Yuri Gurevich, "Average Case Complexity", ICALP'91, 18th International Colloquium on Automata, Languages and Programming, Madrid, Springer Lecture Notes in Computer Science 510, 1991, 615--628.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are several subtle and important issues regarding defining a robust notion of average polynomial time, and we discuss them below. These issues were either mentioned explicitly or hinted at by Levin [Lev86] and various aspects of the issues have been elaborated on by Johnson [Joh84] Gurevich [Gur89, Gur91a, Gur91b], Venkatesan [Ven91] and Impagliazzo [Imp95] From this, Levin s definition of average polynomial time (given here as Definition 2.1) can be derived naturally and can be well justified. Model Independence. Let Sigma n = fx : jxj = ng. Let A be a subset of Sigma n and jAj proportional to 2 ....

....requirement, as shown in footnote 3, based on which the following lemma is straightforward. Lemma 2.1 A function f is polynomial on average iff there are constants k 0 and c such that, for all l 2 IR , f(x) ljxj) k ] c=l. Further discussion of the balancing issue can be found in [Gur89, Gur91b]. Regarding the distribution issue, Impagliazzo [Imp95] observed that Definition 2.1 is equivalent to taking the average on instances up to length n since, when n is sufficiently large, jxj n] is greater than a fixed positive constant. Lemma 2.2 A function f is polynomial on average iff there ....

Y. Gurevich. Average case complexity. In Proceedings of the 18th International Colloquium on Automata, Languages and Programming, vol 510 of Lecture Notes in Computer Science, Springer-Verlag, pages 615--628, 1991.

....are equivalent under polynomial time Turing reductions, as, for example, in the case of NP complete problems. Since polynomial time algorithms do not exist for NP complete problems, unless P = NP, it seems reasonable to look for algorithms that are efficient in the average case [Lev86] see [Gur91] or [Wan] for a survey) In practical applications, instances occur with certain probabilities. To model this process, we use probabilistic Turing machines that output instances in time polynomial in the length of the generated instances [BCGL92] Besides designing efficient algorithms it is also ....

....generator for another NP search problem to which the given one reduces. 2 Preliminaries In this paper we use the standard notations and definitions of computational complexity theory (see, e.g. BDG95] An introduction to the theory of computational average case complexity can be found in [Gur91, BCGL92, Wan]. All languages considered here are over the alphabet Sigma = f0; 1g. The length of a string x 2 Sigma is denoted by jxj. For a set A of strings, let A =n = A Sigma n and A n = S n k=0 A =k . We denote the cardinality of a set A by kAk. We use the pairing function h Delta; ....

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Y. Gurevich. Average case complexity. Journal of Computer and System Sciences, 42(3):346--398, 1991.

....implies that of Ben David et al. since, otherwise, we can construct a polynomial time computable function which the standard distribution of its pre images is not computable in polynomial time. To handle broader classes of distributions whose ranges vary on real numbers between 0 and 1, Gurevich [6] proposed an approximation scheme to polynomial time computable distributions: a distribution is called polynomial time computable if we have a polynomial time algorithm which, on a pair of inputs x and 0 i , outputs an approximation of the value (x) within a factor of 2 Gammai . In this ....

....in average case complexity theory. The first step along this line is due to Levin [11] He introduced the notion of many one reducibility between distributional problems (or randomized problems) by requiring an additional polynomial domination relation between distributions. As proven in [6], distributional problems which can be solved in polynomial time on the average are invariant to polynomial domination relations. In Levin s definition, domination relations play a special role of reducibility between distributions in measuring the complexity of these distributions. For ....

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Y. Gurevich, Average case complexity, J. Comput. System Sci., 42 (1991), pp.346--398.

....reductions to randomized decision problems [Lev86] In the same paper, Levin presented a randomized decision problem which is complete for DistNP: the randomized Tiling problem. This was the beginning of the average case NP completeness theory, which was studied intensively in the latter years [Gur91, SY92, WB93, WB95, Wan95]. In contrast to the worst case, where meanwhile hundreds of NP complete problems exist, only a few average case NP complete problems are known. Many of them are bounded versions of undecidable problems, for example the randomized versions of the halting problem, the Post correspondence problem or ....

.... 2 Preliminaries We assume that the reader is familiar with the standard notions of computational complexity theory (see for example [BDG88, BDG90, Pap94] For an introduction into formal languages and parsing theory we refer to [HU79, AU73] An introduction to average case theory can be found in [Gur91, Wan, WB95]. All languages considered in this note are over the alphabet Sigma = f0; 1g. The length of a string x 2 Sigma is denoted by jxj. For a set A of strings, let A =n = A Sigma n and A n = S n k=0 A =k . We denote the cardinality of a set A by kAk. We use the pairing function ....

