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Proof-Labeling Schemes: Broadcast, Unicast and in Between

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Stabilization, Safety, and Security of Distributed Systems (SSS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10616))

Abstract

We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case \(r=1\), each node may only send the same message to all its neighbors (the broadcast model), in the case \(r\ge \varDelta \), where \(\varDelta \) is the largest node degree in the system, each neighbor may be sent a distinct message (the unicast model), and in general, for \(1\le r\le \varDelta \), each of the r messages is destined to a subset of the neighbors.

We show that message compression linear in r is possible for verifying fundamental problems such as the agreement between edge endpoints on the edge state. Some problems, including verification of maximal matching, exhibit a large gap in complexity between \(r=1\) and \(r>1\). For some other important predicates, the verification complexity is insensitive to r, e.g., the question whether a subset of edges constitutes a spanning-tree. We also consider the congested clique model. We show that the crossing technique [5] for proving lower bounds on the verification complexity can be applied in the case of congested clique only if \(r=1\). Together with a new upper bound, this allows us to determine the verification complexity of MST in the broadcast clique.

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Notes

  1. 1.

    Given a graph \(G=(V,E)\), the induced subgraph \(H=(V_H,E_H)\) over the set of nodes \(V_H\subseteq V\) satisfies that \(E_H=E\cap (V_H\times V_H)\).

  2. 2.

    We note that using simple port numbering requires agreement with the neighbors, which is costly, as we prove in Theorem 2.

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Authors and Affiliations

  1. School of Electrical Engineering, Tel Aviv University, 6997801, Tel Aviv, Israel

    Boaz Patt-Shamir & Mor Perry

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  1. Boaz Patt-Shamir
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  2. Mor Perry
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Correspondence to Mor Perry .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, United Kingdom

    Paul Spirakis

  2. Chalmers University of Technology, Gothenburg, Sweden

    Philippas Tsigas

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Patt-Shamir, B., Perry, M. (2017). Proof-Labeling Schemes: Broadcast, Unicast and in Between. In: Spirakis, P., Tsigas, P. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2017. Lecture Notes in Computer Science(), vol 10616. Springer, Cham. https://doi.org/10.1007/978-3-319-69084-1_1

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  • DOI: https://doi.org/10.1007/978-3-319-69084-1_1

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