Exploring Gafni’s Reduction Land: From Ω^k to Wait-Free Adaptive $(2p-\lceil\frac{p}{k}\rceil)$-Renaming Via k-Set Agreement

Abstract

The adaptive renaming problem consists in designing an algorithm that allows p processes (in a set of n processes) to obtain new names despite asynchrony and process crashes, in such a way that the size of the new renaming space M be as small as possible. It has been shown that M=2p–1 is a lower bound for that problem in asynchronous atomic read/write register systems.

This paper is an attempt to circumvent that lower bound. To that end, considering first that the system is provided with a k-set object, the paper presents a surprisingly simple adaptive M-renaming wait-free algorithm where $M=2p-\lceil\frac{p}{k}\rceil$. To attain this goal, the paper visits what we call Gafni’s reduction land, namely, a set of reductions from one object to another object as advocated and investigated by Gafni. Then, the paper shows how a k-set object can be implemented from a leader oracle (failure detector) of a class denoted Ω^k. To our knowledge, this is the first time that the failure detector approach is investigated to circumvent the M=2p–1 lower bound associated with the adaptive renaming problem. In that sense, the paper establishes a connection between renaming and failure detectors.