The established methodologies for studying computational complexity can be applied to the new problems posed by very large-scale integrated (VLSI) circuits. This thesis develops a ''VLSI model of computation'' and derives upper and lower bounds on the silicon area and time required to solve the problems of sorting and discrete Fourier transformation. In particular, the area A and time T taken by any VLSI chip using any algorithm to perform an N-point Fourier transform must satisfy AT2 ≥ c N2 log2 N, for some fixed c > 0. A more general result for both sorting and Fourier transformation is that AT2x = Ω(N1 + x log2x N) for any x in the range 0 < x < 1. Also, the energy dissipated by a VLSI chip during the solution of either of these problems is at least Ω(N3/2 log N). The tightness of these bounds is demonstrated by the existence of nearly optimal circuits for both sorting and Fourier transformation. The circuits based on the shuffle-exchange interconnection pattern are fast but large: T = O(log2 N) for Fourier transformation, T = O(log3 N) for sorting; both have area A of at most O(N2 / log1/2 N). The circuits based on the mesh interconnection pattern are slow but small: T = O(N1/2 loglog N), A = O(N log2 N).

Les modèles géométriques de calcul permettent d'effectuer des calculs à l'aide de primitives géométriques. Parmi eux, le modèle des machines à signaux se distingue par sa simplicité, ainsi que par sa puissance à réaliser efficacement de nombreux calculs. Nous nous proposons ici d'illustrer et de démontrer cette aptitude, en particulier dans le cas de processus massivement parallèles. Nous montrons d'abord à travers l'étude de fractales que les machines à signaux sont capables d'une utilisation massive et parallèle de l'espace. Une méthode de programmation géométrique modulaire est ensuite proposée pour construire des machines à partir de composants géométriques de base -- les modules -- munis de certaines fonctionnalités. Cette méthode est particulièrement adaptée pour la conception de calculs géométriques parallèles. Enfin, l'application de cette méthode et l'utilisation de certaines des structures fractales résultent en une résolution géométrique de problèmes difficiles comme les problèmes de satisfaisabilité booléenne SAT et Q-SAT. Ceux-ci, ainsi que plusieurs de leurs variantes, sont résolus par machines à signaux avec une complexité en temps intrinsèque au modèle, appelée profondeur de collisions, qui est polynomiale, illustrant ainsi l'efficacité et le pouvoir de calcul parallèle des machines à signaux.

An Euler diagram C = {c_1, c_2,..., c_n} is a collection of n simple closed curves (i.e., Jordan curves) that partition the plane into connected subsets, called regions, each of which is enclosed by a unique combination of curves. Typically, Euler diagrams are used to visualize the distribution of discrete characteristics across a sample population; in this case, each curve represents a characteristic and each region represents the sub-population possessing exactly the combination of containing curves' properties. Venn diagrams are a subclass of Euler diagrams in which there are 2^n regions representing all possible combinations of curves (e.g., two partially overlapping circles). In this dissertation, we study the Euler Diagram Generation Problem (EDGP), which involves constructing an Euler diagram with a prescribed set of regions. We describe a graph-theoretic model of an Euler diagram's structure and use this model to develop necessary-and-sufficient existence conditions. We also use the graph-theoretic model to prove that the EDGP is NP-complete. In addition, we study the related Area-Proportional Euler Diagram Generation Problem (w-EDGP), which involves constructing an Euler diagram with a prescribed set of regions, each of which has a prescribed area. We develop algorithms for constructing area-proportional Euler diagrams composed of up to three circles and rectangles, as well as diagrams with an unbounded number of curves and a region of common intersection. Finally, we present implementations of our algorithms that allow the dynamic manipulation and real-time construction of area-proportional Euler diagrams.

We present several new results on the combinatorial properties of a locally restricted version of monomino-domino coverings of rectilinear regions. These are monomino-domino tatami coverings, and the restriction is that no four tiles may meet at any point. The global structure that the tatami restriction induces has numerous implications, and provides a powerful tool for solving enumeration problems on tatami coverings. Among these we address the enumeration of coverings of rectangles, with various parameters, and we develop algorithms for exhaustive generation of coverings, in constant amortised time per covering. We also con- sider computational complexity on two fronts; firstly, the structure shows that the space required to store a covering of the rectangle is linear in its longest dimension, and secondly, it is NP-complete to decide whether an arbitrary polyomino can be tatami-covered only with dominoes. / Graduate / 0984 / 0405 / alejandro.erickson@gmail.com

