Improved Deterministic Connectivity in Massively Parallel Computation

TL;DR

This paper introduces a deterministic connectivity algorithm for the Massively Parallel Computation model that matches the efficiency of randomized algorithms while significantly reducing local computation time, enabling more practical implementations.

Contribution

It presents the first efficient, deterministic connectivity algorithm in low-memory MPC with nearly linear local computation time, improving upon previous derandomization methods.

Findings

Matches randomized algorithm parameters in deterministic setting

Significantly reduces local computation time to nearly linear

Provides a simpler, more efficient derandomization approach

Abstract

A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity $O(\log D + \log \log_{\frac m n} n)$ and $O(m)$ space, for graphs on $n$ vertices with $m$ edges and diameter $D$. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present a deterministic connectivity algorithm that matches all the parameters of the randomized algorithm and, in addition, significantly reduces the local computation time to nearly linear. Our derandomization method is based on reducing the amount of randomness needed to allow for a simpler efficient search. While similar randomness reduction approaches have been used before, our result is not only strikingly simpler, but it is the first to have efficient local computation. This is why we believe it to serve as a starting point for the systematic development of computation-efficient derandomization approaches in low-memory MPC.