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Randomized proof-labeling schemes

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Abstract

Proof-labeling schemes, introduced by Korman et al. (Distrib Comput 22(4):215–233, 2010. https://doi.org/10.1007/s00446-010-0095-3), are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces verification complexity exponentially while guaranteeing probability of correctness arbitrarily close to one. We also present a novel message-size lower bound technique that applies to deterministic as well as randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size.

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Notes

  1. The existence of p is guaranteed by the Bertrand–Chebyshev theorem, stating that for every \(n>1\) there is always at least one prime p such that \(n<p<2n\).

  2. Although we always refer to connected graphs in this paper, here we emphasize again that this is the set of graphs on which our schemes work. Distributed algorithms are relevant for disconnected graphs when the input graph is a sub-graph of the communication network (e.g., the congested clique [41]).

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Acknowledgements

The authors are thankful to the reviewers for their valuable comments. Thanks to insights from Reviewer #1, the deterministic upper bound of Theorem 7 has been improved from \(O(\log n)\) to \(O(\log c)\), matching the lower bound.

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Authors and Affiliations

  1. Institut de Recherche en Informatique Fondamentale, CNRS and University Paris Diderot, Paris, France

    Pierre Fraigniaud

  2. School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel

    Boaz Patt-Shamir & Mor Perry

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  1. Pierre Fraigniaud
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  3. Mor Perry
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Correspondence to Mor Perry.

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A preliminary version of this paper appeared in Proceedings of 34th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), Donostia-San Sebastin, Spain, July 21-23, 2015.

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Fraigniaud, P., Patt-Shamir, B. & Perry, M. Randomized proof-labeling schemes. Distrib. Comput. 32, 217–234 (2019). https://doi.org/10.1007/s00446-018-0340-8

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