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Making Local Algorithms Wait-Free: the Case of Ring Coloring

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Abstract

When considering distributed computing, reliable message-passing synchronous systems on the one side, and asynchronous failure-prone shared-memory systems on the other side, remain two quite independently studied ends of the reliability/asynchrony spectrum. The concept of locality of a computation is central to the first one, while the concept of wait-freedom is central to the second one. The paper proposes a new DECOUPLED model in an attempt to reconcile these two worlds. It consists of a synchronous and reliable communication graph of nnodes, and on top a set of asynchronous crash-prone processes, each attached to a communication node. To illustrate the DECOUPLED model, the paper presents an asynchronous 3-coloring algorithm for the processes of a ring. From the processes point of view, the algorithm is wait-free. From a locality point of view, each process uses information only from processes at distance \(O(\log ^{*} n)\) from it. This local wait-free algorithm is based on an extension of the classical Cole and Vishkin’s vertex coloring algorithm in which the processes are not required to start simultaneously.

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Notes

  1. CV86 was designed for trees in the PRAM model. It can be easily adapted to failure-free message-passing synchronous systems, for a ring, or a chain of processes

  2. Assuming n ≥ 2, \(\log ^{*} n\) is the number of times the function “\(\log _{2}\)” needs to be applied in the invocation \(\log _{2}(\log _{2}(\log _{2} ...(\log _{2} n)....))\) to obtain value 1. Let us remember that \(\log ^{*}(\)approx. number of atoms in the universe ) = 5.

  3. We use the “time” and “clock tick” terminology for the communication component, to prevent confusion with the “round” terminology used in the description of the CV86 and AST-CV algorithms.

  4. The assumption that processes know the global time is made only to simplify the description of our algorithms. All that a process pi needs to know is the relative order of wake up with respect to its neighbors, which can be deduced from the content of the buffers at wake up time sti.

  5. Usually, local refers to time complexity O(1). Here we adopt the more broad definition of time complexity o(D), where D is the diameter of the graph.

  6. If pi is both a left end and a right end of a unit-segment, it forms its own unit-segment.

  7. Being asynchronous, the waiting of pi during an arbitrary long (but finite) period does not modify its allowed behavior.

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Acknowledgments

C. Delporte, H. Fauconnier, and M. Raynal, were partially supported by the French ANR project DESCARTES, devoted to abstraction layers in distributed computing. A. Castañeda was supported in part by UNAM PAPIIT-DGAPA projects IA101015 and IA102417. S. Rajsbaum was supported in part by UNAM PAPIIT-DGAPA project IN107714.

All the authors want to thank INRIA for its support in the context of the INRIA-UNAM “Équipe Associée” LiDiCo (At the Limits of Distributed Computing).

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Authors and Affiliations

  1. Instituto de Matemáticas, UNAM, Mexico City, México

    Armando Castañeda & Sergio Rajsbaum

  2. IRIF, Université Paris Diderot, Paris, France

    Carole Delporte-Gallet & Hugues Fauconnier

  3. IUF & IRISA Université de Rennes, Rennes, France

    Michel Raynal

Authors
  1. Armando Castañeda
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  2. Carole Delporte-Gallet
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  3. Hugues Fauconnier
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  4. Sergio Rajsbaum
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  5. Michel Raynal
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Correspondence to Michel Raynal.

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This article is part of the Topical Collection on Special Issue on Stabilization, Safety, and Security of Distributed Systems (SSS 2016)

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Castañeda, A., Delporte-Gallet, C., Fauconnier, H. et al. Making Local Algorithms Wait-Free: the Case of Ring Coloring. Theory Comput Syst 63, 344–365 (2019). https://doi.org/10.1007/s00224-017-9772-y

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