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International Colloquium on Structural Information and Communication Complexity

SIROCCO 2016: Structural Information and Communication Complexity pp 3–18Cite as

How Many Cooks Spoil the Soup?

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9988))

Abstract

In this work, we study the following basic question: “How much parallelism does a distributed task permit?” Our definition of parallelism (or symmetry) here is not in terms of speed, but in terms of identical roles that processes have at the same time in the execution. We initiate this study in population protocols, a very simple model that not only allows for a straightforward definition of what a role is, but also encloses the challenge of isolating the properties that are due to the protocol from those that are due to the adversary scheduler, who controls the interactions between the processes. We (i) give a partial characterization of the set of predicates on input assignments that can be stably computed with maximum symmetry, i.e., \(\varTheta (N_{min})\), where \(N_{min}\) is the minimum multiplicity of a state in the initial configuration, and (ii) we turn our attention to the remaining predicates and prove a strong impossibility result for the parity predicate: the inherent symmetry of any protocol that stably computes it is upper bounded by a constant that depends on the size of the protocol.

Supported in part by the School of EEE/CS of the University of Liverpool, NeST initiative, and the EU IP FET-Proactive project MULTIPLEX under contract no 317532. The full version can be found at: https://arxiv.org/abs/1604.07187.

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Notes

  1. 1.

    By “schedule” we mean an “execution” throughout.

  2. 2.

    We shall use throughout the paper \(N_i\) to denote the number of nodes with input/state i.

  3. 3.

    In this work, we only require protocols to preserve their symmetry up to stability. This means that a protocol is allowed to break symmetry arbitrarily after stability, e.g., even elect a unique leader, without having to pay for it. We leave as an interesting open problem the comparison of this convention to the apparently harder requirement of maintaining symmetry forever.

  4. 4.

    Always meaning the imaginary symmetry-maximizing scheduler when lower-bounding the symmetry.

  5. 5.

    Whenever we use an unordered pair in a rule, like \(\{b,c\}\), we mean that the property under consideration concerns both (b, c) and (c, b).

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Acknowledgements

We would like to thank Dimitrios Amaxilatis for setting up and running experiments for the evaluation of the observed symmetry.

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Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Othon Michail & Paul G. Spirakis

  2. Computer Technology Institute and Press “Diophantus” (CTI), Patras, Greece

    Othon Michail & Paul G. Spirakis

Authors
  1. Othon Michail
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  2. Paul G. Spirakis
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Correspondence to Othon Michail .

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Editors and Affiliations

  1. Aalto University, Espoo, Finland

    Jukka Suomela

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Michail, O., Spirakis, P.G. (2016). How Many Cooks Spoil the Soup?. In: Suomela, J. (eds) Structural Information and Communication Complexity. SIROCCO 2016. Lecture Notes in Computer Science(), vol 9988. Springer, Cham. https://doi.org/10.1007/978-3-319-48314-6_1

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  • DOI: https://doi.org/10.1007/978-3-319-48314-6_1

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