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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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Satyanarayana V. Lokam1
24 Jul 2009
TL;DR: This work surveys several techniques for proving lower bounds in Boolean, algebraic, and communication complexity based on certain linear algebraic approaches to study robustness measures of matrix rank that capture the complexity in a given model.
Abstract: We survey several techniques for proving lower bounds in Boolean, algebraic, and communication complexity based on certain linear algebraic approaches. The common theme among these approaches is to study robustness measures of matrix rank that capture the complexity in a given model. Suitably strong lower bounds on such robustness functions of explicit matrices lead to important consequences in the corresponding circuit or communication models. Many of the linear algebraic problems arising from these approaches are independently interesting mathematical challenges.

126 citations

Journal ArticleDOI
TL;DR: A simple variation of the (restricted) conditional complexity measure investigated by Martin-Lof is noted because of interesting characteristics not shared by the measures proposed by Kolmogorov.
Abstract: Kolmogorov in 1965 proposed two related measures of information content (alternately, measures of complexity) based on the size of a program which when processed by a suitable algorithm (machine) yields the desired object. The main emphasis was placed on a conditional complexity measure. In this paper a simple variation of the (restricted) conditional complexity measure investigated by Martin-Lof is noted because of interesting characteristics not shared by the measures proposed by Kolmogorov. The characteristics suggest situations in which this variant is the most desirable measure to employ. The interpretation of the measure offers some desirable general qualities; also the measure is relatively advantageous when working with entities of low complexity and maintains the important properties of the Kolmogorov conditional complexity measure when concerned with high complexity.

125 citations

DissertationDOI
01 Jan 2003
TL;DR: The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations.
Abstract: We present an algorithm for computing the closest point, transform to an explicitly described manifold on a rectilinear grid in low dimensional spaces. The closest point transform finds the closest point on a manifold and the Euclidean distance to a manifold for the points in a grid. We consider manifolds composed of simple geometric shapes, such as, a set of points, piecewise linear curves or triangle meshes. The algorithm solves the eikonal equation |∇u| = 1 with the method of characteristics. For many problems, the computational complexity of the algorithm is linear in both the number of grid points and the complexity of the manifold. Many query problems can be aided by using orthogonal range queries (ORQ). There are several standard data structures for performing ORQ's in 3-D, including kd-trees, octrees, and cell arrays. We develop additional data structures based on cell arrays. We study the characteristics of each data structure and compare their performance. We present a new algorithm for solving the single-source, non-negative weight, shortest-paths problem. Dijkstra's algorithm solves this problem with computational complexity O ((E + V)log V) where E is the number of edges and V is the number of vertices. The new algorithm, called Marching with a Correctness Criterion (MCC), has computational complexity O (E + RV), where R is the ratio of the largest to smallest edge weight. Sethian's Fast Marching Method (FMM) may be used to solve static Hamilton-Jacobi equations. It has computational complexity O (N log N), where N is the number of grid points. The FMM has been regarded as an optimal algorithm because it is closely related to Dijkstra's algorithm. The new shortest-paths algorithm discussed above can be used to develop an ordered, upwind, finite difference algorithm for solving static Hamilton-Jacobi equations. This algorithm requires difference schemes that difference not only in coordinate directions, but in diagonal directions as well. It has computational complexity O(RN), where R is the ratio of the largest to smallest propagation speed and N is the number of grid points.

123 citations

Proceedings ArticleDOI
09 Jun 2003
TL;DR: The following new lower bounds in two concrete complexity models are shown: in the two-party communication complexity model, it is shown that the tribes function on n inputs has two-sided error randomized complexity, while its nondeterminstic complexity and co-nondeterministic complexity are both Θ(√n).
Abstract: We show the following new lower bounds in two concrete complexity models:(1) In the two-party communication complexity model, we show that the tribes function on n inputs[6] has two-sided error randomized complexity Ω(n), while its nondeterminstic complexity and co-nondeterministic complexity are both Θ(√n). This separation between randomized and nondeterministic complexity is the best possible and it settles an open problem in Kushilevitz and Nisan[17], which was also posed by Beame and Lawry[5].(2) In the Boolean decision tree model, we show that the recursive majority-of-three function on 3h inputs has randomized complexity Ω((7/3)h). The deterministic complexity of this function is Θ(3h), and the nondeterministic complexity is Θ(2h). Our lower bound on the randomized complexity is a substantial improvement over any lower bound for this problem that can be obtained via the techniques of Saks and Wigderson [23], Heiman and Wigderson[14], and Heiman, Newman, and Wigderson[13]. Recursive majority is an important function for which a class of natural algorithms known as directional algorithms does not achieve the best randomized decision tree upper bound.These lower bounds are obtained using generalizations of information complexity, which quantifies the minimum amount of information that will have to be revealed about the inputs by every correct algorithm in a given model of computation.

122 citations

Journal ArticleDOI
TL;DR: It is argued that for many problems in this setting, parameterized computational complexity rather than NP-completeness is the appropriate tool for studying apparent intractability and a new result is described for the Longest Common Subsequence problem.
Abstract: Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NP-completeness is the appropriate tool for studying apparent intractability. At issue in the theory of parameterized complexity is whether a problem can be solved in time O(n alpha) for each fixed parameter value, where alpha is a constant independent of the parameter. In addition to surveying this complexity framework, we describe a new result for the Longest Common Subsequence problem. In particular, we show that the problem is hard for W[t] for all t when parameterized by the number of strings and the size of the alphabet. Lower bounds on the complexity of this basic combinatorial problem imply lower bounds on more general sequence alignment and consensus discovery problems. We also describe a number of open problems pertaining to the parameterized complexity of problems in computational biology where small parameter values are important.

120 citations


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