To further facilitate mitochondrial DNA (mtDNA) sequence analysis for human identity testing, a better understanding of its mutation rate is needed. Prior to the middle 1990's the mutation rate applied to a forensic or evolutionary analysis was determined by phylogenetic means, This method involved calculating genetic distances as determined by amino acid or DNA sequence variability within or between species. The mutation rate as determined by this method ranged from 0.025-0.26 nucleotide substitutions/ site/ myr (million years). With the recent advent of mtDNA analysis as a tool in human identity testing an increased number of observations have recently come to light calling into question the mutation rate derived from the phylogenetic method. The mutation rate as observed from forensic analysis appears to be much higher than that calculated phylogenetically. This is an area that needs to be resolved in human identity testing. Mutations that occur within a maternal lineage can lead to a possible false exclusion of an individual as belonging to that lineage. A greater understanding of the actual rate of mutation within a given maternal lineage can assist in determining criteria for including or excluding individuals as belonging to that lineage. The method used to assess the mutation rate in this study was to compare mtDNA sequences derived from the HVI and HVII regions of the D-loop from several different maternal lineages. The sequence information was derived from five unrelated families consisting of thirty-five individuals. One intergenerational mutational event was found. This derives to approximately 1.9 nucleotide substitutions/ site/ myr. This mutation rate was very consistent with several other similar studies. This increased mutation rate needs to be considered by forensic testing laboratories performing mtDNA sequence analysis prior to formulating any conclusive results.

Randomness has proved to be a powerful tool in all of computation. It is pervasive in areas such as networking, machine learning, computer graphics, optimization, computational number theory and is "necessary" for cryptography. Though randomized algorithms and protocols assume access to "truly" random bits, in practice, they rely on the output of "imperfect" sources of randomness such as pseudo-random number generators or physical sources. Hence, from a theoretical standpoint, it becomes important to view randomness as a resource and to study the following fundamental questions pertaining to it: Extraction: How do we generate "high quality" random bits from "imperfect" sources? Randomization: How do we use randomness to obtain efficient algorithms? Derandomization: How (and when) can we "remove" our dependence on random bits? In this thesis, we consider important problems in these three prominent and diverse areas pertaining to randomness. In randomness extraction, we present extractors for "oblivious bit fixing sources". In (a non-traditional use of) randomization, we have obtained results in machine learning (learning juntas) and proved hardness of lattice problems. While in derandomization, we present a deterministic algorithm for a fundamental problem called "identity testing". In this thesis we also initiate a complexity theoretic study of Hilbert's 17th problem. Here identity testing is used in an interesting manner. A common theme in this work has been the use of tools from areas such as number theory in a variety of ways, and often the techniques themselves are quite interesting.

