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20
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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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.
A unified framework for testing linearinvariant properties
 In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science
, 2010
"... In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and F ..."
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Cited by 7 (4 self)
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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linearinvariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following: 1. We introduce a simple combinatorial condition, which we call subspaceheredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. 2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with onesided error, thus addressing a challenge posed by Sudan recently. 3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.
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 7 (3 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
Efficient testing of sparse GF(2) polynomials
 In Automata, Languages and Programming: ThirtyFifth International Colloquium (ICALP
, 2008
"... Abstract. We give the first algorithm that is both queryefficient and timeefficient for testing whether an unknown function f: {0, 1} n →{−1, 1} is an ssparse GF(2) polynomial versus ǫfar from every such polynomial. Our algorithm makes poly(s,1/ǫ) blackbox queries to f and runs in time n · poly ..."
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Cited by 4 (1 self)
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Abstract. We give the first algorithm that is both queryefficient and timeefficient for testing whether an unknown function f: {0, 1} n →{−1, 1} is an ssparse GF(2) polynomial versus ǫfar from every such polynomial. Our algorithm makes poly(s,1/ǫ) blackbox queries to f and runs in time n · poly(s,1/ǫ). The only previous algorithm for this testing problem [DLM + 07] used poly(s,1/ǫ) queries, but had running time exponential in s and superpolynomial in 1/ǫ. Our approach significantly extends the “testing by implicit learning ” methodology of [DLM + 07]. The learning component of that earlier work was a bruteforce exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie [SS96]. A crucial element of this work, which enables us to simulate the membership queries required by [SS96], is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of “lowinfluence ” sets of variables. 1
Learning kModal Distributions via Testing
"... A kmodal probability distribution over the domain {1,..., n} is one whose histogram has at most k “peaks ” and “valleys. ” Such distributions are natural generalizations of monotone (k = 0) and unimodal (k = 1) probability distributions, which have been intensively studied in probability theory and ..."
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A kmodal probability distribution over the domain {1,..., n} is one whose histogram has at most k “peaks ” and “valleys. ” Such distributions are natural generalizations of monotone (k = 0) and unimodal (k = 1) probability distributions, which have been intensively studied in probability theory and statistics. In this paper we consider the problem of learning an unknown kmodal distribution. The learning algorithm is given access to independent samples drawn from the kmodal distribution p, and must output a hypothesis distribution ˆp such that with high probability the total variation distance between p and ˆp is at most ɛ. We give an efficient algorithm for this problem that runs in time poly(k, log(n), 1/ɛ). For k ≤ Õ( √ log n), the number of samples used by our algorithm is very close (within an Õ(log(1/ɛ)) factor) to being informationtheoretically optimal. Prior to this work computationally efficient algorithms were known only for the cases k = 0, 1 [Bir87b, Bir97]. A novel feature of our approach is that our learning algorithm crucially uses a new property testing algorithm as a key subroutine. The learning algorithm uses the property tester to efficiently decompose the kmodal distribution into k (near)monotone distributions, which are easier to learn. 1
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 2 (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.
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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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,
Approximating and Testing kHistogram Distributions in Sublinear Time
"... A discrete distribution p, over [n], is a khistogram if its probability distribution function can be represented as a piecewise constant function with k pieces. Such a function is represented by a list of k intervals and k corresponding values. We consider the following problem: given a collection ..."
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A discrete distribution p, over [n], is a khistogram if its probability distribution function can be represented as a piecewise constant function with k pieces. Such a function is represented by a list of k intervals and k corresponding values. We consider the following problem: given a collection of samples from a distribution p, find a khistogram that (approximately) minimizes the ℓ2 distance to the distribution p. We give time and sample efficient algorithms for this problem. We further provide algorithms that distinguish distributions that have the property of being a khistogram from distributions that are ɛfar from any khistogram in the ℓ1 distance and ℓ2 distance respectively. 1.
A Brief Introduction to Property Testing
"... Abstract. This short article provides a brief description of the main issues that underly the study of property testing. It is meant to serve as a general introduction to a collection of surveys and extended abstracts ..."
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Abstract. This short article provides a brief description of the main issues that underly the study of property testing. It is meant to serve as a general introduction to a collection of surveys and extended abstracts
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"... Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespeci ed property or is ɛfar from the property (for a given distance parameter ɛ, and for a prespeci ed distance measu ..."
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Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespeci ed property or is ɛfar from the property (for a given distance parameter ɛ, and for a prespeci ed distance measure). The tester is given query access to the input, and is required to run in sublinear time. In this thesis we focus on testing properties of directed graphs (digraphs). In particular we present the following results (where n is the number of vertices in the graph, d is the maximum degree, and davg is the average degree): 1. We present a testing algorithm for the property of Eulerianity in boundeddegree digraphs, which runs in time 1 Õ(1/ɛ). For unboundeddegree digraphs we show a lower bound of Ω ( √ n/ɛ), and give a testing algorithm that runs in time Õ( √ n/ɛ 3/2). 2. We consider the property of kedgeconnectivity in digraphs and present testing algorithms for both boundeddegree digraphs and unboundeddegree digraphs, that run in time