Is a LOCAL algorithm computable?

TL;DR

This paper investigates the impact of computability assumptions in the LOCAL model on solving locally checkable labeling problems, revealing how knowledge of the graph size influences problem complexity.

Contribution

It demonstrates the existence of an LCL problem with different complexities depending on computability and knowledge of graph size, clarifying the role of computability in distributed algorithms.

Findings

An LCL problem solvable in O(log n) rounds in uncomputable LOCAL model.

The same problem solvable in O(log n) rounds if an upper bound on n is known.

The problem requires Ω(√n) rounds in the computable model without knowledge of n.

Abstract

Common definitions of the "standard" LOCAL model tend to be sloppy and even self-contradictory on one point: do the nodes update their state using an arbitrary function or a computable function? So far, this distinction has been safe to neglect, since problems where it matters seem contrived and quite different from e.g. typical local graph problems studied in this context. We show that this question matters even for locally checkable labeling problems (LCLs), perhaps the most widely studied family of problems in the context of the LOCAL model. Furthermore, we show that assumptions about computability are directly connected to another aspect already recognized as highly relevant: whether we have any knowledge of $n$, the size of the graph. Concretely, we show that there is an LCL problem $\Pi$ with the following properties: 1. $\Pi$ can be solved in $O(\log n)$ rounds if the LOCAL model is uncomputable. 2. $\Pi$ can be solved in $O(\log n)$ rounds in the computable model if we know any upper bound on $n$. 3. $\Pi$ requires $\Omega(\sqrt{n})$ rounds in the computable model if we do not know anything about $n$. We also show that the connection between computability and knowledge of $n$ holds in general: for any LCL problem $\Pi$, if you have any bound on $n$, then $\Pi$ has the same round complexity in the computable and uncomputable models.