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On the Effect of the Proximity Parameter on Property Testers

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Computational Complexity and Property Testing

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12050))

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Abstract

This work refers to the effect of the proximity parameter on the operation of (standard) property testers. Its bottom-line is that, except in pathological cases, the effect of the proximity parameter is restricted to determining the query complexity of the tester. The point is that, in non-pathological cases, the mapping of the proximity parameter to the query complexity can be reversed in an adequate sense.

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Notes

  1. 1.

    For example, if the domain is a finite field, then one may need to provide its representation (and not merely its size), especially when no standard representation can be assumed (e.g., as in the case that the field has \(2^n\) elements). Another example refers to the bounded-degree graph model (cf. [5]), where one should also provide the degree bound rather than just its product with the number of vertices; actually, one typically provides the degree bound and the number of vertices (and does not provide their multiple).

  2. 2.

    This crucial fact was overlooked in [7], as pointed out in [1, 8]. Actually, the assertion itself is inaccurate; one may consider alternative models (cf. [3]) in which the tester is given sampling access to the domain (instead of its description).

  3. 3.

    We refer to the standard definition (as in, e.g., [4, 9]), and not to the definition of a proximity-oblivious tester (cf. [6]).

  4. 4.

    Indeed, we use \({\lfloor {s\epsilon }\rfloor }+u > (s\epsilon -1)+u = s\epsilon -(1-u)\) and the hypothesis that \(q(s,\epsilon ')\le q(s,\epsilon '')\) for every \(\epsilon '\ge \epsilon ''>0\). Actually, typically \({\widehat{q}}(s,\epsilon )=q(s,\epsilon )\) holds, since \(s\epsilon -(1-u)\approx s\epsilon \), let alone that \({\lfloor {s\epsilon }\rfloor }+u > s\epsilon \) may hold. (This is indeed the case when \(\epsilon \) is a multiple of 1/s (as advocated in the previous comment).).

References

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

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  1. Oded Goldreich
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Correspondence to Oded Goldreich .

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  1. Weizmann Institute of Science, Rehovot, Israel

    Oded Goldreich

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Goldreich, O. (2020). On the Effect of the Proximity Parameter on Property Testers. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-43662-9_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43661-2

  • Online ISBN: 978-3-030-43662-9

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