An Equivariance Theorem with Applications to Renaming (Preliminary Version)

TL;DR

This paper establishes a topological framework using equivariance to analyze the renaming problem in distributed systems, linking solvability to the existence of certain chain maps, and deriving impossibility results.

Contribution

It introduces a novel topological approach to the renaming problem, connecting lower bounds to the non-existence of equivariant chain maps.

Findings

Lower bounds are characterized by topological obstructions.

Non-existence of equivariant chain maps implies impossibility of renaming algorithms.

Provides a new topological perspective on distributed computing limitations.

Abstract

In the renaming problem, each process in a distributed system is issued a unique name from a large name space, and the processes must coordinate with one another to choose unique names from a much smaller name space. We show that lower bounds on the solvability of renaming in an asynchronous distributed system can be formulated as a purely topological question about the existence of an equivariant chain map from a topological disk to a topological annulus. Proving the non-existence of such a map implies the non-existence of a distributed renaming algorithm in several related models of computation.