Can Distributed Uniformity Testing Be Local?

Meir, Uri; Minzer, Dor; Oshman, Rotem

doi:10.1145/3293611.3331613

Cited by 2 publications

(6 citation statements)

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“…Thus, the sample complexity we achieve is not only optimal with respect to collision-based testing, but also near-optimal with respect to any uniformity tester. This is evident from a variety of known lower bounds in some of the models considered in this paper (See [25, 16,23,13,4] for these lower bounds 11 ). The full discussion of collision-based testing vs. arbitrary testers appears at the end of this section, in hope to give a clearer picture of how strong collision-based testing can be.…”

Section: Limitations Of the Methodsmentioning

confidence: 88%

“…In the past few years there has been a growing interest in distribution testing under various computational models, including collaborative testing in multiparty model, the streaming model, privacy aspects of testing, and others ( [6,7,17,9,2,23,4,5,1,24,18,8] and more). We mention in more details the works concerning uniformity testers in models we pursue.…”

Section: Related Workmentioning

confidence: 99%

“…In addition to these, quite a few works had the focus of showing impossibility results both in the classic and the entire variety of models. For out interest, we mention [16,25] for the classic version, as well as [23] for the simultaneous model (with multiple samples per machine), and [13,4] for the streaming model.…”

Section: Related Workmentioning

confidence: 99%

“…We give our focus to the variant posed in [17]: what is the number of samples per player needed to test uniformity? The "single collision" algorithm devised for that question only requires one bit from each player, and indeed it was shown to be optimal in [23]. However, this algorithm is somewhat delicate: first, the range of the parameter k is limited (it cannot be too high or too low, with regards to n, ); second, if the number of players k is not accurately known to all players, the algorithm breaks.…”

Section: Simultaneous Modelmentioning

confidence: 99%

“…Remark 8. As apparent from [6,7,23], when messages are r bits long, the correct value to look for in the sample complexity is 2 r (and sometimes √ 2 r ). For this reason, when discussing communication, we explicitly write expressions such as log Θ(1/ 4 ) instead of a more general Θ (log(1/ )).…”

Section: Corollary 7 In the Simultaneous Model One Can Test Uniformit...mentioning

confidence: 99%

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Comparison Graphs: a Unified Method for Uniformity Testing

Meir

2020

Preprint

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Distribution testing can be described as follows: q samples are being drawn from some unknown distribution P over a known domain [n]. After the sampling process, a decision must be made about whether P holds some property, or is far from it. The most studied problem in the field is arguably uniformity testing, where one needs to distinguish the case that P is uniform over [n] from the case that P is -far from being uniform (in 1). The sample complexity of a property is the amount of necessary and sufficient samples to test for this property, and it is known to be Θ √ n/ 2 when testing for uniformity. This problem was recently considered in various restricted models that pose, for example, communication or memory constraints. In more than one occasion, the known optimal solution boils down to counting collisions among the drawn samples (each two samples that have the same value add one to the count). This idea dates back to the first uniformity tester, and was coined the name "collision-based tester".In this paper, we introduce the notion of comparison graphs and use it to formally define a generalized collision-based tester. Roughly speaking, the edges of the graph indicate the tester which pairs of samples should be compared (that is, the original tester is induced by a clique, where all pairs are being compared). We prove a structural theorem that gives a sufficient condition for a comparison graph to induce a good uniformity tester. As an application, we develop a generic method to test uniformity, and devise nearly-optimal uniformity testers under various computational constraints. We improve and simplify a few known results, and introduce a new model in which the method also produces an efficient tester.The idea behind our method is to translate the computational constraints of a certain model to ones on the comparison graph, which paves the way to finding a good graph: a set of pairs that can be compared in this model, and induces a uniformity tester. We believe that in future consideration of uniformity testing in new models, our method can be used to obtain efficient testers with minimal effort.

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