On the metric complexity of continuous-time systems

In this paper, metric complexities of certain classes of continuous-time systems are studied, using the time-domain sampling approach and the concepts of Kolmogorov, Gel'fand and sampling n-widths for certain classes of Sobolev space. A sampling theorem is obtained which extends Shannon's sampling theorem to systems with possibly non-band-limited spectra. The theorem demonstrates that continuous-time systems in certain Sobolev spaces can be approximately reconstructed causally from their sampled systems. The Kolmogorov, Gel'fand and sampling n-widths of various uncertainty sets in the Sobolev spaces are derived. The results show that the sampling approach is in fact asymptotically optimal, when the sampling interval is selected to minimize the loss of information in the sampling process, for the modelling of systems in such Sobolev spaces.     

On the complexity of computing determinants

We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in and bit operations; here denotes the largest entry in absolute value and the exponent adjustment by +o(1) captures additional factors for positive real constants C₁, C₂, C₃. The bit complexity results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n^3.2+o(1) and O(n^2.697263) ring additions, subtractions and multiplications.To B. David Saunders on the occasion of his 60th birthday     

The complexity of computing the permanent

It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations. Related counting problems are also considered. The reductions used are characterized by their nontrivial use of arithmetic.     

Best partitions of a structure in a given computing environment

For every analysis problem requiring partitioning, there is at least one best one which produces least cost of computing in a given computing environment. Such a partitioning may be identified easily, if one would have a table listing the required number of partitions and their sizes as a function of the available rapid access memory (RAM), and the knowledge of the prevailing unit costs of RAM and peripheral processing units (PPU) in the computing facility. The tables containing partitioning information as a function of available RAM may be generated by small time-shared software which require to scan only once the usual input data of the analysis problem. The generation of such tables costs practically nothing, yet, if the structural analysis software is designed to take advantage of the information provided by them, the analysis cost may be reduced drastically. This paper presents a small time-shared software, called RAMPART, written in BASIC language of HP2000F, to produce tables for partitioning information as a function of RAM request, assuming that the analysis will be carried out by the displacement method using Cholesky decomposition.     

Approximating and computing behavioural distances in probabilistic transition systems

In an earlier paper we presented a pseudometric on the states of a probabilistic transition system, yielding a quantitative notion of behavioural equivalence. The behavioural pseudometric was defined via the terminal coalgebra of a functor based on a metric on Borel probability measures. In the present paper we give a polynomial-time algorithm, based on linear programming, to calculate the distances between states up to a prescribed degree of accuracy.     

The complexity of computing maximal word functions

Maximal word functions occur in data retrieval applications and have connections with ranking problems, which in turn were first investigated in relation to data compression [21]. By the maximal word function of a languageL ^*, we mean the problem of finding, on inputx, the lexicographically largest word belonging toL that is smaller than or equal tox.In this paper we present a parallel algorithm for computing maximal word functions for languages recognized by one-way nondeterministic auxiliary pushdown automata (and hence for the class of context-free languages).This paper is a continuation of a stream of research focusing on the problem of identifying properties others than membership which are easily computable for certain classes of languages. For a survey, see [24].     

On the complexity of computing the diameter of a polytope

We show that it isD ^P -hard to determine the combinatorial diameter of a polytope specified by linear inequalities with integer data. Our result partially resolves a long-term open question.     

密码学中信息论及计算复杂性的研究

针对信息论和计算复杂性是密码学的两个重要理论基础,而信息的加密与破译又和信息论密切相关.研究了信息的传输和保密问题,并对保密系统进行了数学描述和分析.讨论了完善保密性、理论保密性与实际保密性,给出了算法复杂度的两个时间算法,探讨了纠错码中的几个NP问题及其密码学作用.通过介绍计算复杂性理论中的几个重要概念,给出了P,NP,CO-NP与PSPACE之间的关系.     

On the number of multilinear partitions and the computing capacity of multiple-valued multiple-threshold perceptrons

We introduce the concept of multilinear partition of a point set V/spl sub/R/sup n/ and the concept of multilinear separability of a function f:Vtwo head right arrowK={0,...,k-1}. Based on well-known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K(2). The (n,k,s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n,k,s)-perceptron, respectively, for V subset R(n) in general position and for V=K(2). Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K(2).     

