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195
Probabilistic checking of proofs: a new characterization of NP
 Journal of the ACM
, 1998
"... Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from ..."
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Cited by 366 (28 self)
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Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NPhard.
Simple Constructions of Almost kwise Independent Random Variables
, 1992
"... We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the dist ..."
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Cited by 272 (41 self)
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We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1))(log log n + k/2 + log k + log 1 ɛ), where ɛ is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ɛ < 1/(k log n)). An additional advantage of our constructions is their simplicity.
Learning Decision Trees using the Fourier Spectrum
, 1991
"... This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each ..."
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Cited by 187 (10 self)
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This work gives a polynomial time algorithm for learning decision trees with respect to the uniform distribution. (This algorithm uses membership queries.) The decision tree model that is considered is an extension of the traditional boolean decision tree model that allows linear operations in each node (i.e., summation of a subset of the input variables over GF (2)). This paper shows how to learn in polynomial time any function that can be approximated (in norm L 2 ) by a polynomially sparse function (i.e., a function with only polynomially many nonzero Fourier coefficients). The authors demonstrate that any function f whose L 1 norm (i.e., the sum of absolute value of the Fourier coefficients) is polynomial can be approximated by a polynomially sparse function, and prove that boolean decision trees with linear operations are a subset of this class of functions. Moreover, it is shown that the functions with polynomial L 1 norm can be learned deterministically. The algorithm can a...
Visual Cryptography
, 1995
"... In this paper we consider a new type of cryptographic scheme, which can decode concealed images without any cryptographic computations. The scheme is perfectly secure and very easy to implement. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a ..."
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Cited by 180 (4 self)
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In this paper we consider a new type of cryptographic scheme, which can decode concealed images without any cryptographic computations. The scheme is perfectly secure and very easy to implement. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a transparency to each one of the n users; any k of them can see the image by stacking their transparencies, but any k  1 of them gain no information about it.
Improved NonApproximability Results
, 1994
"... We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in... ..."
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Cited by 118 (15 self)
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We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in...
Construction of asymptotically good, lowrate errorcorrecting codes through pseudorandom graphs
 IEEE Transactions on Information Theory
, 1992
"... A new technique, based on the pseudorandom properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping ..."
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Cited by 115 (22 self)
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A new technique, based on the pseudorandom properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF (2)) as well. Although these concatenated codes lie below Zyablov bound, they are still superior to previouslyknown explicit constructions in the zerorate neighborhood.
ChernoffHoeffding Bounds for Applications with Limited Independence
 SIAM J. Discrete Math
, 1993
"... ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the rando ..."
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Cited by 103 (10 self)
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ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the random variables, thereby importing a variety of standard results to the case of limited independence for free. Additional methods are also presented, and the aggregate results are sharp and provide a better understanding of the proof techniques behind these bounds. They also yield improved bounds for various tail probability distributions and enable improved approximation algorithms for jobshop scheduling. The "limited independence" result implies that a reduced amount of randomness and weaker sources of randomness are sufficient for randomized algorithms whose analyses use the ChernoffHoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routi...
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 102 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
Checking the Correctness of Memories
 Algorithmica
, 1995
"... We extend the notion of program checking to include programs which alter their environment. In particular, we consider programs which store and retrieve data from memory. The model we consider allows the checker a small amount of reliable memory. The checker is presented with a sequence of reques ..."
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Cited by 96 (10 self)
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We extend the notion of program checking to include programs which alter their environment. In particular, we consider programs which store and retrieve data from memory. The model we consider allows the checker a small amount of reliable memory. The checker is presented with a sequence of requests (online) to a data structure which must reside in a large but unreliable memory. We view the data structure as being controlled by an adversary. We want the checker to perform each operation in the input sequence using its reliable memory and the unreliable data structure so that any error in the operation of the structure will be detected by the checker with high probability. We present checkers for various data structures. We prove lower bounds of log n on the amount of reliable memory needed by these checkers where n is the size of the structure. The lower bounds are information theoretic and apply under various assumptions. We also show timespace tradeoffs for checking random access memories as a generalization of those for coherent functions. 1