New combinatorial topology bounds for renaming: The upper bound

In the <i>renaming</i> task, <i>n</i>+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,…, <i>K</i>. To rule out trivial solutions, a protocol must be <i>anonymous</i>: 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 <i>K</i>≥ 2<i>n</i>. Several algebraic topology proofs of a lower bound stating that no such protocol exists when <i>K</i> < 2<i>n</i> have been published. In a companion article, we present the first completely combinatorial renaming lower bound proof stating if <i>n</i> + 1 is a primer power, then renaming is not wait-free solvable when <i>K</i> < 2<i>n</i>. In this article, we show that if <i>n</i> + 1 is not a primer power, then there exists a wait-free renaming protocol for <i>K</i> = 2<i>n</i>−1. Therefore the renaming lower bound for <i>K</i> < 2<i>n</i> is incorrect. More precisely, our main theorem states that there exists a wait-free renaming protocol for <i>K</i> < 2<i>n</i> if and only if <i>n</i> + 1 is not a prime power. We prove this result using the known equivalence of <i>K</i>-renaming for <i>K</i> = 2<i>n</i> − 1 and the <i>weak symmetry breaking</i> 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.