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Average-case complexity
About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.
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4 citations
Journal Article•
TL;DR: This work offers an alternative measure of complexity for approximations-optimization tasks by defining a hierarchy on the set of inputs to a learning task, so that natural inputs occupy only bounded levels of this hierarchy and that there are algorithms that handle in polynomial time each such bounded level.
Abstract: We address a fundamental problem of complexity theory -the inadequacy of worst-case complexity for the task of evaluating the computational resources required for real life problems. While being the best known measure and enjoying the support of a rich and elegant theory, worst-case complexity seems gives rise to over-pessimistic complexity values. Many standard task, that are being carried out routinely in machine learning applications, are NP-hard, that is, infeasible from the worst-case-complexity perspective. In this work we offer an alternative measure of complexity for approximations-optimization tasks. Our approach is to define a hierarchy on the set of inputs to a learning task, so that natural ('real data') inputs occupy only bounded levels of this hierarchy and that there are algorithms that handle in polynomial time each such bounded level.
4 citations
Posted Content•
TL;DR: A more general definition of generic-case complexity is proposed, based on using a random process for generating inputs of an algorithm and using the time needed to generate an input as a way of measuring the size of that input.
Abstract: We propose a more general definition of generic-case complexity, based on using a random process for generating inputs of an algorithm and using the time needed to generate an input as a way of measuring the size of that input.
4 citations
11 Jul 1990
TL;DR: It is shown that Deterministic Bounded Halting is complete for 〈P, L-samplable〉 and that Graph Reachability is complete with respect to average case complexity classes.
Abstract: The average case complexity classes 〈P, L-samplable〉 and 〈NL, L-samplable〉 are defined and we show that Deterministic Bounded Halting is complete for 〈P, L-samplable〉 and that Graph Reachability is complete for 〈NL, L-samplable〉.
4 citations
TL;DR: For an extensive range of infinite words, and the associated symbolic dynamical systems, the minimal and maximal complexity functions the authors get by replacing finite words by finite patterns, or words with holes are computed.
Abstract: For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes. Given a language L on a finite alphabet A, the complexity function p L (n) counts for every n the number of factors of length n of L; this is a very useful notion, both inside word combinatorics and for the study of symbolic dynamical systems, see for example the survey [7]; of particular interest are the infinite words which are determined by the complexity of their language, those words for which p L (n) ≤ n for at least one n are ultimately periodic [15], while the Sturmian words, of complexity n + 1 for all n, are natural codings of rotations, see [6, 16], or Chapter 6 of [17], and Section 4 below. Note that the complexity is exponential when the language has positive topological entropy, and has not been widely used for that range of languages. To study further the combinatorial properties of infinite words, the notion of maximal pattern complexity, denoted by p *
4 citations