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 i+1 |0 ≤ i ≤ n−1 2 } 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 i+1 |0 ≤ i ≤ n−1 2 } are not relatively prime. We prove This is the first part of the full version of the work presented in the ACM PODC 2008 [9]. Partly supported by PAPIIT-DGAPA project IN116808 and Macroproyecto para las Tecnologías de la Información, UNAM.A. Castañeda (B) · S. Rajsbaum 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].

Convergence and covering on graphs for wait-free robots

“…Coordination problems in distributed computing can be about reaching agreement, often referred to as colorless problems [8] such as consensus, loop agreement, set agreement, graph convergence, or more generally robot convergence, or they can deal with reaching disagreement, which is usually much more difficult to analyze [19] as in weak symmetry breaking [11,19,25], renaming [4,12] or committee decision [13].…”

Structure theory of flip graphs with applications to Weak Symmetry Breaking

“…Several groups of researchers have studied the solvability of the WSB by means of comparison-based IIS protocols. Due primarily to the work of Herlihy and Shavit (1999), as well as Castañeda and Rajsbaum (2008, 2010, 2012, it is known that the WSB is solvable if and only if the number of processes is not a prime power, see also Attiya and Paz (2012) for a countingbased argument for the impossibility part. This makes n ¼ 6 the smallest number of processes for which this task is solvable.…”

“…The construction of k, when the number is not a prime power, was then done by see Castañeda and Rajsbaum (2012), using the following method. First, boundary values are assigned, making sure that this obstruction value is 0.…”

“…l to denote edges belonging to the matching, while denotes all other edges. Castañeda and Rajsbaum (2008, 2010, 2012 sb ðnÞ ¼ Oðn qþ3 Þ, if n is not a prime power and Attiya et al (2013) q is the largest prime power in the prime factorization of n sb ðnÞ ! 2 Kozlov (2016) sb ð6tÞ 3, for all t !…”

Tight Bounds for Asynchronous Renaming

“…[Attiya et al 1990], [Bar-Noy and Dolev 1989], [Burns and Peterson 1989], [Moir and Anderson 1995], [Herlihy and Shavit 1999], [Afek and Merritt 1999], [Attiya and Fouren 2001], [Eberly et al 1998], [Panconesi et al 1998], has studied the solvability and complexity of renaming in an asynchronous environment. In particular, tight, or strong deterministic renaming, where the size of the namespace is exactly n, is known to be impossible [Herlihy and Shavit 1999], [Castañeda and Rajsbaum 2010]. In fact, (n + t − 1) is the best achievable namespace size when t processes may crash [Castañeda and Rajsbaum 2010], [Castañeda and Rajsbaum 2012].…”