Almost Optimal Distributed Algorithms for Large-Scale Graph Problems.
Gopal Pandurangan, Peter Robinson, Michele Scquizzato · arXiv (Cornell University) · 2015
Motivated by the increasing need to understand the algorithmic foundations of distributed large-scale graph computations, we study a number of 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 graph is assumed to be randomly partitioned among the $k$ machines, a common implementation in many real-world systems. The communication is point-to-point, and the goal is to minimize the time complexity, i.e., the number of communication rounds of the computation. Our main result is an (almost) optimal distributed randomized algorithm for graph connectivity. Our algorithm runs in $\tilde{O}(n/k^2)$ rounds ($\tilde{O}$ notation hides a $\text{polylog}(n)$ factor and an additive $\text{polylog}(n)$ term). This improves over the best previously known bound of $\tilde{O}(n/k)$ [Klauck et al., SODA 2015], and is optimal (up to a polylogarithmic factor) in view of the existing lower bound of $\tilde{\Omega}(n/k^2)$. Our improved algorithm uses a bunch of techniques, including linear graph sketching, which for the first time is applied in a non-trivial way in distributed computing. We then present fast randomized algorithms for computing minimum spanning trees, (approximate) min-cuts, and for many graph verification problems. All these algorithms take $\tilde{O}(n/k^2)$ rounds, and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of $\tilde{\Omega}(n/k^2)$ for many graph verification problems.