The region approach for computing relative neighbourhood graphs in the Lp metric

The following geometrical proximity concepts are discussed: relative closeness and geographic closeness. Consider a setV={v ₁,v ₂, ...,v _n } of distinct points in atwo-dimensional space. The pointv _j is said to be arelative neighbour ofv _i ifd _p (v _i ,v _j )≤max{d _p (v _j ,v _k ),d _p (v _j ,v _k )} for allv _k ∈V, whered _p denotes the distance in theL _p metric, 1≤p≤∞. After dividing the space around the pointv _i into eight sectors (regions) of equal size, a closest point tov _i in some region is called anoctant (region, orgeographic) neighbour ofv _i. For anyL _p metric, a relative neighbour ofv _i is always an octant neighbour in some region atv _i. This gives a direct method for computing all relative neighbours, i.e. for establishing therelative neighbourhood graph ofV. For every pointv _i ofV, first search for the octant neighbours ofv _i in each region, and then for each octant neighbourv _j found check whether the pointv _j is also a relative neighbour ofv _i. In theL _p metric, 1<p<∞, the total number of octant neighbours is shown to be θ(n) for any set ofn points; hence, even a straightforward implementation of the above method runs in θn ²) time. In theL ₁ andL _∞ metrics the method can be refined to a θ(n logn+m) algorithm, wherem is the number of relative neighbours in the output,n-1≤m≤n(n-1). TheL ₁ (L _∞) algorithm is optimal within a constant factor.     

On the complexity of computing the profinite closure of a rational language

Profinite topology is used in the classification of rational languages. In particular, several important decidability problems, related to the Malcev product, reduce to the computation of the closure of a rational language in the profinite topology. It is known that, given a rational language by a deterministic automaton, computing a deterministic automaton accepting its profinite closure can be done with an exponential upper bound. This paper is dedicated the study of a lower bound for this problem: we prove that, in some cases, if the alphabet contains at least three letters, it requires an exponential time.     

On computing all north-east nearest neighbors in the L1 metric

Given a collection of n points in the plane, we exhibit an algorithm that computes the nearest neighbor in the north-east (first quadrant) of each point, in the L₁ metric. By applying a suitable transformation to the input points, the same procedure can be used to compute the L₁ nearest neighbor in any given octant of each point. This is the basis of an algorithm for computing the minimum spanning tree of the n points in the L₁ metric. All three algorithms run in O(n lg n) total time and O(n) space.     

A novel scalability metric about iso-area of performance for parallel computing

Scalability is an important performance metric of parallel computing, but the traditional scalability metrics only try to reflect the scalability for parallel computing from one side, which makes it difficult to fully measure its overall performance. This paper studies scalability metrics intensively and completely. From lots of performance parameters of parallel computing, a group of key ones is chosen and normalized. Further the area of Kiviat graph is used to characterize the overall performance of parallel computing. Thereby a novel scalability metric about iso-area of performance for parallel computing is proposed and the relationship between the new metric and the traditional ones is analyzed. Finally the novel metric is applied to address the scalability of the matrix multiplication Cannon’s algorithm under LogP model. The proposed metric is significant to improve parallel computing architecture and to tune parallel algorithm design.     

On the optimal choice of quality metric in image compression: a soft computing approach

Images take lot of computer space; in many practical situations, we cannot store all original images, we have to use compression. Moreover, in many such situations, compression ratio provided by even the best lossless compression is not sufficient, so we have to use lossy compression. In a lossy compression, the reconstructed image ? is, in general, different from the original image I. There exist many different lossy compression methods, and most of these methods have several tunable parameters. In different situations, different methods lead to different quality reconstruction, so it is important to select, in each situation, the best compression method. A natural idea is to select the compression method for which the average value of some metric d(I,?) is the smallest possible. The question is then: which quality metric should we choose? In this paper, we show that under certain reasonable symmetry conditions, L ^p metrics d(I,?)=∫|I(x)??(x)| ^p dx are the best, and that the optimal value of p can be selected depending on the expected relative size r of the informative part of the image.     

On the complexity of computing and learning with multiplicative neural networks.

In a great variety of neuron models, neural inputs are combined using the summing operation. We introduce the concept of multiplicative neural networks that contain units that multiply their inputs instead of summing them and thus allow inputs to interact nonlinearly. The class of multiplicative neural networks comprises such widely known and well-studied network types as higher-order networks and product unit networks. We investigate the complexity of computing and learning for multiplicative neural networks. In particular, we derive upper and lower bounds on the Vapnik-Chervonenkis (VC) dimension and the pseudo-dimension for various types of networks with multiplicative units. As the most general case, we consider feedforward networks consisting of product and sigmoidal units, showing that their pseudo-dimension is bounded from above by a polynomial with the same order of magnitude as the currently best-known bound for purely sigmoidal networks. Moreover, we show that this bound holds even when the unit type, product or sigmoidal, may be learned. Crucial for these results are calculations of solution set components bounds for new network classes. As to lower bounds, we construct product unit networks of fixed depth with super-linear VC dimension. For sigmoidal networks of higher order, we establish polynomial bounds that, in contrast to previous results, do not involve any restriction of the network order. We further consider various classes of higher-order units, also known as sigma-pi units, that are characterized by connectivity constraints. In terms of these, we derive some asymptotically tight bounds. Multiplication plays an important role in both neural modeling of biological behavior and computing and learning with artificial neural networks. We briefly survey research in biology and in applications where multiplication is considered an essential computational element. The results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks.