Ability to Count Is Worth $\Theta(\Delta)$ Rounds

TL;DR

This paper establishes a tight linear lower bound on the number of communication rounds needed in a specific distributed computing model, resolving a long-standing open problem and completing the classification of model complexities.

Contribution

It proves a matching linear-in- lower bound, confirming the optimality of previous simulation results involving  rounds.

Findings

Linear lower bound matches previous upper bounds

Closes the gap in understanding model complexities

Confirms optimality of existing simulation results

Abstract

Hella et al. (PODC 2012, Distributed Computing 2015) identified seven different models of distributed computing - one of which is the port-numbering model - and provided a complete classification of their computational power relative to each other. However, one of their simulation results involves an additive overhead of $2\Delta-2$ communication rounds, and it was not clear, if this is actually optimal. In this paper we give a positive answer: there is a matching linear-in-$\Delta$ lower bound. This closes the final gap in our understanding of the models, with respect to the number of communication rounds.