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22
Hundreds of Impossibility Results for Distributed Computing
 Distributed Computing
, 2003
"... We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models. The parameters of the models considered include synchrony, faulttolerance, different communication media, and randomization. The resource bounds refe ..."
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Cited by 52 (5 self)
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We survey results from distributed computing that show tasks to be impossible, either outright or within given resource bounds, in various models. The parameters of the models considered include synchrony, faulttolerance, different communication media, and randomization. The resource bounds refer to time, space and message complexity. These results are useful in understanding the inherent difficulty of individual problems and in studying the power of different models of distributed computing.
The Price of Anonymity: Optimal Consensus despite Asynchrony, Crash and Anonymity
, 2008
"... This paper addresses the consensus problem in asynchronous systems prone to process crashes, where additionally the processes are anonymous (they cannot be distinguished one from the other: they have no name and execute the same code). To circumvent the three computational adversaries (asynchrony, f ..."
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Cited by 14 (3 self)
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This paper addresses the consensus problem in asynchronous systems prone to process crashes, where additionally the processes are anonymous (they cannot be distinguished one from the other: they have no name and execute the same code). To circumvent the three computational adversaries (asynchrony, failures and anonymity) each process is provided with a failure detector of a class denoted ψ, that gives it an upper bound on the number of processes that are currently alive (in a nonanonymous system, the classes ψ and Pthe class of perfect failure detectors are equivalent). The paper first presents a simple ψbased consensus algorithm where the processes decide in 2t + 1 asynchronous rounds (where t is an upper bound on the number of faulty processes). It then shows one of its main results, namely, 2t + 1 is a lower bound for consensus in the anonymous systems equipped with ψ. The second contribution addresses earlydecision. The paper presents and proves correct an earlydeciding algorithm where the processes decide in min(2f + 2, 2t + 1) asynchronous rounds (where f is the actual number of process failures). This leads to think that anonymity doubles the cost (wrt synchronous systems) and it is conjectured that min(2f + 2, 2t + 1) is the corresponding lower bound. The paper finally considers the kset agreement problem in anonymous systems. It first shows that the
Looking for the Weakest Failure Detector for kSet Agreement in Messagepassing Systems
 Is Πk the End of the Road?, INRIA, 2009, http://hal.inria.fr/inria00384993/en/, PI
, 1929
"... Abstract: In the kset agreement problem, each process (in a set of n processes) proposes a value and has to decide a proposed value in such a way that at most k different values are decided. While this problem can easily be solved in asynchronous systems prone to t process crashes when k> t, it ..."
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Cited by 13 (3 self)
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Abstract: In the kset agreement problem, each process (in a set of n processes) proposes a value and has to decide a proposed value in such a way that at most k different values are decided. While this problem can easily be solved in asynchronous systems prone to t process crashes when k> t, it cannot be solved when k ≤ t. Since several years, the failure detectorbased approach has been investigated to circumvent this impossibility. While the weakest failure detector class to solve the kset agreement problem in read/write sharedmemory systems has recently been discovered (PODC 2009), the situation is different in messagepassing systems where the weakest failure detector classes are known only for the extreme cases k = 1 (consensus) and k = n − 1 (set agreement). This paper introduces a candidate for the general case. It presents a new failure detector class, denoted Πk, and shows Π1 = Σ × Ω (the weakest class for k = 1), and Πn−1 = L (the weakest class for k = n − 1). Then, the paper investigates the structure of Πk and shows it is the combination of two failures detector classes denoted Σk and Ωk (that generalize the previous “quorums ” and “eventual leaders ” failure detectors classes). Finally, the paper proves that Σk is a necessary requirement (as far as information on failure is concerned) to solve the kset agreement problem in messagepassing systems. The paper presents also a Πn−1based algorithm that solves the (n − 1)set agreement problem. This algorithm provides us with a new algorithmic insight on the way the (n − 1)set agreeement problem can be solved in asynchronous messagepassing systems (insight from the point of view of the nonpartitioning constraint defined by Σn−1).
Tight bounds for kset agreement with limited scope accuracy failure detectors
 Distributed Computing
"... In a system with limitedscope failure detectors, there are q clusters of processes such that some correct process in each cluster is never suspected by any process in that cluster. The failure detector class Sx,q satisfies this property all the time, while ⋄Sx,q satisfies it eventually. This paper ..."
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Cited by 13 (1 self)
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In a system with limitedscope failure detectors, there are q clusters of processes such that some correct process in each cluster is never suspected by any process in that cluster. The failure detector class Sx,q satisfies this property all the time, while ⋄Sx,q satisfies it eventually. This paper gives the first tight bounds for the kset agreement task in asynchronous messagepassing models augmented with failure detectors from either the Sx,q or ⋄Sx,q classes. For Sx,q, we show that any kset agreement protocol that tolerates f failures must satisfy f<k+ x − q, wherexisthe combined size of the k largest clusters. This result establishes for the first time that the protocol of Mostéfaoui and Raynal for the Sx = Sx,1 failure detector is optimal. For ⋄Sx,q, we show that any kset agreement protocol that tolerates f failures must satisfy f<min ( n+1,k+ x − q). We give a novel protocol 2 that matches our lower bound, disproving a conjecture of Mostéfaoui and Raynal for the ⋄Sx = ⋄Sx,1 failure detector. Our lower bounds exploit techniques borrowed from Combinatorial Topology, demonstrating for the first time that this approach is applicable to models that encompass failure detectors. 1
Towards a topological characterization of asynchronous complexity
 In Proceedings of the 16th Annual ACM Symposium on Principles of Distributed Computing
, 1997
"... Abstract. This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity. We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate ..."
