Fast Distributed Algorithms for Connectivity and MST in Large Graphs [article]

Gopal Pandurangan, Peter Robinson, Michele Scquizzato
2016 arXiv   pre-print
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 ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n ≫ k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal
more » ... is to minimize 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 Õ(n/k^2) rounds (Õ notation hides a (n) factor and an additive (n) term). This improves over the best previously known bound of Õ(n/k) [Klauck et al., SODA 2015], and is optimal (up to a polylogarithmic factor) in view of an existing lower bound of Ω̃(n/k^2). Our improved algorithm uses a bunch of techniques, including linear graph sketching, that prove useful in the design of efficient distributed graph algorithms. Using the connectivity algorithm as a building block, we then present fast randomized algorithms for computing minimum spanning trees, (approximate) min-cuts, and for many graph verification problems. All these algorithms take Õ(n/k^2) rounds, and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of Ω̃(n/k^2) rounds for many graph verification problems by leveraging lower bounds in random-partition communication complexity.
arXiv:1503.02353v3 fatcat:n2zc6n42vvefthbwwoqzo2242i