Infinite Networks, Halting and Local Algorithms

TL;DR

This paper explores the capabilities of local algorithms in infinite networks, proving that any universally halting distributed algorithm in such networks must be local, contrasting with finite networks where nonlocal algorithms can halt universally.

Contribution

It establishes a theoretical link between infinite networks and local algorithms, showing that in infinite networks, all universally halting algorithms are inherently local.

Findings

In infinite networks, all universally halting algorithms are local.

Finite networks allow nonlocal algorithms to halt universally.

The study connects logic and distributed computing in network analysis.

Abstract

The immediate past has witnessed an increased amount of interest in local algorithms, i.e., constant time distributed algorithms. In a recent survey of the topic (Suomela, ACM Computing Surveys, 2013), it is argued that local algorithms provide a natural framework that could be used in order to theoretically control infinite networks in finite time. We study a comprehensive collection of distributed computing models and prove that if infinite networks are included in the class of structures investigated, then every universally halting distributed algorithm is in fact a local algorithm. To contrast this result, we show that if only finite networks are allowed, then even very weak distributed computing models can define nonlocal algorithms that halt everywhere. The investigations in this article continue the studies in the intersection of logic and distributed computing initiated in (Hella et al., PODC 2012) and (Kuusisto, CSL 2013).