What can be decided locally without identifiers?

TL;DR

This paper investigates whether unique node identifiers are necessary for distributed local decision problems across various models, showing that in most cases, identifiers are essential for deciding certain network properties.

Contribution

The paper fully characterizes when local decision can be made without identifiers across different computational and identifier size models, extending prior work.

Findings

Identifiers are not needed when unbounded and uncomputable functions are allowed.

In most models, identifiers are crucial for local decision of certain properties.

Classical computability theory techniques are used to construct decision algorithms.

Abstract

Do unique node identifiers help in deciding whether a network $G$ has a prescribed property $P$? We study this question in the context of distributed local decision, where the objective is to decide whether $G \in P$ by having each node run a constant-time distributed decision algorithm. If $G \in P$, all the nodes should output yes; if $G \notin P$, at least one node should output no. A recent work (Fraigniaud et al., OPODIS 2012) studied the role of identifiers in local decision and gave several conditions under which identifiers are not needed. In this article, we answer their original question. More than that, we do so under all combinations of the following two critical variations on the underlying model of distributed computing: ($B$): the size of the identifiers is bounded by a function of the size of the input network; as opposed to ($\neg B$): the identifiers are unbounded. ($C$): the nodes run a computable algorithm; as opposed to ($\neg C$): the nodes can compute any, possibly uncomputable function. While it is easy to see that under ($\neg B, \neg C$) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present. Our constructions use ideas from classical computability theory.