Of Choices, Failures and Asynchrony: The Many Faces of Set Agreement

Abstract

Set agreement is a fundamental problem in distributed computing in which processes collectively choose a small subset of values from a larger set of proposals. The impossibility of fault-tolerant set agreement in asynchronous networks is one of the seminal results in distributed computing. In synchronous networks, too, the complexity of set agreement has been a significant research challenge that has now been resolved. Real systems, however, are neither purely synchronous nor purely asynchronous. Rather, they tend to alternate between periods of synchrony and periods of asynchrony. Nothing specific is known about the complexity of set agreement in such a "partially synchronous" setting.In this paper, we address this challenge, presenting the first (asymptotically) tight bound on the complexity of set agreement in such systems. We introduce a novel technique for simulating, in a fault-prone asynchronous shared memory, executions of an asynchronous and failure-prone message-passing system in which some fragments appear synchronous to some processes.We use this simulation technique to derive a lower bound on the round complexity of set agreement in a partially synchronous system by a reduction from asynchronous wait-free set agreement. Specifically, we show that every set agreement protocol requires at least t k + 2 synchronous rounds to decide. We present an (asymptotically) matching algorithm that relies on a distributed asynchrony detection mechanism to decide as soon as possible during periods of synchrony. From these two results, we derive the size of the minimal window of synchrony needed to solve set agreement.By relating synchronous, asynchronous and partially synchronous environments, our simulation technique is of independent interest. In particular, it allows us to obtain a new lower bound on the complexity of early deciding k-set agreement complementary to that of Gafni et al. (in SIAM J. Comput. 40(1):63-78, 2011), and to re-derive the combinatorial topology lower bound of : [570][571][572][573][574][575][576][577][578][579][580] 2009) in an algorithmic way.

Tight Bounds for Adopt-Commit Objects

“…While this is only a small improvement in complexity, we show a matching lower bound on the individual step complexity of any anonymous implementation (including randomized implementations against an oblivious adversary) of an m-valued adopt-commit object that supports at least Ω log m log log m processes. Our lower bound also implies a lower bound of Ω log m log log m on the 1 The definition of adopt-commit objects in [2] uses the term agreement for this property. We use agreement instead for the stronger unconditional agreement property of consensus objects.…”

A modular approach to shared-memory consensus, with applications to the probabilistic-write model

“…We call an object annotated with a decision bit in this way a deciding object. The adopt and commit values for the decision bit are used for compatibility with the definition of adopt-commit objects given by [2].…”

“…The consensus problem [1][2][3] is to devise a protocol so that n processes, each with a private input, can agree on a single common output that is equal to some process's input. For asynchronous deterministic processes, consensus is known to be impossible with a single crash failure in either a message-passing [4] or shared-memory [5] model.…”

Tight bounds for anonymous adopt-commit objects

“…An adopt-commit object [2] or ratifier [3] is a one-shot shared-memory object that represents the adopt-commit * Supported in part by NSF grant CCF-0916389. † Supported in part by the Natural Science and Engineering Research Council of Canada.…”

“…This means that all processes run the same code. Differences between the behaviour of two different processes can arise only as a result of different input values, (different supplies of random bits, in the case of a randomized protocol), and when 1 The definition of adopt-commit objects in [2] uses the term agreement for this property. We use agreement instead for the stronger unconditional agreement property of consensus objects.…”