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Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

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Abstract

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously \(\tilde{O}(m)\) messages and \(\tilde{O}(\sqrt{n} + D)\) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send \(\tilde{\varOmega}(m)\) messages and incur \(\tilde{\varOmega}(\sqrt{n} + D)\) time in worst case.

Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of \(\tilde{\varOmega}(m)\) messages and \(\tilde{\varOmega}(\sqrt{n} + D)\) time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously \(\tilde{O}(m)\) messages and \(\tilde{O}(\sqrt{n} + D)\) time. Specifically, the best known time-optimal algorithm (using \({\tilde{O}}(\sqrt {n} + D)\) time) requires O(m+n 3/2) messages, and the best known message-optimal algorithm (using \({\tilde{O}}(m)\) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.

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Notes

  1. \(\tilde{\varOmega}\) (respectively \(\tilde{O}\)) is a relaxed variant of the Ω (rep., O) notation that ignores polylog factors.

  2. Equivalently, we may consider also disconnected graphs, and require T to be a spanning forest of G.

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Authors and Affiliations

  1. Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Liah Kor & David Peleg

  2. CNRS and University Paris Diderot – Paris 7, Paris, France

    Amos Korman

Authors
  1. Liah Kor
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  2. Amos Korman
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  3. David Peleg
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Corresponding author

Correspondence to Amos Korman.

Additional information

L.K. and D.P. are supported by a grant from the United States-Israel Binational Science Foundation (BSF).

A.K. is supported by the ANR project DISPLEXITY, and by the INRIA project GANG.

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Kor, L., Korman, A. & Peleg, D. Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification. Theory Comput Syst 53, 318–340 (2013). https://doi.org/10.1007/s00224-013-9479-7

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