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Anonymous obstruction-free (n, k)-set agreement with \(n-k+1\) atomic read/write registers

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Abstract

The k-set agreement problem is a generalization of the consensus problem. Namely, assuming that each process proposes a value, every non-faulty process must decide one of the proposed values, under the constraint that at most k different values are decided. This is a hard problem in the sense that it cannot be solved in a pure read/write asynchronous system, in which k or more processes may crash. One way to sidestep this impossibility result consists in weakening the termination property, requiring only that a process decides if it executes alone during a long enough period of time. This is the well-known obstruction-freedom progress condition. Consider a system of n anonymous asynchronous processes that communicate through atomic read/write registers, and such that any number of them may crash. This paper addresses and solves the challenging open problem of designing an obstruction-free k-set agreement algorithm with only \((n-k+1)\) atomic registers. From a shared memory cost point of view, our algorithm is the best algorithm known to date, thereby establishing a new upper bound on the number of registers needed to solve this problem. For the consensus case \((k=1)\), the proposed algorithm is up to an additive factor of 1 close to the best known lower bound. Further, the paper extends this algorithm to obtain an x-obstruction-free solution to the k-set agreement problem that employs \((n-k+x)\) atomic registers (with \(1 \le x\le k<n\)), as well as a space-optimal solution for the repeated version of k-set agreement. Using this last extension, we prove that n registers are enough for every colorless task that is obstruction-free solvable with identifiers and any number of registers.

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Notes

  1. Let us observe here that single-writer/multi-reader registers are meaningless in anonymous systems. This is due to the fact that, as processes have no identity, it is not possible to link each of them to some specific registers (for which the process would be the only writer).

  2. In an anonymous system, the input values are the only way to distinguish processes.

  3. Let us notice that this is the only place in the proof where the algorithm requires more than \((n-1)\) MWMR atomic registers.

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Acknowledgements

The authors want to thank Rati Gelashvili for fruitful comments on a preliminary version of the paper, which gave rise to Sect. 8. They want to thank also the referees for their constructive comments, which helped them improve and simplify the presentation. This work was partially supported by the Franco-German DFG/ANR project DISCMAT devoted to connections between mathematics and distributed computing, and the French ANR project DESCARTES devoted to distributed software engineering.

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Authors and Affiliations

  1. IRISA, Université de Rennes, 35042, Rennes, France

    Zohir Bouzid

  2. Institut Universitaire de France, Paris, France

    Michel Raynal

  3. Telecom SudParis and CNRS, Université Paris-Saclay, Evry, France

    Pierre Sutra

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  1. Zohir Bouzid
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  2. Michel Raynal
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  3. Pierre Sutra
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Correspondence to Michel Raynal.

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A preliminary version of parts of this article appeared in [9].

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Bouzid, Z., Raynal, M. & Sutra, P. Anonymous obstruction-free (n, k)-set agreement with \(n-k+1\) atomic read/write registers. Distrib. Comput. 31, 99–117 (2018). https://doi.org/10.1007/s00446-017-0301-7

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