Near-Optimal Massively Parallel Graph Connectivity

TL;DR

This paper introduces a randomized parallel algorithm for graph connectivity in the MPC model that is nearly optimal, significantly improving round complexity for graphs with various diameters, and matching a conditional lower bound.

Contribution

The paper presents a new MPC algorithm for graph connectivity that achieves near-optimal round complexity across a wide range of graph diameters, improving upon previous results.

Findings

Achieves $O( ext{log } D)$ rounds for graphs with diameter $D$ in $[ ext{log}^{ ext{epsilon}} n, n]$

Uses $O( ext{log log } n)$ rounds for all other graphs

Matches the conditional lower bound based on the TwoCycle conjecture

Abstract

Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is $n^\delta$ for some desirably small constant $\delta \in (0, 1)$. We present an algorithm that for graphs with diameter $D$ in the wide range $[\log^{\epsilon} n, n]$, takes $O(\log D)$ rounds to identify the connected components and takes $O(\log \log n)$ rounds for all other graphs. The algorithm is randomized, succeeds with high probability, does not require prior knowledge of $D$, and uses an optimal total space of $O(m)$. We complement this by showing a conditional lower-bound based on the widely believed TwoCycle conjecture that $\Omega(\log D)$ rounds are indeed necessary in this setting. Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni et al. [FOCS 2018] who presented an algorithm with $O(\log D \cdot \log \log n)$ round-complexity. Our algorithm improves this result for the whole range of values of $D$ and almost settles the problem due to the conditional lower-bound. Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of the (CRCW) PRAM in asymptotically the same number of rounds.