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An adaptivity hierarchy theorem for property testing
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Canonne, Clément L. and Gur, Tom (2018) An adaptivity hierarchy theorem for property testing. computational complexity, 27 (4). pp. 671-716. doi:10.1007/s00037-018-0168-4 ISSN 1016-3328.
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Official URL: http://dx.doi.org/10.1007/s00037-018-0168-4
Abstract
Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of adaptive testing algorithms, wherein each query may be determined by the answers received to prior queries, and their non-adaptive counterparts, in which all queries are independent of answers obtained from previous queries.
In this work, we investigate the role of adaptivity in property testing at a finer level. We first quantify the degree of adaptivity of a testing algorithm by considering the number of “rounds of adaptivity” it uses. More accurately, we say that a tester is k-(round) adaptive if it makes queries in k+1 rounds, where the queries in the i ’th round may depend on the answers obtained in the previous i−1 rounds. Then, we ask the following question:
Does the power of testing algorithms smoothly grow with the number of rounds of adaptivity?
We provide a positive answer to the foregoing question by proving an adaptivity hierarchy theorem for property testing. Specifically, our main result shows that for every n∈N and 0≤k≤n0.33 there exists a property Pn,k of functions for which (1) there exists a k -adaptive tester for Pn,k with query complexity O~(k) , yet (2) any (k−1) -adaptive tester for Pn,k must make Ω~(n/k2) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||||
| Journal or Publication Title: | computational complexity | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 1016-3328 | ||||||||
| Official Date: | December 2018 | ||||||||
| Dates: |
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| Volume: | 27 | ||||||||
| Number: | 4 | ||||||||
| Page Range: | pp. 671-716 | ||||||||
| DOI: | 10.1007/s00037-018-0168-4 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Copyright Holders: | © Springer International Publishing AG, part of Springer Nature 2018 |
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