Optimal Memory-Anonymous Symmetric Deadlock-Free Mutual Exclusion

Abstract

The notion of an anonymous shared memory (recently introduced 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. Hence, there is permanent disagreement on the location names among processes. In this context, the PODC paper presented -among other results-a symmetric deadlock-free mutual exclusion (mutex) algorithm for two processes and a necessary condition on the size m of the anonymous memory for the existence of a symmetric deadlock-free mutex algorithm in an n-process system. This condition states that m must be greater than 1 and belong to the set M (n) = {m : ∀ ℓ : 1 < ℓ ≤ n : gcd(ℓ, m) = 1} (symmetric means that, while each process has its own identity, process identities can only be compared with equality).The present paper answers several open problems related to symmetric deadlock-free mutual exclusion in an n-process system (n ≥ 2) where the processes communicate through m registers. It first presents two algorithms. The first considers that the registers are anonymous read/write atomic registers and works for any m greater than 1 and belonging to the set M (n). Hence, it shows that this condition on m is both necessary and sufficient. The second algorithm considers that the registers are anonymous read/modify/write atomic registers. It assumes that m ∈ M (n). These algorithms differ in their design principles and their costs (measured as the number of registers which must contain the identity of a process to allow it to enter the critical section). The paper also shows that the condition m ∈ M (n) is necessary for deadlock-free mutex on top of anonymous read/modify/write atomic 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 anonymous registers read/write or read/modify/write.

Brief Announcement: Fully Anonymous Shared Memory Algorithms

“…Remark It is worth noticing that, from a distributed computing understanding and computability point of view, the condition m ∈ M (n) shows that, as far as deadlock-free mutual exclusion using RMW registers is concerned, there is no computability gap between full anonymity (as addressed here) and register-restricted anonymity (addressed in [3]). Both require m ∈ M (n).…”

“…identifiers for an local identifiers local identifiers external observer for process pi for process pj R [1] Ri [2] Rj [3] R [2] Ri [3] Rj [1] R [3] Ri [1] Rj [2] permutation fi() : [ In both models, atomic [20] means that the operations on the registers appear as if they have been executed sequentially, each operation appearing between its start event and its end event, and for any x ∈ {1, ...m}, each read operation of a register R[x] returns the value v, where v is the last value written in R[x] by a write or a successful compare&swap(R[x], −, −) operation (we also say that the execution is linearizable [19]). We notice that the RMW model is at least as strong as the RW model.…”

“…Two memory-anonymous symmetric deadlock-free mutual exclusion algorithms are presented in [3]. One is for the RW register model, the other one for the RMW register model.…”

“…As far as the mutex algorithm is concerned, it is also shown that m ∈ M (n), where M (n) = { m such that ∀ ℓ : 1 < ℓ ≤ n: gcd(ℓ, m) = 1} is a necessary and sufficient condition on the size of the anonymous memory for such an algorithm. As already mentioned, the mutual exclusion problem can be solved for non-anonymous processes in both the anonymous RW register model and the anonymous RMW register model [3]. However, there is no mutual exclusion algorithm when the processes are anonymous, even when using non-anonymous RW registers.…”

“…-The lower bound result in [3], which states that m ∈ M (n) is a necessary and sufficient condition for symmetric deadlock-free mutual exclusion for n non-anonymous processes and anonymous RMW registers. As already noticed, algorithms for anonymous processes are, by definition, also symmetric.…”