Deterministic Massively Parallel Connectivity

TL;DR

This paper introduces a deterministic algorithm for graph connectivity in the MPC model that matches the best randomized algorithms, achieving $O(\,log D + \,log \,log n)$ rounds with low local space.

Contribution

It presents the first deterministic MPC algorithm for graph connectivity that matches the efficiency of the best randomized algorithms.

Findings

Deterministic algorithm achieves $O(\,log D + \,log \,log n)$ rounds.

Matches the performance of the best randomized algorithms.

Operates under low local space constraints.

Abstract

We consider the problem of designing fundamental graph algorithms on the model of Massive Parallel Computation (MPC). The input to the problem is an undirected graph $G$ with $n$ vertices and $m$ edges, and with $D$ being the maximum diameter of any connected component in $G$. We consider the MPC with low local space, allowing each machine to store only $\Theta(n^\delta)$ words for an arbitrarily constant $\delta > 0$, and with linear global space (which is equal to the number of machines times the local space available), that is, with optimal utilization. In a recent breakthrough, Andoni et al. (FOCS 18) and Behnezhad et al. (FOCS 19) designed parallel randomized algorithms that in $O(\log D + \log \log n)$ rounds on an MPC with low local space determine all connected components of an input graph, improving upon the classic bound of $O(\log n)$ derived from earlier works on PRAM algorithms. In this paper, we show that asymptotically identical bounds can be also achieved for deterministic algorithms: we present a deterministic MPC low local space algorithm that in $O(\log D + \log \log n)$ rounds determines all connected components of the input graph.