Optimal Memory-Anonymous Symmetric Deadlock-Free Mutual Exclusion

Abstract

The notion of an anonymous shared memory, introduced by Taubenfeld in PODC 2017, considers that processes use different names for the same memory location. As an example, a location name A used by a process p and a location name B ≠ A used by another process q can correspond to the very same memory location X, and similarly for the names B used by p and A used by q which may (or may not) correspond to the same memory location Y ≠ X. In this context, the PODC paper presented a 2-process symmetric deadlock-free mutual exclusion (mutex) algorithm and a necessary condition on the size m of the anonymous memory for the existence of such an n-process algorithm. This condition states that m must be belongs to M(n) {1} where M(n)= {m: ∀ ℓ: (1) < ℓ ≤ n: gcd(ℓ,m)=1). Symmetric means here that,process identities define a specific data type which allows a process to check only if two identities are equal or not. The present paper presents two optimal deadlock-free symmetric mutual exclusion algorithms for n-process systems where communication is through m registers. The first algorithm, which considers anonymous read/write registers, works for any m which is ≥ n and belongs to the set M(n). It follows that this condition on m is both necessary and sufficient for symmetric deadlock-free mutual exclusion in this anonymity context, and this algorithm is optimal with respect to m The second algorithm, which considers anonymous read/modify/write atomic registers, works for any m∈ M(n), which is shown to be necessary and sufficient for anonymous read/modify/write registers. It follows that, when m > 1, m ∈ M(n) is a tight characterization of the size of the anonymous shared memory needed to solve deadlock-free mutex, be the registers read/write or read/modify/write.