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Register a new userSet-Consensus Collections are Decidable
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Added by
Petr Kuznetsov
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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A natural way to measure the power of a distributed-computing model is to characterize the set of tasks that can be solved in it. %the model. In general, however, the question of whether a given task can be solved in a given model is undecidable, even if we only consider the wait-free shared-memory model. In this paper, we address this question for restricted classes of models and tasks. We show that the question of whether a collection $C$ of emph{$(ell,j)$-set consensus} objects, for various $ell$ (the number of processes that can invoke the object) and $j$ (the number of distinct outputs the object returns), can be used by $n$ processes to solve wait-free $k$-set consensus is decidable. Moreover, we provide a simple $O(n^2)$ decision algorithm, based on a dynamic programming solution to the Knapsack optimization problem. We then present an emph{adaptive} wait-free set-consensus algorithm that, for each set of participating processes, achieves the best level of agreement that is possible to achieve using $C$. Overall, this gives us a complete characterization of a read-write model defined by a collection of set-consensus objects through its emph{set-consensus power}.
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The set consensus problem has played an important role in the study of distributed systems for over two decades. Indeed, the search for lower bounds and impossibility results for this problem spawned the topological approach to distributed computing, which has given rise to new techniques in the design and analysis of protocols. The design of efficient solutions to set consensus has also proven to be challenging. In the synchronous crash failure model, the literature contains a sequence of solutions to set consensus, each improving upon the previous ones. This paper presents an unbeatable protocol for nonuniform k-set consensus in the synchronous crash failure model. This is an efficient protocol whose decision times cannot be improved upon. Moreover, the description of our protocol is extremely succinct. Proving unbeatability of this protocol is a nontrivial challenge. We provide two proofs for its unbeatability: one is a subtle constructive combinatorial proof, and the other is a topological proof of a new style. These two proofs provide new insight into the connection between topological reasoning and combinatorial reasoning about protocols, which has long been a subject of interest. In particular, our topological proof reasons in a novel way about subcomplexes of the protocol complex, and sheds light on an open question posed by Guerraoui and Pochon (2009). Finally, using the machinery developed in the design of this unbeatable protocol, we propose a protocol for uniform k-set consensus that beats all known solutions by a large margin.
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