When and How Process Groups Can Be Used to Reduce the Renaming Space

Abstract

Considering the M-renaming problem and process groups, this paper investigates the following question: Is there a relation between the number of groups and the size of the new name space M? This question can be rephrased as follows: Can the initial partitioning of the processes into m groups allows the size of the renaming space M to be reduced, and if yes, how much?

This paper answers the previous questions. Let n denote the number of processes. Assuming that the processes are initially partitioned into m = n − ℓ non-empty groups, such that each process knows only its identity and its group number, the paper first presents a wait-free M-renaming algorithm whose size of the new name space is M = n + 2ℓ − 1. For $\frac{n}{2} < m \leq n-1$ (i.e. $1\leq \ell < \frac{n}{2}$), we have M < 2n − 1, which shows that, when the number of groups is greater than $\frac{n}{2}$, groups allow to circumvent the renaming lower bound in read/write systems. Then, on the lower bound size, the paper shows that there are pairs of values (n,m) such that there is no read/write wait-free M-renaming algorithm for which M ≤ 2n − 2. This impossibility result breaks our hope to have a renaming algorithm providing a new name space whose size would decrease “regularly” as the number of groups increases from 1 to n. Finally, the paper considers the case where each group includes at least s processes. This algorithm shows that, when m is such that $\frac{n}{s+1}< m < \frac{n}{s}$, there is an M-renaming algorithm where M = 3n − (s + 1)m − 1 = n(2 − s) + (s + 1)ℓ − 1. Hence, the paper leaves open the following question: For any n and s = 1, does the predicate $m > \frac{n}{2}$ define a threshold on the number of groups which allows the 2n − 2 lower bound on the renaming space size to be bypassed?