Bounds for Renaming : The Upper Bound ∗
نویسندگان
چکیده
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. showed in 1990 that renaming has a wait-free solution when K ≥ 2n. Several algebraic topology proofs of a lower bound stating that no such protocol exists whenK < 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 forK < 2n is incorrect, exhibiting a wait-free renaming protocol forK = 2n−1. More precisely, our main theorem states that there exists a wait-free renaming protocol for K < 2n if and only if the set of integers { ( n+1 i+1 ) |0 ≤ i ≤ bn−1 2 c} are relatively prime. This paper is the journal version of the upper bound result presented in the ACM PODC 2008 conference. Namely, we show here that a protocol for renaming exists when K < 2n, if n is such that { ( n+1 i+1 ) |0 ≤ i ≤ bn−1 2 c} are relatively prime. 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.
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