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Cited by 9 (0 self)
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Abstract. This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity. We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate snapshot (IIS) model of Borowsky and Gafni. We do so by introducing a novel form of topological tool called the nonuniform chromatic subdivision. Building on the framework of Herlihy and Shavit’s topological computability model, our theorem states that the time complexity of any asynchronous algorithm is directly proportional to the level of nonuniform chromatic subdivisions necessary to allow a simplicial map from a task’s input complex to its output complex. To show the power of our theorem, we use it to derive a new tight bound on the time to achieve n process approximate agreement in the IIS model: � max input−min input � logd, where d = 3 for two processes ɛ and d = 2 for three or more. This closes an intriguing gap between the known upper and lower bounds implied by the work of Aspnes and Herlihy. More than the new bounds themselves, the importance of our asynchronous complexity theorem is that the algorithms and lower bounds it allows us to derive are intuitive and simple, with topological proofs that require no mention of concurrency at all.
The Combined Power of Conditions and Information on Failures to Solve Asynchronous Set Agreement
, 2008
"... To cope with the impossibility of solving agreement problems in asynchronous systems made up of n processes and prone to t process crashes, system designers tailor their algorithms to run fast in “normal” circumstances. Two orthogonal notions of “normality” have been studied in the past through fa ..."
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Cited by 8 (5 self)
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To cope with the impossibility of solving agreement problems in asynchronous systems made up of n processes and prone to t process crashes, system designers tailor their algorithms to run fast in “normal” circumstances. Two orthogonal notions of “normality” have been studied in the past through failure detectors that give processes information about process crashes, and through conditions that restrict the inputs to an agreement problem. This paper investigates how the two approaches can benefit from each other to solve the kset agreement problem, where processes must agree on at most k of their input values (when k = 1 we have the famous consensus problem). It proposes novel failure detectors for solving kset agreement, and a protocol that combines them with conditions, establishing a new bridge among asynchronous, synchronous and partially synchronous systems with respect to agreement problems. The
From a static impossibility to an adaptive lower bound: the complexity of early deciding set agreement
 In Proceedings of the 37 th ACM Symposium on Theory of Computing (STOC’05
, 2005
"... Set agreement, where processors decisions constitute a set of outputs, is notoriously harder to analyze than consensus where the decisions are restricted to a single output. This is because the topological questions that underly set agreement are not about simple connectivity as in consensus. Analyz ..."
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Cited by 7 (3 self)
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Set agreement, where processors decisions constitute a set of outputs, is notoriously harder to analyze than consensus where the decisions are restricted to a single output. This is because the topological questions that underly set agreement are not about simple connectivity as in consensus. Analyzing set agreement inspired the discovery of the relation between topology and distributed algorithms, and consequently the impossibility of asynchronous set agreement. Yet, the application of topological reasoning has been to the static case, that of asynchronous and synchronous tasks. It is not known yet for example, how to characterize starvationfree solvability of nonterminating tasks. Nonterminating tasks are dynamic entities with no defined end. In a similar vain, early deciding synchronous set agreement,
Failure Detectors to Solve Asynchronous kSet Agreement: a Glimpse of Recent Results
"... Abstract: In the kset agreement problem, each process proposes a value and has to decide a value in such a way that a decided value is a proposed value and at most k different values are decided. This problem can easily be solved in synchronous systems or in asynchronous systems prone to t process ..."
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Cited by 5 (1 self)
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Abstract: In the kset agreement problem, each process proposes a value and has to decide a value in such a way that a decided value is a proposed value and at most k different values are decided. This problem can easily be solved in synchronous systems or in asynchronous systems prone to t process crashes when t < k. In contrast, it has been shown that kset agreement cannot be solved in asynchronous systems when k ≤ t. Hence, since several years, the failure detectorbased approach has been investigated to circumvent this impossibility. This approach consists in enriching the underlying asynchronous system with an additional module per process that provides it with information on failures. Hence, without becoming synchronous, the enriched system is no longer fully asynchronous. This paper surveys this approach in both asynchronous shared memory systems and asynchronous message passing systems. It presents and discusses recent results and associated kset agreement algorithms.
A Note on Set Agreement with Omission Failures
 Electronic Notes in Theoretical Computing Science
"... This paper considers the kset agreement problem in a synchronous distributed system model with sendomission failures in which at most f processes can fail by sendomission. We show that, in a system of n +1 processes (n +1>f), no algorithm can solve kset agreement in rounds. Our lower ..."
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Cited by 4 (1 self)
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This paper considers the kset agreement problem in a synchronous distributed system model with sendomission failures in which at most f processes can fail by sendomission. We show that, in a system of n +1 processes (n +1>f), no algorithm can solve kset agreement in rounds. Our lower bound proof uses topological techniques to characterize subsets of executions of our model. The characterization has a surprisingly regular structure which leads to a simple and succinct proof. We also show that the lower bound is tight by exhibiting a new algorithm that solves kset agreement in + 1 rounds.
A new synchronous lower bound for set agreement. Submitted for publication
 In Proceedings of DISC 2001
, 2001
"... Abstract. We have a new proof of the lower bound that kset agreement requires ⌊f/k ⌋ + 1 rounds in a synchronous, messagepassing model with f crash failures. The proof involves constructing the set of reachable states, proving that these states are highly connected, and then appealing to a wellkn ..."
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Cited by 2 (2 self)
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Abstract. We have a new proof of the lower bound that kset agreement requires ⌊f/k ⌋ + 1 rounds in a synchronous, messagepassing model with f crash failures. The proof involves constructing the set of reachable states, proving that these states are highly connected, and then appealing to a wellknown topological result that high connectivity implies that set agreement is impossible. We construct the set of reachable states in an iterative fashion using a round operator that we define, and our proof of connectivity is an inductive proof based on this iterative construction and using simple properties of the round operator. This is the shortest and simplest proof of this lower bound we have seen. 1