We study the question of determining whether an unknown function has a particular property or is ffl-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being k-colorable or having a ae-clique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edge-queries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, k-connected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is ffl-away from having the property. For example, the 2-Connectivity algorithm rejects (w.h.p.) any N-vertex d-degree graph for which more ...

this technical aspect (as in the bounded-degree model the closest graph having the property must have at most dN edges and degree bound d as well).

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Locally testable codes are error-correcting codes that admit very efficient codeword tests. Specifically, using

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We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is ffl-far from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).

We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We consider (randomized) algorithms which may query the function at arguments of their choice, and seek algorithms which query the function at relatively few places. We focus on combinatorial properties, and specifically on graph properties. The two standard representations of graphs -- by adjacency matrices and by incidence lists -- yield two different models for testing graph properties. In the first model, most appropriate for dense graphs, distance between N-vertex graphs is measured as the fraction of edges on which the graphs disagree over N 2 . In the second model, most appropriate for bounded-degree graphs, distance between N-vertex d-degree graphs is measured as the fraction of edges on ...

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Zero-knowledge proofs are proofs that are both convincing and yet yield nothing beyond the validity of the assertion being proven. Since their introduction about twenty years ago, zero-knowledge proofs have attracted a lot of attention and have, in turn, contributed to the development of other areas of cryptography and complexity theory.