New combinatorial topology bounds for renaming: the lower bound

Abstract

In the renaming task n + 1 processes start with unique input names taken from a large space and must choose unique output names taken from a smaller name space, 0, 1, . . . , K. To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process id. Attiya et al. showed in 1990 that renaming has a wait-free solution when K ≥ 2n. Several proofs of a lower bound stating that no such protocol exists when K < 2n have been published. We presented in the ACM PODC 2008 conference the following two results. First, we presented the first completely combinatorial lower bound proof stating that no such a protocol exists when K < 2n. This bound holds for infinitely many values of n. Second, for the other values of n, we proved that the lower bound for K < 2n is incorrect, exhibiting a wait-free renaming protocol for K = 2n−1. More precisely, we presented a theorem stating that there exists a wait-free renaming protocol for K < 2n if and only if the set of integers ${\{ {n+1 \choose i+1} | 0 \leq i \leq \lfloor \frac{n-1}{2} \rfloor \}}$ are relatively prime. This paper is the first part of the full version of the results presented in the ACM PODC 2008 conference. It includes only the lower bound. Namely, we show here that no protocol for renaming exists when K <  2n, if n is such that ${\{ {n+1 \choose i+1} | 0 \leq i \leq \lfloor \frac{n-1}{2}\rfloor \}}$ are not relatively prime. We prove this result using the known equivalence of K-renaming for K = 2n−1 and the weak symmetry breaking task. In this task processes have no input values and the output values are 0 or 1, and it is required that in every execution in which all processes participate, at least one process decides 0 and at least one process decides 1. The full version of the upper bound appears in a companion paper [10].