An adaptivity hierarchy theorem for property testing

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 counterp...

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Bibliographic Details
Published in:Computational complexity Vol. 27; no. 4; pp. 671 - 716
Main Authors: Canonne, Clément L., Gur, Tom
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.12.2018
Springer Nature B.V
Subjects:
Adaptive algorithms
Algorithm Analysis and Problem Complexity
Algorithms
Computational Mathematics and Numerical Analysis
Computer Science
Queries
Theorems
sublinear-time algorithms
complexity theory
property testing
F.1.3 [Computation by Abstract Devices] Complexity Measures and Classes
ISSN:1016-3328, 1420-8954
Online Access:Get full text
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Summary: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 ≤ n 0.33 there exists a property P n , k of functions for which (1) there exists a k -adaptive tester for P n , k with query complexity O ~ ( k ) , yet (2) any ( k - 1 ) -adaptive tester for P n , k must make Ω ~ ( n / k 2 ) queries. In addition, we show that such a qualitative adaptivity hierarchy can be witnessed for testing natural properties of graphs.
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ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-018-0168-4