New combinatorial topology bounds for renaming
Armando Castañeda, Sergio Rajsbaum · Journal of the ACM · 2012
In the renaming task, n +1 processes start with unique input names 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. [1990] showed that renaming has a wait-free solution when K ≥ 2 n . Several algebraic topology proofs of a lower bound stating that no such protocol exists when K < 2 n have been published. In a companion article, we present the first completely combinatorial renaming lower bound proof stating if n + 1 is a primer power, then renaming is not wait-free solvable when K < 2 n . In this article, we show that if n + 1 is not a primer power, then there exists a wait-free renaming protocol for K = 2 n −1. Therefore the renaming lower bound for K < 2 n is incorrect. More precisely, our main theorem states that there exists a wait-free renaming protocol for K < 2 n if and only if n + 1 is not a prime power. We prove this result using the known equivalence of K -renaming for K = 2 n − 1 and the weak symmetry breaking 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 1 and at least one process decides 0.