Introduction to Property Testing

Goldreich, Oded

doi:10.1017/9781108135252

Cited by 233 publications

(229 citation statements)

References 172 publications

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“…Particular cases include when X is the line [ CS14,BCS18]. We refer the reader to [Gol17, Chapter 4] for more on monotonicity testing, or for an overview of the field of property testing (as introduced in [RS96, GGR98]) in general.…”

Section: Related Workmentioning

confidence: 99%

Finding Monotone Patterns in Sublinear Time

Ben‐Eliezer

Canonne

Letzter

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed k ∈ N and ε > 0, we show that the non-adaptive query complexity of finding a length-k monotone subsequence of f : [n] → R, assuming that f is ε-far from free of such subsequences, is Θ((log n) log 2 k ). Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made (log n) O(k 2 ) non-adaptive queries; and the only lower bound known, of Ω(log n) queries for the case k = 2, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

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Section: Related Workmentioning

confidence: 99%

Finding Monotone Patterns in Sublinear Time

Ben‐Eliezer

Canonne

Letzter

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…We believe that this model is one of the most natural models, and it is also most relevant to computer science applications. Similarly as it has been done in [15,Chapter 10.5.3], we would advocate further study of this model because of its importance, its applications, and the variety (and beauty) of techniques used to advance this topic.…”

Section: Discussionmentioning

confidence: 91%

A Characterization of Graph Properties Testable for General Planar Graphs with one-Sided Error (It's all About Forbidden Subgraphs)

Czumaj

Sohler

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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The problem of characterizing testable graph properties (properties that can be tested with a number of queries independent of the input size) is a fundamental problem in the area of property testing. While there has been some extensive prior research characterizing testable graph properties in the dense graphs model and we have good understanding of the bounded degree graphs model, no similar characterization has been known for general graphs, with no degree bounds. In this paper we take on this major challenge and consider the problem of characterizing all testable graph properties in general planar graphs.We consider the model in which a general planar graph can be accessed by the random neighbor oracle that allows access to any given vertex and access to a random neighbor of a given vertex. We show that, informally, a graph property P is testable with one-sided error for general planar graphs if and only if testing P can be reduced to testing for a finite family of finite forbidden subgraphs. While our presentation focuses on planar graphs, our approach extends easily to general minor-free graphs.Our analysis of the necessary condition relies on a recent construction of canonical testers in the random neighbor oracle model that is applied here to the one-sided error model for testing in planar graphs. The sufficient condition in the characterization reduces the problem to the task of testing Hfreeness in planar graphs, and is the main and most challenging technical contribution of the paper: we show that for planar graphs (with arbitrary degrees), the property of being H-free is testable with one-sided error for every finite graph H, in the random neighbor oracle model.

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“…17 Evidently, if one does not wish to allow vertices of weight 0, then one can instead assign to v a weight tending to 0; or, more accurately, a weight that is small enough with respect to (the inverse of) the sample complexity of an alleged tester for P (in a proof by contradiction that such a tester does not exist). 18 We note that if P is non-extendable but hereditary, then one can easily obtain infinitely many examples showing that P is not testable (rather than just the one example given in the proof of Proposition 4.1). Indeed, instead of adding just one vertex to G 1 , one can add to G 1 any number k of vertices (for a large k), and give these new vertices weight o(1/k), while distributing the remaining weight uniformly among the vertices of G 1 (note that such an assignment is precisely what the setting of Theorem 9 forbids).…”

Section: On Variations Of the Vdf Model And Related Problemsmentioning

confidence: 99%

“…The adjacency matrix model was first defined and studied in [20], where the area of property testing was first introduced. This model has been extensively studied in the past two decades, see Chapter 8 of [18]. For a selected (but certainly not comprehensive) list of works on the dense graph model of property testing, see [2,21,23].…”

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confidence: 99%

Testing graphs against an unknown distribution

Gishboliner

Shapira

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graphs vertex set. A graph property P is said to be testable if the local statistics of a graph can allow one to distinguish between graphs satisfying P and those that are far from satisfying it.Goldreich recently introduced a generalization of this model in which one endows the vertex set of the input graph with an arbitrary and unknown distribution, and asked which of the properties that can be tested in the classical model can also be tested in this more general setting. We completely resolve this problem by giving a (surprisingly "clean") characterization of these properties. To this end, we prove a removal lemma for vertex weighted graphs which is of independent interest. Introduction Background and the main resultProperty testers are fast randomized algorithms whose goal is to distinguish (with high probability, say, 2/3) between objects satisfying some fixed property P and those that are ε-far from satisfying it. Here, ε-far means that an ε-fraction of the input object should be modified in order to obtain an object satisfying P. The study of such problems originated in the seminal papers of Rubinfeld and Sudan [28], Blum, Luby and Rubinfeld [9], and Goldreich, Goldwasser and Ron [20]. Problems of this nature have been studied in so many areas that it will be impossible to survey them here. Instead, the reader is referred to the recent monograph [18] for more background and references. While this area studies questions in theoretical computer science, it has several strong connections with central problems in extremal combinatorics, most notably to the regularity method and the removal lemma, see Subsection 1.2.The classical property testing model assumes that one can uniformly sample entries of the input. In distribution-free testing one assumes that the input is endowed with some arbitrary and unknown distribution D, which also affects the way one defines the distance to satisfying a property. As discussed in [19], one motivation for this model is that it can handle settings in which one cannot produce uniformly distributed entries from the input. Another motivation is that the distribution D can assign higher weight/importance to parts of the input which we want to have higher impact on the distance to satisfying the given property. Until very recently, problems of this type were studied almost exclusively in the setting of testing properties of functions, see [10,11,15,17,24]. Let us mention that distribution-free testing is similar in spirit to the celebrated PAC learning model of Valiant [31], see also the discussion in [27].Our investigation here concerns a distribution-free variant of the adjacency matrix model, also known as the dense graph model. The adjacency matrix model was first defined and studied in [20], where the area of property testing was first...

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Introduction to Property Testing

Cited by 233 publications

References 172 publications

Finding Monotone Patterns in Sublinear Time

Finding Monotone Patterns in Sublinear Time

A Characterization of Graph Properties Testable for General Planar Graphs with one-Sided Error (It's all About Forbidden Subgraphs)

Testing graphs against an unknown distribution

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