Cette thèse explore deux territoires distincts de l’informatique fondamentale : la complexité et la compression. Plus précisément, dans une première partie, nous étudions la puissance des circuits arithmétiques non commutatifs, qui calculent des polynômes non commutatifs en plusieurs indéterminées. Pour cela, nous introduisons plusieurs modèles de calcul, restreints dans leur manière de calculer les monômes. Ces modèles en généralisent d’autres, plus anciens et largement étudiés, comme les programmes à branchements. Les résultats sont de trois sortes. Premièrement, nous donnons des bornes inférieures sur le nombre d’opérations arithmétiques nécessaires au calcul de certains polynômes tels que le déterminant ou encore le permanent. Deuxièmement, nous concevons des algorithmes déterministes fonctionnant en temps polynomial pour résoudre le problème du test d’identité polynomiale. Enfin, nous construisons un pont entre la théorie des automates et les circuits arithmétiques non commutatifs, ce qui nous permet de dériver de nouvelles bornes inférieures en utilisant une mesure reposant sur le rang de la matrice dite de Hankel, provenant de la théorie des automates. Une deuxième partie concerne l’analyse de l’algorithme de compression sans perte Lempel-Ziv. Pourtant très utilisé, sa stabilité est encore mal établie. Vers la fin des années 90s, Jack Lutz popularise la question suivante, connue sous le nom de « one-bit catastrophe » : « étant donné un mot compressible, est-il possible de le rendre incompressible en ne changeant qu’un seul bit ? ». Nous montrons qu’une telle catastrophe est en effet possible. Plus précisément, en donnant des bornes optimales sur la variation de la taille de la compression, nous montrons qu’un mot « très compressible » restera toujours compressible après modification d’un bit, mais que certains mots « peu compressibles » deviennent en effet incompressibles. / This thesis explores two territories of computer science: complexity and compression. More precisely, in a first part, we investigate the power of non-commutative arithmetic circuits, which compute multivariate non-commutative polynomials. For that, we introduce various models of computation that are restricted in the way they are allowed to compute monomials. These models generalize previous ones that have been widely studied, such as algebraic branching programs. The results are of three different types. First, we give strong lower bounds on the number of arithmetic operations needed to compute some polynomials such as the determinant or the permanent. Second, we design some deterministic polynomial-time algorithm to solve the white-box polynomial identity problem. Third, we exhibit a link between automata theory and non-commutative arithmetic circuits that allows us to derive some old and new tight lower bounds for some classes of non-commutative circuits, using a measure based on the rank of a so-called Hankel matrix. A second part is concerned with the analysis of the data compression algorithm called Lempel-Ziv. Although this algorithm is widely used in practice, we know little about its stability. Our main result is to show that an infinite word compressible by LZ’78 can become incompressible by adding a single bit in front of it, thus closing a question proposed by Jack Lutz in the late 90s under the name “one-bit catastrophe”. We also give tight bounds on the maximal possible variation between the compression ratio of a finite word and its perturbation—when one bit is added in front of it.

La complexité algorithmique est l'étude des ressources nécessaires — le temps, la mémoire, … — pour résoudre un problème de manière algorithmique. Dans ce cadre, la théorie de la complexité algébrique est l'étude de la complexité algorithmique de problèmes de nature algébrique, concernant des polynômes.Dans cette thèse, nous étudions différents aspects de la complexité algébrique. D'une part, nous nous intéressons à l'expressivité des déterminants de matrices comme représentations des polynômes dans le modèle de complexité de Valiant. Nous montrons que les matrices symétriques ont la même expressivité que les matrices quelconques dès que la caractéristique du corps est différente de deux, mais que ce n'est plus le cas en caractéristique deux. Nous construisons également la représentation la plus compacte connue du permanent par un déterminant. D'autre part, nous étudions la complexité algorithmique de problèmes algébriques. Nous montrons que la détection de racines dans un système de n polynômes homogènes à n variables est NP-difficile. En lien avec la question « VP = VNP ? », version algébrique de « P = NP ? », nous obtenons une borne inférieure pour le calcul du permanent d'une matrice par un circuit arithmétique, et nous exhibons des liens unissant ce problème et celui du test d'identité polynomiale. Enfin nous fournissons des algorithmes efficaces pour la factorisation des polynômes lacunaires à deux variables. / Computational complexity is the study of the resources — time, memory, …— needed to algorithmically solve a problem. Within these settings, algebraic complexity theory is the study of the computational complexity of problems of algebraic nature, concerning polynomials. In this thesis, we study several aspects of algebraic complexity. On the one hand, we are interested in the expressiveness of the determinants of matrices as representations of polynomials in Valiant's model of complexity. We show that symmetric matrices have the same expressiveness as the ordinary matrices as soon as the characteristic of the underlying field in different from two, but that this is not the case anymore in characteristic two. We also build the smallest known representation of the permanent by a determinant.On the other hand, we study the computational complexity of algebraic problems. We show that the detection of roots in a system of n homogeneous polynomials in n variables in NP-hard. In line with the “VP = VNP ?”question, which is the algebraic version of “P = NP?” we obtain a lower bound for the computation of the permanent of a matrix by an arithmetic circuit, and we point out the links between this problem and the polynomial identity testing problem. Finally, we give efficient algorithms for the factorization of lacunary bivariate polynomials.