Group Renaming

Abstract

We study the group renaming task, which is a natural generalization of the renaming task. An instance of this task consists of n processors, partitioned into m groups, each of at most g processors. Each processor knows the name of its group, which is in { 1, ..., M }. The task of each processor is to choose a new name for its group such that processors from different groups choose different new names from {1, ..., ℓ}, where ℓ< M. We consider two variants of the problem: a tight variant, in which processors of the same group must choose the same new group name, and a loose variant, in which processors from the same group may choose different names. Our findings can be briefly summarized as follows:

• 1 We present an algorithm that solves the tight variant of the problem with ℓ= 2m − 1 in a system consisting of g-consensus objects and atomic read/write registers. In addition, we prove that it is impossible to solve this problem in a system having only (g − 1)-consensus objects and atomic read/write registers.

• 1 We devise an algorithm for the loose variant of the problem that only uses atomic read/write registers, and has $\ell = 3n - \sqrt{n} - 1$. The algorithm also guarantees that the number of different new group names chosen by processors from the same group is at most $\min\{g, 2m, 2\sqrt{n}\}$. Furthermore, we consider the special case when the groups are uniform in size and show that our algorithm is self-adjusting to have ℓ= m (m + 1) / 2, when $m < \sqrt{n}$, and $\ell = 3n / 2 + m - \sqrt{n}/2 - 1$, otherwise.