Group Renaming

We study the group renaming problem, which is a natural generalization of the renaming problem. An instance of this problem 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, . . . , `}, where ` < 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 several different names. Our findings can be briefly summarized as follows: 1. We present an algorithm that solves the tight variant of the problem with ` = 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. 2. We devise an algorithm for the loose variant of the problem that only uses atomic read/write registers, and has ` = 3n−√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√n}. Furthermore, we consider the special case when the groups are uniform in size and show that our algorithm is self-adjusting to have ` = m(m + 1)/2, when m < √ n, and ` = 3n/2 + m−√n/2− 1, otherwise. ∗An extended abstract of this paper appeared in Proceedings of the 12th International Conference on Principles of Distributed Systems, pages 58–72, 2008. †School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email: [email protected]. ‡School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email: [email protected]. Supported by the Binational Science Foundation, by the Israel Science Foundation, and by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848. §School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email: [email protected]. ¶AT&T Labs, 180 Park Ave., Florham Park, NJ 07932, USA. Email: [email protected]. ‖The Interdisciplinary Center, P.O. Box 167, Herzliya 46150, Israel. Email: [email protected].