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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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Proceedings ArticleDOI
01 Sep 2015
TL;DR: This work considers the problem of interpolating a high-degree sparse polynomial, where the sparsity is in the number of monomial terms with non-zero coefficients, and proposes a probabilistic algorithm that requires only O(k) evaluations of a polynomials with complex coefficients, on the unit circle at specified points and has a complexity O( k log k), where k is theSparsity of thePolynomial.
Abstract: We consider the problem of interpolating a high-degree sparse polynomial, where the sparsity is in the number of monomial terms with non-zero coefficients. We propose a probabilistic algorithm that requires only O(k) evaluations of a polynomial with complex coefficients, on the unit circle at specified points and has a complexity O(k log k), where k is the sparsity of the polynomial. Thus the evaluation complexity as well as the computational complexity are independent of the maximum degree n in contrast to existing algorithms in the literature. We extend our algorithm to polynomials defined over the finite field using fast algorithms in the literature to compute discrete logs for certain field sizes.

2 citations

Posted Content
TL;DR: Using the sieve method of combinatorics, the k-error linear complexity distribution of the 2-periodic binary sequences is investigated based on Games-Chan algorithm to derive counting functions for the number and the complete counting functions on the 4- error linear complexity of the sequence.
Abstract: The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated based on Games-Chan algorithm. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity less than $2^n$ are characterized. Second, for $k=3,4$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$ are presented. Third, for $k=4,5$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity less than $2^n$ are derived. As a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the 3-error linear complexity are obtained, and the complete counting functions on the 4-error linear complexity of $2^n$-periodic binary sequences are obvious.

2 citations

Posted Content
TL;DR: A general notion of information-related complexity applicable to both natural and man-made systems is proposed and is shown to generalize the existing notion of statistical complexity when the system in question can be described by a discrete-time stochastic process.
Abstract: A general notion of information-related complexity applicable to both natural and man-made systems is proposed. The overall approach is to explicitly consider a rational agent performing a certain task with a quantifiable degree of success. The complexity is defined as the minimum (quasi-)quantity of information that's necessary to complete the task to the given extent -- measured by the corresponding loss. The complexity so defined is shown to generalize the existing notion of statistical complexity when the system in question can be described by a discrete-time stochastic process. The proposed definition also applies, in particular, to optimization and decision making problems under uncertainty in which case it gives the agent a useful measure of the problem's "susceptibility" to additional information and allows for an estimation of the potential value of the latter.

2 citations

Posted Content
TL;DR: It is argued that Kolmogorov complexity calculus can be more useful if it is refined to include an important practical case of simple binary strings, and shows that under such restrictions some error terms can disappear from the standard complexity calculus.
Abstract: Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the reference computer. We argue that this approach can be more useful if it is refined to include an important practical case of simple binary strings. Kolmogorov complexity calculus may be developed for this case if we restrict the class of available reference computers. The interesting problem is to define a class of computers which is restricted in a {\it natural} way modeling the real-life situation where only a limited class of computers is physically available to us. We give an example of what such a natural restriction might look like mathematically, and show that under such restrictions some error terms, even logarithmic in complexity, can disappear from the standard complexity calculus. Keywords: Kolmogorov complexity; Algorithmic information theory.

2 citations

Journal ArticleDOI
TL;DR: The results described concern methods for obtaining lower bounds, synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.
Abstract: This paper contains a review of the authors’ results in the theory of algorithm complexity. The results described concern methods for obtaining lower bounds (containing almost all exponential lower bounds on monotone complexity of monotone functions), synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.

2 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
20216
202010
20199
201810
201732