Abstract
An immediate snapshot object is a high level communication object, built on top of a read/write distributed system in which all except one processes may crash. This object provides the processes with a single operation, denoted \(\mathsf{write\_snapshot}()\), which allows the invoking process to write a value and obtain a set of pairs \(\langle \)process id, value\(\rangle \) satisfying some set containment properties, that represent a snapshot of the values written to the object, occurring immediately after the write step.
Considering an n-process model in which up to t processes may crash, this paper introduces first the k-resilient immediate snapshot object, which is a natural generalization of the basic immediate snapshot (which corresponds to the case \(k=t=n-1\)). In addition to the set containment properties of the basic immediate snapshot, a k-resilient immediate snapshot object requires that each set returned to a process contains at least \((n-k)\) pairs.
The paper first shows that, for \(k,t<n-1\), k-resilient immediate snapshot is impossible in asynchronous read/write systems. Then it investigates a model of computation where the processes communicate with each other by accessing k-immediate snapshot objects, and shows that this model is stronger than the t-crash model. Considering the space of x-set agreement problems (which are impossible to solve in systems such that \(x\le t\)), the paper shows then that x-set agreement can be solved in read/write systems enriched with k-immediate snapshot objects for \(x=\mathsf{max}(1,t+k-(n-2))\). It also shows that, in these systems, k-resilient immediate snapshot and consensus are equivalent when \(1\le t<n/2\) and \(t\le k\le (n-1)-t\). Hence, the paper establishes strong relations linking fundamental distributed computing objects (one related to communication, the other to agreement), which are impossible to solve in pure read/write systems.
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Notes
- 1.
- 2.
As already indicated, this work was initiated in [9]. Considering k-IS in a system in which up to k processes may crash, this preliminary result showed that, somehow surprisingly, while there is a deterministic \((n-1)\)-resilient algorithm implementing an \((n-1)\)-IS object in an \((n-1)\)-crash read/write system, there is no t-resilient algorithm that implements a t-IS object when \(1\le t<n-1\).
- 3.
An equivalent formulation of the Immediacy property is: \(\forall ~i,j:~ \big ((\langle i,- \rangle \in view_j) \wedge (\langle j,- \rangle \in view_i)\big ) \Rightarrow (view_i = view_j)\).
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Acknowledgments
The authors want to thank the referees for their constructive comments. This work has been partially supported by the French ANR project DESCARTES (16-CE40-0023-03) devoted to layered and modular structures in distributed computing, and the UNAM-PAPIIT projects IN107714, IN106520.
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Delporte, C., Fauconnier, H., Rajsbaum, S., Raynal, M. (2020). k-Immediate Snapshot and x-Set Agreement: How Are They Related?. In: Devismes, S., Mittal, N. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2020. Lecture Notes in Computer Science(), vol 12514. Springer, Cham. https://doi.org/10.1007/978-3-030-64348-5_8
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