New combinatorial topology upper and lower bounds for renaming

Castañeda, Armando; Rajsbaum, Sergio

doi:10.1145/1400751.1400791

Cited by 42 publications

(55 citation statements)

References 17 publications

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“…Turns out that just because si has an output for pi we can show that the wait-free task T on 2t + 1 simulators induced by WeakRenaming (n + t, n, n + t − 1), is equivalent WeakSymmetryBreaking (2(t + 1) − 1, t + 1) (WSB) which is a task on 2(t + 1) − 1 processors and just asks that for participating set of size t + 1 all processor output 0 or 1, but at least one processor outputs 0 and at least one processor outputs 1. This task was recently been shown to be unsolvable for certain values of t and solvable for others [7]. We can immediately infer that the same holds for WeakRenaming (n + t, n, n + t − 1).…”

Section: Introductionmentioning

confidence: 57%

The extended BG-simulation and the characterization of t-resiliency

Gafni

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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A distributed task T on n processors is an input/output relation between a collection of processors' inputs and outputs. While all tasks are solvable if no processor may ever crash, the FLP result revealed that the possibility of a failure of just a single processor precludes a solution to the task of consensus. That is consensus is not solvable 1-resiliently. Yet, some nontrivial tasks are wait-free solvable, i.e. n − 1-resiliently. What tasks are solvable if at most t < n processors may crash? I.e. what tasks are solvable t-resiliently?The Herlihy-Shavit condition characterizes wait-free solvability, i.e., when t = n − 1. The Borowsky-Gafni (BG) simulation extends this characterization to the t-resilient case for the case "colorless" tasks -tasks like consensus in which one processor can adopt the output of any other processor. It does this by reducing questions about t-resilient solvability, to a question of wait-free solvability. The latter question has been characterized.In this paper, we amend the BG-simulation to result in the Extended-BG-simulation, an extension that yields a full characterization of t-resilient solvability: A task T on n processors is solvable t-resiliently iff all tasks T on t + 1 simulators s0, . . . , st created as follows are wait-free solvable. Simulator si is given an input of processor pi as well as the input to a set of processors of size n − (t + 1) with ids higher than i. Simulator si outputs for pi as well as for a (possibly different) set of processors of size n − (t + 1) with ids higher than i. The input/output of the t + 1 simulators have to be a projection of a single original input/output tuple-pair in T .We demonstrate the convenience that the characterization provides, in two ways. First, we prove a new equivalence result: We show that n processes can solve t-resiliently weak renaming with n + (t + 1) − 2 names, where n > 1 and 0 < t < n, iff weak-renaming on t + 1 processors is wait-free solvable with 2t names. Second, we reproduce the result that the solvability of n-processors tasks, t-resiliently, for t > 1 and n > 2, is undecidable, by a simple reduction to

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Section: Introductionmentioning

confidence: 57%

The extended BG-simulation and the characterization of t-resiliency

Gafni

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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“…It has been shown in [29] that WSB and (2n−2)-renaming are equivalent. It has been shown in [17] that (2n − 2)-renaming is not read/write wait-free solvable when { n i…”

Section: Hierarchy Results Gsb Tasks Of Intermediate Difficultymentioning

confidence: 99%

The universe of symmetry breaking tasks

Imbs

Rajsbaum²,

Raynal

2011

Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing

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Processes in a concurrent system need to coordinate using a shared memory or a message-passing subsystem in order to solve agreement tasks such as, for example, consensus or set agreement. However, often coordination is needed to "break the symmetry" of processes that are initially in the same state, for example, to get exclusive access to a shared resource, to get distinct names or to elect a leader.This paper introduces and studies the family of generalized symmetry breaking (GSB) tasks, that includes election, renaming and many other symmetry breaking tasks. Differently from agreement tasks, a GSB task is "inputless", in the sense that processes do not propose values; the task specifies only the symmetry breaking requirement, independently of the system's initial state (where processes differ only on their identifiers). Among many various characterizing the family of GSB tasks, it is shown that (non adaptive) perfect renaming is universal for all GSB tasks.Key-words: Agreement, Coordination, Decision task, Election, Disagreement, Distributed computability, Renaming, k-Set agreement, Symmetry Breaking, Universal construction, Wait-freedom. L'univers des tâches "Symmetry Breaking"Résumé : Dans un système réparti, les processus ont besoin de coordination en utilisant un sous-système de mémoire partagée ou de passage de message pour pouvoir résoudre des problèmes tels que le consensus ou l'accord ensembliste. Dans certains cas, la coordination est nécéssaire pour "casser la symétrie" entre des processus qui ont le mêmeétat initial.Ce rapport introduit la famille des tâches "generalized symmetry breaking" (GSB) qui inclut l'élection, le renommage et de nombreuses autres tâches qui cassent la symétrie. IntroductionProcesses of a distributed system need to coordinate through a communication medium (shared memory or message-passing subsystem) in order to solve various forms of agreement problems. If no coordination is ever needed in the computation, then we have a set of centralized, independent programs rather than a global distributed computation. Agreement coordination is one of the main issues of distributed computing. As an example, consensus is a very strong form of agreement where processes have to agree on the input of some process. It is a fundamental problem in distributed computing, and the cornerstone when one has to implement a replicated state machine, e.g. [20,37,40].Considering a shared memory asynchronous system where processes may fail by crashing, we are interested here in tasks [39], defined by an input/output relation ∆, and where processes start with private input values forming an input vector I and, after communication, individually decide on output values forming an output vector O, satisfying the specification of the considered task, i.e., O ∈ ∆(I). Several specific agreement tasks have been studied in detail, such as consensus [25] and set agreement [21]. Indeed, the importance of agreement is such that it has been studied deeply, from a more general perspective, defining familie...

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“…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.…”

Section: Previous Workmentioning

confidence: 99%

“…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 !…”

Section: Appendix: Path Building Kitmentioning

confidence: 99%

Structure theory of flip graphs with applications to Weak Symmetry Breaking

Kozlov

2017

J Appl. and Comput. Topology

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This paper is devoted to advancing the theoretical understanding of the iterated immediate snapshot (IIS) complexity of the Weak Symmetry Breaking task (WSB). Our rather unexpected main theorem states that there exist infinitely many values of n, such that WSB for n processes is solvable by a certain explicitly constructed 3-round IIS protocol. In particular, the minimal number of rounds, which an IIS protocol needs in order to solve the WSB task, does not go to infinity, when the number of processes goes to infinity. Our methods can also be used to generate such values of n. We phrase our proofs in combinatorial language, while avoiding using topology. To this end, we study a certain class of graphs, which we call flip graphs. These graphs encode adjacency structure in certain subcomplexes of iterated standard chromatic subdivisions of a simplex. While keeping the geometric background in mind for an additional intuition, we develop the structure theory of matchings in flip graphs in a purely combinatorial way. Our bound for the IIS complexity is then a corollary of this general theory. As an afterthought of our result, we suggest to change the overall paradigm. Specifically, we think, that the bounds on the IIS complexity of solving WSB for n processes should be formulated in terms of the size of the solutions of the associated Diophantine equation, rather than in terms of the value n itself.

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New combinatorial topology upper and lower bounds for renaming

Cited by 42 publications

References 17 publications

The extended BG-simulation and the characterization of t-resiliency

The extended BG-simulation and the characterization of t-resiliency

The universe of symmetry breaking tasks

Structure theory of flip graphs with applications to Weak Symmetry Breaking

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