Génération aléatoire uniforme de mots de langages rationnels

RésuméNous donnons deux algorithmes de génération aléatoire et uniforme de mots, qui s'appliquent à des classes particulières de langages rationnels. Leur efficacité est mesurée en termes de complexité logarithmique, en fonction de la longueur n des mots engendrés. Le premier algorithme est dédié aux langages dont les séries génératrices possèdent un unique pôle, éventuellement multiple; sa complexité en temps est de l'ordre de n log n, et l'espace mémoire occupé est en log n. Le second algorithme est réservé aux langages dont les séries génératrices possèdent la propriété suivante: il existe un unique pôle de plus petit module, et ce pôle est simple. Après un pré-traitement en temps polynomial en n, le tirage aléatoire de tout mot s'effectue en temps moyen et espace linéaires.AbstractThe problem of generating uniformly at random words of a given language has been the subject of extensive study in the last few years. An important part of that work is devoted to the generation of words of context-free languages (see, e.g., [6, 8, 9, 12]). For a given integer n > 0, the words of length n > 0 of any unambiguous context-free language can be generated uniformly at random by using algorithms derived from the general method which was introduced by Wilf [14, 15] and systematized by Flajolet et al. [7]. Clearly, this can be applied to the set of rational languages, which constitute an important special case of context-free languages.Most authors use the uniform measure of complexity (see [1]) in order to compute the complexity of the algorithms of generation. This measure is based on the following hypotheses: any simple arithmetic operation (addition, multiplication) has time cost 0(1), and a constant amount of memory space is taken by any number. Thus, we know that words of any rational language can be generated by using an algorithm which, with respect to the uniform measure of complexity, runs in linear time (in terms of the length of the words) and constant space [9]. This measure is realistic only if there is a reasonable bound on the numbers involved in the operations. However, the classical random generation algorithms involve operations on numbers which grow exponentially in terms of the length of the words to be generated. Moreover, the programs which make use of these algorithms are generally used to generate very large words, for example for the purpose of studying the asymptotic behavior of some parameters. Therefore, the uniform measure does not reflect the real behavior of such programs. It turns out that the logarithmic measure of complexity is much more realistic: one assumes that the space taken by a number k is O(log k), and that any simple arithmetic operation can be done in time O(log k). It is with respect to this measure that we will evaluate the performance of algorithms in this paper.Our goal is to design efficient algorithms (in terms of logarithmic complexity) to generate uniformly at random words from certain classes of rational languages. We consider rational languages defined by their minimal finite deterministic automata. When computing complexity, neither the size of the automaton nor the cardinality of the alphabet are taken in account.In Section 2 we present some background on rational languages and their generating series. We describe briefly the classical method for generating words of such languages and we study its logarithmic complexity. We show that it is at best quadratic for most languages. This is due mainly to computations on numbers which grow exponentially with the length of the words to be generated. In order to improve significantly the efficiency of the algorithms, we must avoid handling of large numbers, or at least decrease substantially the frequency of computations on such numbers. Another alternative, briefly discussed in [7] and [12], is to compute with floating point numbers instead of integers. In this case, the logarithmic complexity is time-linear. However, using floating point numbers leads inevitably to approximations which prevent the exact uniformity of the generation.In Sections 3 and 4 we show that, in some cases, we can avoid computations on large numbers entirely or almost entirely, while keeping the exact uniformity of the generation. We determine two classes of rational languages for which this is the case.Section 3 concerns languages whose associated generating series have a unique singularity. We present a simple version of the classical algorithm, which totally avoids handling of large numbers. The logarithmic complexity of the method is O(n log n) in time and O(log n) in memory space.Section 4 focuses on languages whose associated generating series have the following property: there exists a unique singularity of minimum modulus, and this singularity is simple. For such languages we give a probabilistic version of the classical algorithm which generates words randomly while avoiding most computations on large numbers. This method needs a preprocessing stage, which can be done in polynomial time and linear space in terms of the length n of the words. Following preprocessing, any word of length n can be generated in average linear time and space

Quantum Algorithms for Matrix Problems and Machine Learning

This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.Science and Engineering Research Board (Department of Science and Technology), Government of Indi