On the Power of Graphical Reconfigurable Circuits

TL;DR

This paper introduces the graphical reconfigurable circuits (GRC) model, demonstrating its ability to efficiently solve many distributed tasks with minimal assumptions, while also establishing fundamental lower bounds for others.

Contribution

The paper defines the GRC model, extending previous models to general graphs, and shows its effectiveness and limitations in solving distributed problems.

Findings

GRC algorithms can solve many tasks in polylogarithmic time.

Certain tasks have proven lower bounds, making GRC algorithms inherently slow.

The GRC model operates under minimal assumptions, using simple communication channels.

Abstract

We introduce the \emph{graphical reconfigurable circuits (GRC)} model as an abstraction for distributed graph algorithms whose communication scheme is based on local mechanisms that collectively construct long-range reconfigurable channels (this is an extension to general graphs of a distributed computational model recently introduced by Feldmann et al.\ (JCB 2022) for hexagonal grids). The crux of the GRC model lies in its modest assumptions: (1) the individual nodes are computationally weak, with state space bounded independently of any global graph parameter; and (2) the reconfigurable communication channels are highly restrictive, only carrying information-less signals (a.k.a.\ \emph{beeps}). Despite these modest assumptions, we prove that GRC algorithms can solve many important distributed tasks efficiently, i.e., in polylogarithmic time. On the negative side, we establish various runtime lower bounds, proving that for other tasks, GRC algorithms (if they exist) are doomed to be slow.