New combinatorial topology bounds for renaming: The upper bound

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≥ 2n. Several algebraic topology proofs of a lower bound stating that no such protocol exists when K < 2n 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 < 2n. In this article, we show that if n + 1 is not a primer power, then there exists a wait-free renaming protocol for K = 2n−1. Therefore the renaming lower bound for K < 2n is incorrect. More precisely, our main theorem states that there exists a wait-free renaming protocol for K < 2n if and only if n + 1 is not a prime power. We prove this result using the known equivalence of K-renaming for K = 2n − 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.