Massively Parallel Ruling Set Made Deterministic

TL;DR

This paper introduces deterministic algorithms for the 2-Ruling Set problem in Massively Parallel Computation, achieving constant rounds in linear MPC and sublogarithmic rounds in sublinear MPC, advancing the state of deterministic distributed algorithms.

Contribution

It provides the first deterministic algorithms for 2-Ruling Set in both linear and sublinear MPC models with improved round complexities.

Findings

Constant-round deterministic algorithm in linear MPC matching recent randomized results.

First sublogarithmic-round deterministic algorithm in sublinear MPC.

Simpler derandomization technique based on bounded independence.

Abstract

We study the deterministic complexity of the $2$-Ruling Set problem in the model of Massively Parallel Computation (MPC) with linear and strongly sublinear local memory. Linear MPC: We present a constant-round deterministic algorithm for the $2$-Ruling Set problem that matches the randomized round complexity recently settled by Cambus, Kuhn, Pai, and Uitto [DISC'23], and improves upon the deterministic $O(\log \log n)$-round algorithm by Pai and Pemmaraju [PODC'22]. Our main ingredient is a simpler analysis of CKPU's algorithm based solely on bounded independence, which makes its efficient derandomization possible. Sublinear MPC: We present a deterministic algorithm that computes a $2$-Ruling Set in $\tilde O(\sqrt{\log n})$ rounds deterministically. Notably, this is the first deterministic ruling set algorithm with sublogarithmic round complexity, improving on the $O(\log \Delta + \log \log^* n)$-round complexity that stems from the deterministic MIS algorithm of Czumaj, Davies, and Parter [TALG'21]. Our result is based on a simple and fast randomness-efficient construction that achieves the same sparsification as that of the randomized $\tilde O(\sqrt{\log n})$-round LOCAL algorithm by Kothapalli and Pemmaraju [FSTTCS'12].