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.

Minimum description length principles for detection and classification of FTP exploits

Abstract: In this paper we build on the principle of "conservation of complexity", analyzed in Evans, S et al. (2001), to measure protocol redundancy and pattern content as a metric for information assurance. We first analyze complexity estimators as a tool for detecting FTP exploits. Results showing the utility of complexity-based information assurance to detect exploits over the file transfer protocol are presented and analyzed. We show that complexity metrics are able to distinguish between FTP exploits and normal sessions within some margin of error. We then derive a new heuristic for complexity estimation using minimum description length principles and develop a new complexity estimator and compression algorithm based on grammar inference using this heuristic. This estimator is used to provide meaningful models of unknown data sets. Finally we demonstrate the capability of our complexity-based approach to classify protocol behavior based on similarity distance metrics from known behaviors.

Complexity theory for operators in analysis

Abstract: We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions, which we call regular functions, as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using regular functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on regular functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions.Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. The latter two cannot be represented succinctly using existing approaches based on infinite sequences, so ours is the first treatment of them. As an interesting example, the task of numerical algorithms for solving the initial value problem of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results could only state how complex the solution can be. We now reformulate them to show that the operator itself is polynomial-space complete.

A tight relationship between generic oracles and type-2 complexity theory

Abstract: We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type-2 classes are distinct.

Counting and enumeration complexity with application to multicriteria scheduling

Abstract: In this paper we tackle an important point of combinatorial optimisation: that of complexity theory when dealing with the counting or enumeration of optimal solutions. Complexity theory has been initially designed for decision problems and evolved over the years, for instance, to tackle particular features in optimisation problems. It has also evolved, more or less recently, towards the complexity of counting and enumeration problems and several complexity classes, which we review in this paper, have emerged in the literature. This kind of problems makes sense, notably, in the case of multicriteria optimisation where the aim is often to enumerate the set of the so-called Pareto optima. In the second part of this paper we review the complexity of multicriteria scheduling problems in the light of the previous complexity results.

The automatic complexity analysis of divide-and-conquer algorithms

Abstract: Current tools performing automatic complexity analysis are capable to deal with function definitions based on structural induction. Divide-and-conquer algorithms with "intelligent" divide function (like quicksort) are not based on structural induction, but on noetherian induction. This paper presents a method of automatic complexity analysis to deal with such kinds of functions.