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46
Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
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Cited by 34 (8 self)
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We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 57 (2011)
, 2011
"... A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts ..."
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Cited by 21 (4 self)
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A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) In other words, Lipschitz functions are not very sensitive to small changes in the input. We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are ɛfar from having the property, that is, differ from every function with the property on at least an ɛ fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f. We consider functions over domains {0, 1} d, {1,..., n} and {1,..., n} d, equipped with ℓ1 distance.
Testing Boolean Function Isomorphism
"... Abstract. Two boolean functions f, g: {0, 1} n → {0, 1} are isomorphic if they are identical up to relabeling of the input variables. We consider the problem of testing whether two functions are isomorphic or far from being isomorphic with as few queries as possible. In the setting where one of the ..."
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Cited by 11 (5 self)
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Abstract. Two boolean functions f, g: {0, 1} n → {0, 1} are isomorphic if they are identical up to relabeling of the input variables. We consider the problem of testing whether two functions are isomorphic or far from being isomorphic with as few queries as possible. In the setting where one of the functions is known in advance, we show that the nonadaptive query complexity of the isomorphism testing problem is ˜ Θ(n). In fact, we show that the lower bound of Ω(n) queries for testing isomorphism to g holds for almost all functions g. In the setting where both functions are unknown to the testing algorithm, we show that the query complexity of the isomorphism testing problem is ˜ Θ(2 n/2). The bound in this result holds for both adaptive and nonadaptive testing algorithms. 1
On Proximity Oblivious Testing
, 2009
"... We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximityobli ..."
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We initiate a systematic study of a special type of property testers. These testers consist of repeating a basic test for a number of times that depends on the proximity parameter, whereas the basic test is oblivious of the proximity parameter. We refer to such basic tests by the term proximityoblivious testers. While proximityoblivious testers were studied before – most notably in the algebraic setting – the current study seems to be the first one to focus on graph properties. We provide a mix of positive and negative results, and in particular characterizations of the graph properties that have constantquery proximityoblivious testers in the two standard models (i.e., the adjacency matrix and the boundeddegree models). Furthermore, we show that constantquery proximityoblivious testers do not exist for many easily testable properties, and that even when proximityoblivious testers exist, repeating them does not necessarily yield the best standard testers for the corresponding property.
Testing low complexity affineinvariant properties
 In Khanna [Kha13
"... Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant n ..."
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Cited by 8 (3 self)
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Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over Fp of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that having low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f: F n p → [R] is testable, if the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis. 1
Proximity Oblivious Testing and the Role of Invariances
"... We present a general notion of properties that are characterized by local conditions that are invariant under a sufficiently rich class of symmetries. Our framework generalizes two popular models of testing graph properties as well as the algebraic invariances studied by Kaufman and Sudan (STOC’08) ..."
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Cited by 7 (1 self)
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We present a general notion of properties that are characterized by local conditions that are invariant under a sufficiently rich class of symmetries. Our framework generalizes two popular models of testing graph properties as well as the algebraic invariances studied by Kaufman and Sudan (STOC’08). Our focus is on the case that the property is characterized by a constant number of local conditions and a rich set of invariances. We show that, in the aforementioned models of testing graph properties, characterization by such invariant local conditions is closely related to proximity oblivious testing (as defined by Goldreich and Ron, STOC’09). In contrast to this relation, we show that, in general, characterization by invariant local conditions is neither necessary nor sufficient for proximity oblivious testing. Furthermore, we show that easy testability is not guaranteed even when the property is characterized by local conditions that are invariant under a 1transitive group of permutations.
Lower Bounds for Testing Properties of Functions on Hypergrid Domains
"... Abstract. We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions f: [n] d → R on the hypergrid: monotonicity, convexity, and the Lipschitz property. Our lower bounds also apply to the more restri ..."
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Cited by 7 (1 self)
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Abstract. We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions f: [n] d → R on the hypergrid: monotonicity, convexity, and the Lipschitz property. Our lower bounds also apply to the more restricted setting of functions f: [n] → R on the line (i.e., to hypergrids with d = 1), where they give optimal lower bounds for all three properties. The lower bound for testing convexity is the first lower bound for that property, and the lower bound for the Lipschitz property is new for tests with 2sided error. We obtain our lower bounds via the connection to communication complexity established by Blais, Brody, and Matulef (2012). Our results are the first to apply this method to functions with nonhypercube domains. A key ingredient in this generalization is the set of Walsh functions, an orthonormal basis of the set of functions f: [n] d → R. 1
Comparing the strength of query types in property testing: the case of testing kcolorability
, 2007
"... Abstract. We study the power of four query models in the context of property testing in general graphs, where our main case study is the problem of testing kcolorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds ..."
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Cited by 7 (3 self)
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Abstract. We study the power of four query models in the context of property testing in general graphs, where our main case study is the problem of testing kcolorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i th neighbor of a particular vertex. We show that while for pair queries, testing kcolorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give upper and lower bounds on the query complexity for onesided error in all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to twosided error algorithms. The problem of testing kcolorability was previously studied in the contexts of dense graphs and of sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.
Algorithms on Evolving Graphs
"... and massive in nature, we define a new general framework for computing with such graphs. In our framework, the graph changes over time andan algorithm can only track these changes by explicitly probing the graph. This framework captures the inherent tradeoff between the complexity of maintaining an ..."
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Cited by 6 (1 self)
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and massive in nature, we define a new general framework for computing with such graphs. In our framework, the graph changes over time andan algorithm can only track these changes by explicitly probing the graph. This framework captures the inherent tradeoff between the complexity of maintaining an uptodateviewof the graph and the quality of results computed with the available view. We apply this framework to two classical graph connectivityproblems, namely, pathconnectivityandminimumspanningtrees, and obtain efficient algorithms.
NonInteractive Proofs of Proximity
, 2013
"... We initiate a study of noninteractive proofs of proximity. These proofsystems consist of a verifier that wishes to ascertain the validity of a given statement, using a short (sublinear length) explicitly given proof, and a sublinear number of queries to its input. Since the verifier cannot even re ..."
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Cited by 6 (1 self)
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We initiate a study of noninteractive proofs of proximity. These proofsystems consist of a verifier that wishes to ascertain the validity of a given statement, using a short (sublinear length) explicitly given proof, and a sublinear number of queries to its input. Since the verifier cannot even read the entire input, we only require it to reject inputs that are far from being valid. Thus, the verifier is only assured of the proximity of the statement to a correct one. Such proofsystems can be viewed as the N P (or more accurately MA) analogue of property testing. We explore both the power and limitations of noninteractive proofs of proximity. We show that such proofsystems can be exponentially stronger than property testers, but are exponentially weaker than the interactive proofs of proximity studied by Rothblum, Vadhan and Wigderson (STOC 2013). In addition, we show a natural problem that has a full and (almost) tight multiplicative tradeoff between the length of the proof and the verifier’s query complexity. On the negative side, we also show that there exist properties for which even a linearlylong (noninteractive) proof of proximity cannot significantly reduce the query complexity.