Fast Distributed Algorithms for Connectivity and MST in Large Graphs

TL;DR

This paper introduces near-optimal distributed algorithms for fundamental graph problems like connectivity and MST in large graphs, significantly reducing communication rounds and leveraging advanced techniques such as graph sketching.

Contribution

The paper presents the first nearly optimal distributed randomized algorithms for graph connectivity, MST, and related problems with round complexity close to theoretical lower bounds.

Findings

Connectivity algorithm runs in O(n/k^2) rounds

Algorithms for MST, min-cuts, and verification also run in O(n/k^2) rounds

Proves lower bounds matching the upper bounds up to polylogarithmic factors

Abstract

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 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 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 $\tilde{O}(n/k^2)$ rounds ($\tilde{O}$ notation hides a $\poly\log(n)$ factor and an additive $\poly\log(n)$ 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 an existing lower bound of $\tilde{\Omega}(n/k^2)$. 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 $\tilde{O}(n/k^2)$ rounds, and are optimal up to polylogarithmic factors. We also show an almost matching lower bound of $\tilde{\Omega}(n/k^2)$ rounds for many graph verification problems by leveraging lower bounds in random-partition communication complexity.