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Fast distributed algorithms for testing graph properties

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Abstract

We initiate a thorough study of distributed property testing—producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general model we obtain faster tests for triangle-freeness and cycle-freeness, and in the sparse model we obtain a faster test for bipartiteness. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time than their counterparts from the sequential querying model of traditional property testing. More importantly, the distributed algorithms we develop for testing graph properties are in many cases much faster than what is known for exactly deciding whether the property holds. The simplest property testing algorithms allow a relatively smooth transition to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

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Notes

  1. Here \(\tilde{\Omega }\) hides factors that are polylogarithmic in n.

  2. The probability of rejection can be improved by standard amplification techniques. In particular, the test can be repeated multiple times, and a vertex will reject if it rejects in any of the invocations.

  3. In the literature there are also investigations of a slightly strengthened general model, where pair queries are also allowed; its discussion is out of the scope for this paper.

  4. This technique was recently independently and concurrently devised in [20] for a different use.

  5. Sometimes in the sparse graph model the allowed number of changes is \(\epsilon dn\), as relates to the maximum possible number of edges; when d is held constant the difference is not essential.

  6. Pipelining means that each vertex has a buffer for each edge, which holds the information (edges between chosen vertices, in our case) it needs to send over that edge. The vertex sends the pieces of information one after the other.

  7. Note that the paths we address in this section are not necessarily simple.

  8. A more involved analysis of multiple prioritized BFS executions was used in [28], allowing all BFS executions to fully finish in a short time without too much delay due to congestion. Since we require a much weaker guarantee, we can avoid the strong full-fledged prioritization algorithm of [28] and settle for a simple rule that keeps one BFS tree alive. Also, the multiple BFS construction of [33] does not fit our demands as it may not reach all desired vertices within the required distance, in case there are many vertices that are closer.

  9. Our lower bound applies even to the less restricted LOCAL model of communication, which does not limit the size of the messages.

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Authors and Affiliations

  1. Department of Computer Science, Technion – Israel Institute of Technology, Haifa, Israel

    Keren Censor-Hillel, Eldar Fischer, Gregory Schwartzman & Yadu Vasudev

Authors
  1. Keren Censor-Hillel
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  2. Eldar Fischer
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  3. Gregory Schwartzman
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  4. Yadu Vasudev
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Corresponding author

Correspondence to Keren Censor-Hillel.

Additional information

A preliminary version of this work appeared in DISC 2016, pages 43–56.

Supported in part by the Israel Science Foundation (Grant 1696/14).

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Censor-Hillel, K., Fischer, E., Schwartzman, G. et al. Fast distributed algorithms for testing graph properties. Distrib. Comput. 32, 41–57 (2019). https://doi.org/10.1007/s00446-018-0324-8

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