by the need to understand the algorithmic foundations of distributed large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the $k$ machines. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. We present (almost) tight bounds for the round complexity of two fundamental graph problems, namely PageRank computation and triangle enumeration. Our tight lower bounds, a main contribution of the paper, are established through an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines for correctly solving a problem. Our approach is generic and might be useful in showing lower bounds in the context of similar problems and similar models. We show a lower bound of $\tilde{\Omega}(n/k^2)$ rounds for computing the PageRank. (Notation $\tilde \Omega$ hides a ${1}/{\text{polylog}(n)}$ factor.) We also present a simple distributed algorithm that computes the PageRank of all the nodes of a graph in $\tilde{O}(n/k^2)$ rounds (notation $\tilde O$ hides a $\text{polylog}(n)$ factor and an additive $\text{polylog}(n)$ term). For triangle enumeration, we show a lower bound of $\tilde{\Omega}(m/k^{5/3})$ rounds, where $m$ is the number of edges of the graph. Our result implies a lower bound of $\tilde\Omega(n^{1/3})$ for the congested clique, which is tight up to logarithmic factors. We also present a distributed algorithm that enumerates all the triangles of a graph in $\tilde{O}(m/k^{5/3})$ rounds.