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25
Algorithmic and Analysis Techniques in Property Testing
"... Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform ..."
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Cited by 27 (4 self)
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Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The selfcorrecting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the densegraphs model), as well as in other contexts;
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 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
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.
Algorithmic aspects of property testing in the dense graphs model. ECCC
, 2008
"... In this paper we consider two basic questions regarding the query complexity of testing graph properties in the adjacency matrix model. The first question refers to the relation between adaptive and nonadaptive testers, whereas the second question refers to testability within complexity that is inv ..."
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In this paper we consider two basic questions regarding the query complexity of testing graph properties in the adjacency matrix model. The first question refers to the relation between adaptive and nonadaptive testers, whereas the second question refers to testability within complexity that is inversely proportional to the proximity parameter, denoted ɛ. The study of these questions reveals the importance of algorithmic design (also) in this model. The highlights of our study are: • A gap between the complexity of adaptive and nonadaptive testers. Specifically, there exists a (natural) graph property that can be tested using Õ(ɛ−1) adaptive queries, but cannot be tested using o(ɛ −3/2) nonadaptive queries. • In contrast, there exist natural graph properties that can be tested using Õ(ɛ−1) nonadaptive queries, whereas Ω(ɛ −1) queries are required even in the adaptive case. We mention that the properties used in the foregoing conflicting results have a similar flavor, although they are of course different. Keywords:
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 5 (0 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.
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
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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Cited by 3 (0 self)
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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.
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 3 (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.
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