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Y. Gurevich. Average case complexity. Journal of Computer and System Sciences, 42(3):346--398, 1991.

....M , is polynomial on average. In [9, 10] Levin gave a robust and still natural definition of polynomial time on average. With his papers Levin initiated research in computational average case complexity. For an introduction to computational average case complexity the reader is referred to [4] or more recently to [19] Thus in average case complexity one considers the time space complexity of a problem under the assumption that the inputs are given under some (fixed) distribution on the instances. In [17] it was proposed to consider the set of decision problems that are in AP for ....

....for E) and are solvable in time polynomial on average, for every polynomialtime computable distribution. 2 Preliminaries In this paper we use the standard definitions and notation of computational complexity. For definitions and notation of average case complexity the reader is referred to [4]. An introduction to resource bounded measure can be found in [12] and in [14] Let Sigma = f0; 1g be fixed, Sigma denote the set of all finite strings (over Sigma) and Sigma n denote all strings of length n. For every set A Sigma let A n = A Sigma n and for a string z we use z ....

Y. Gurevich. Average case complexity. J. Comput. Syst. Sci., 42(3):346--398, 1991. A special issue on FOCS'87.

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Yuri Gurevich, "Average Case Complexity", ICALP'91, 18th International Colloquium on Automata, Languages and Programming, Madrid, Springer Lecture Notes in Computer Science 510, 1991, 615--628.

....The second notion is the notion of domination. Given two probability distributions and , say that dominates if is equal to or greater than modulo a polynomial factor. In other words, dominates if there exists a polynomial p such that Pr [fxg] Pr [fxg] Delta p(jxj) It can be checked [Le1,Gu1] that (X; is Ptime on average if there exists such that dominates and (X; is Ptime on average. There seems to be a consensus that natural probability distributions are dominated by Ptime computable ones. Thus, Challenger may draw instances with respect to a Ptime computable probability ....

....many attempts in order to force Solver to work hard. Roughly speaking, Challenger needs t = 2 i attempts to draw a string with i leading zeroes and to force Solver to work t 2 = 4 i steps. ffl A: Very good. You force me to disclose the official definition of polynomiality on average [Le1,Gu1]. The definition is slightly different from what you (most justifiably) assumed it to be. Let n be a function that assigns probabilities to strings of length n. The connection between this case and the case of one distribution over all strings is discussed at length in [Gu1] Further, let T be a ....

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Yuri Gurevich. "Average case complexity". To appear in a special issue of J. Computer and System Sciences with selected papers of FOCS'87. [Available to be emailed on request in the form of a latex file.]

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Yuri Gurevich, "Average Case Complexity", ICALP'91, 18th International Colloquium on Automata, Languages and Programming, Madrid, Springer Lecture Notes in Computer Science 510, 1991, 615--628.

....seems ad hoc. In the first part of this paper (Sections 2 4) we attempt to derive a simpler version of it from (more or less) first principles. Even though that original notion of reduction, deterministic in nature, was sufficient to establish the completeness of a number of natural problems [Le, Gu1, Gu2], it turned out to be too restrictive. Many randomized decision problems of interest are flat in the following technical sense: There exists 0 such that the probability of any instance of sufficiently large size n is bounded from above by 2 Gamman . However, no randomized decision ....

....technical sense: There exists 0 such that the probability of any instance of sufficiently large size n is bounded from above by 2 Gamman . However, no randomized decision problem with a flat domain is complete unless deterministic exponential time equals nondeterministic exponential time [Gu1]. To overcome this difficulty, Levin suggested more general randomizing reduction. Versions of randomizing reduction were defined and successfully used in [VL] and [BCGL] A simple version of randomizing reduction was defined in greater detail and proved transitive in [Gu1] That version was too ....

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Yuri Gurevich, "Average Case Complexity", J. Computer and System Sciences (a special issue on FOCS'87) to appear.

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Yuri Gurevich. Average case complexity. In Proceedings of the 18th International Colloquium on Automata, Languages and Programming, volume 510 of Lecture Notes in Computer Science, Springer, pages 615628, 1991.

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Y. Gurevich, Average Case Complexity, Internat. Symp. on Information Theory, IEEE, Proc. 1985.

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Y. Gurevich, Average Case Complexity, Internat. Symp. on Information Theory, IEEE, Proc. 1985.

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Y. Gurevich. Average case complexity. In Proceedings of the 18th Annual Colloquium on Automata, Languages and Programming, vol. 510 of Lecture Notes in Computer Science, Springer-Verlag, pages 615--628, 1991.

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