This thesis examines graph polynomials and particularly their complexity. We give short proofs of two results from Gessel and Sagan (1996) which present new evaluations of the Tutte polynomial concerning orientations. A theorem of Massey et al (1997) gives an expression concerning the average size of a forest in a graph. We generalise this result to any simplicial complex. We answer a question posed by Kleinschmidt and Onn (1995) by showing that the language of partitionable simplicial complexes is in NP. We prove the following result concerning the complexity of the Tutte polynomial: Theorem 1. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers x and y, and evaluate the Tutte polynomial, T(G;x,y). The rank generating function S of a graphic 2-polymatroid was introduced by Oxley and Whittle (1993). It has many similarities to the Tutte polynomial and we prove the following results. Theorem 2. Evaluating S at a fixed point (u,v) is #P-hard unless uv=1 when there is a polynomial time algorithm. Theorem 3. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers u and v, and evaluate S(G;u,v). We consider a class of graphs $S$, which are those graphs which are obtainable from a graph with no edges using the unsigned version of Reidemeister moves. We examine the relationship between this class and other similarly defined classes such as the delta-wye graphs. There remain many open questions such as whether S contains every graph. However we have an invariant of the moves, based on the Tutte polynomial, which allows us to determine from which graph with no edges, if any, a particular graph can be obtained. Finally we consider a new polynomial on weighted graphs which is motivated by the study of weight systems on chord diagrams. We give three states model and a recipe theorem. An unweighted version of this polynomial is shown to contain as specialisations, a wide range of graph invariants, such as the Tutte polynomial, the polymatroid polynomial of Oxley and Whittle (1993) and the symmetric function generalisation of the chromatic polynomial introduced by Stanley (1995). We close with a discussion of complexity issues proving hardness results for very restricted classes of graphs.

The first part of this research program concerns the development of customized and easily implementable Probably Approximately Correct (PAC)-learning algorithms for episodic tasks over acyclic state spaces. The defining characteristic of our algorithms is that they take explicitly into consideration the acyclic structure of the underlying state space and the episodic nature of the considered learning task. The first of the above two attributes enables a very straightforward and efficient resolution of the ``exploration vs exploitation' dilemma, while the second provides a natural regenerating mechanism that is instrumental in the dynamics of our algorithms. Some additional characteristics that distinguish our algorithms from those developed in the past literature are (i) their direct nature, that eliminates the need of a complete specification of the underlying MDP model and reduces their execution to a very simple computation, and (ii) the unique emphasis that they place in the efficient implementation of the sampling process that is defined by their PAC property. More specifically, the aforementioned PAC-learning algorithms complete their learning task by implementing a systematic episodic sampling schedule on the underlying acyclic state space. This sampling schedule combined with the stochastic nature of the transitions taking place, define the need for efficient routing policies that will help the algorithms complete their exploration program while minimizing, in expectation, the number of executed episodes. The design of an optimal policy that will satisfy a specified pattern of arc visitation requirements in an acyclic stochastic graph, while minimizing the expected number of required episodes, is a challenging problem, even under the assumption that all the branching probabilities involved are known a priori. Hence, the sampling process that takes place in the proposed PAC-learning algorithms gives rise to a novel, very interesting stochastic control/scheduling problem, that is characterized as the problem of the Optimal Node Visitation (ONV) in acyclic stochastic digraphs. The second part of the work presented herein seeks the systematic modelling and analysis of the ONV problem. The last part of this research program explores the computational merits obtained by heuristical implementations that result from the integration of the ONV problem developments into the PAC-algorithms developed in the first part of this work. We study, through numerical experimentation, the relative performance of these resulting heuristical implementations in comparison to (i) the initial version of the PAC-learning algorithms, presented in the first part of the research program, and (ii) standard Q-learning algorithm variations provided in the RL literature. The work presented in this last part reinforces and confirms the driving assumption of this research, i.e., that one can design customized RL algorithms of enhanced performance if the underlying problem structure is taken into account.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.

Most signal processing methods were developed for continuous signals. Digital devices, such as the computer, process only discrete signals. This dissertation proposes new techniques to accurately define and efficiently implement an important signal processing method---the time--frequency distribution (TFD)---using discrete signals. The TFD represents a signal in the joint time--frequency domain. Because these distributions are a function of both time and frequency they, unlike traditional signal processing methods, can display frequency content that changes over time. TFDs have been used successfully in many signal processing applications as almost all real-world signals have time-varying frequency content. Although TFDs are well defined for continuous signals, defining and computing a TFD for discrete signals is problematic. This work overcomes these problems by making contributions to the definition, computation, and application of discrete TFDs. The first contribution is a new discrete definition of TFDs. A discrete TFD (DTFD) should be free from the sampling-related distortion known as aliasing and satisfy all the important mathematical properties that the continuous TFD satisfies. Many different DTFD definitions exist but none come close to attaining this ideal. I propose three new components which make up the DTFD: 1) a new discrete Wigner--Ville distribution (DWVD) definition which satisfies all properties, 2) a new discrete analytic signal which minimises aliasing in the DWVD, and 3) a new method to define and convolve the discrete kernel with the DWVD to produce the DTFD. The result: a DTFD definition that, relative to the existing definitions, better approximates the ideal DTFD. The second contribution is two sets of computationally efficient algorithms to compute the proposed DTFD. The first set of algorithms computes the DTFD exactly; the second set requires less memory than the first set by computing time- and, or frequency-decimated versions of the DTFD. Both sets of algorithms reduce the computational load by exploiting symmetries in the DTFD and by constructing kernel-specific algorithms for four different kernel types. The third, and final, contribution is a biomedical application for the proposed DTFD and algorithms. This application is to accurately detect seizure events in newborn electroencephalogram (EEG) signals. Existing detection methods do not perform well enough for use in a clinical setting. I propose a new method which is more robust than existing methods and show how using the proposed DTFD, comparative to an existing DTFD, improves detection performance for this method. In summary, this dissertation makes practical contributions to the area of time--frequency signal processing by proposing an improved DTFD definition, efficient DTFD algorithms, and an improved newborn EEG seizure detection method using DTFDs.