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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 12 (3 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 Computability by Width Two OBDDs
"... Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by ..."
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Cited by 1 (1 self)
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Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a readonce width2 Ordered Binary Decision Diagram (OBDD), also known as a branching program, where the order of the variables is known. Width2 OBDDs generalize two classes of functions that have been studied in the context of property testing linear functions (over GF (2)) and monomials. In both these cases membership can be tested in time that is linear in 1/ɛ. Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number, n, of variables in the tested function, we show that (onesided error) testing for computability by a width2 OBDD requires Ω(log(n)) queries, and give an algorithm (with onesided error) that tests for this property and performs Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is “far ” (for some definition of far) from every object with that property [RS96,
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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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