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Despite the benefit of the memory hierarchy, it is still essential, in order to reduce accesses to higher levels of memory, to have an efficient usage of registers and local memories (also called scratchpad memories) present in most embedded processors, graphical processors (GPUs) and network processors. During the compilation, from a source language to an executable code, there are two optimizations that are of utmost importance: the register allocation and the local memory allocation. In this thesis's report we are interested in decoupled approaches, solving separately the allocation and assignment problems, that helps to improve the quality of the register and local memory allocations. In the first part of this thesis we are interested in two aspects of the register allocation problem: the improvements of the just-in-time (JIT) register allocation and the spill minimization problem. We introduce the split register allocation which leverages the decoupled approach to improve register allocation in the context of JIT compilation. We experimentally validate the effectiveness of split register allocation and its portability with respect to register count variations, relying on annotations whose impact on the bytecode size is negligible. We introduce a new decoupled approach, called iterated-optimal allocation, which focus on the spill minimization problem. The iterated-optimal allocation algorithm achieves results close to optimal while offering pseudo-polynomial guarantees for SSA programs and fast allocations on general programs. In the second part of this thesis, we study how a decoupled local memory allocation can be proposed in light of recent progresses in register allocation. We first validate our intuition for decoupled approach to local memory allocation. Then, we study the local memory allocation in a more theoretical way setting the junction between local memory allocation for linearized programs and weighted interval graph coloring. We design and analyze a new variant of the ship-building problem called the submarine-building problem. We show that this problem is NP-complete on interval graphs, while it is solvable in linear time for proper interval graphs, equivalent to unit interval graphs. The submarine-building problem is the first problem that is known to be NP-complete on interval graphs, while it is solvable in linear time for unit interval graphs. In the third part of this thesis, we propose a heuristic-based solution, the clustering allocator, which decouples the local memory allocation problem and aims to minimize the allocation cost. The clustering allocator while devised for local memory allocation, it appears to be a very good solution to the register allocation problem. After many years of separation, this new algorithm seems to be a bridge to reconcile the local memory allocation and the register allocation problems.

The classical graph edge coloring problem deals in coloring the edges of a given graph with minimum number of colors such that no two adjacent edges in the graph, get the same color in the proposed coloring. In the following work, we look at the other end of the spectrum where in our goal is to maximize the number of colors used for coloring the edges of the graph under some vertex specific constraints. We deal with the MAXIMUM EDGE COLORING problem which is defined as the following –For an integer q ≥2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. The question is very well motivated by the problem of channel assignment in wireless networks. This problem is NP-hard for q ≥ 2, and has been well-studied from the point of view of approximation. This problem has not been studied in the parameterized context before. Hence as a next step, this thesis investigates the parameterized complexity of this problem where the standard parameter is the solution size. The main focus of the work is the special case of q=2 ,i.e. MAXIMUM EDGE 2-COLORING which is theoretically intricate and practically relevant in the wireless networks setting. We first show an exponential kernel for the MAXIMUM EDGE q-COLORING problem where q is a fixed constant and q ≥ 2.We do a more specific analysis for the kernel of the MAXIMUM EDGE 2-COLORING problem. The kernel obtained here is still exponential in size but is better than the kernel obtained for MAXIMUM EDGE q-COLORING problem in case of q=2. We then show a fixed parameter tractable algorithm for the MAXIMUM EDGE 2-COLORING problem with a running time of O*∗(kO(k)).We also show a fixed parameter tractable algorithm for the MAXIMUM EDGE q-COLORING problem with a running time of O∗(kO(qk) qO(k)). The fixed parameter tractability of the dual parametrization of the MAXIMUM EDGE 2-COLORING problem is established by arguing a linear vertex kernel for the problem. We also show that the MAXIMUM EDGE 2-COLORING problem remains hard on graphs where the maximum degree is a constant and also on graphs without cycles of length four. In both these cases, we obtain quadratic kernels. A closely related variant of the problem is the question of MAX EDGE{1,2-}COLORING. For this problem, the vertices in the input graph may have different qε,{1.2} values and the goal is to use at least k colors for the edge coloring of the graph such that every vertex sees at most q colors, where q is either one or two. We show that the MAX EDGE{1,2}-COLORING problem is W[1]-hard on graphs that have no cycles of length four.

Nous étudions dans cette thèse différents problèmes d'évitabilité des répétitions dans les mots infinis. Soulevée par Thue et motivée par ses travaux sur les mots sans carrés, la problématique s'est développée au cours du XXe siècle, et est aujourd'hui devenue un des grands domaines de recherche en combinatoire des mots. En 1972, Dejean proposa une importante conjecture, dont la validation étape par étape s'est terminée récemment (2009). La conjecture concerne le seuil des répétitions d'un alphabet, i.e., la borne inférieure des exposants évitables sur cet alphabet. La notion de seuil, comme frontière entre évitabilité et non-évitabilité d'un ensemble donné de mots, est le fil directeur de nos travaux. Nous nous intéressons d'abord à une généralisation du seuil des répétitions (nous donnons des encadrements de sa valeur). Cette notion permet d'ajouter, pour décrire l'ensemble des répétitions à éviter, au paramètre de l'exposant, celui de la longueur des répétitions. Puis, nous étudions des problèmes d'existence de mots dans lesquels, simultanément, certaines répétitions sont interdites et d'autres sont forcées. Nous répondons, pour l'alphabet ternaire, à la question : quels réels sont l'exposant critique d'un mot infini sur un alphabet fixé? Nous introduisons ensuite une notion de haute répétitivité, et établissons une description partielle des couples d'exposants paramètrant une double contrainte de haute répétitivité et d'évitabilité. Pour finir, nous utilisons des résultats et techniques issus de ces problématiques pour résoudre une question de coloration de graphes : nous introduisons un seuil des répétitions, calqué sur celui connu pour les mots, et donnons sa valeur pour deux classes de graphes, les arbres et les graphes de subdivisions. / In this thesis we study various problems on repetition avoidance in infinite words. Raised by Thue and motivated by his work on squarefree words, the topic developed during the 20th century, and has nowadays become a principal area of research in combinatorics on words. In 1972, Dejean proposed an important conjecture whose verification in steps was completed recently (2009). The conjecture concerns the repetition threshold for an alphabet, i.e., the infimum of the avoidable exponents for that alphabet. The notion of threshold as a borderline between avoidability and unavoidability for a given set of words is the guiding line of our work. First, we focus on a generalization of the repetition threshold. This concept allows us to include, in addition to the exponent, the length of the repetitions as a parameter in the description of the set of repetitions to avoid. We obtain various bounds in that respect. We then study existence problems for words in which simultaneously some repetitions are forbidden, and others are forced. For the ternary alphabet, we answer the question: what real numbers are the critical exponent of some infinite word over a given alphabet? Also, we introduce a notion of highly repetitive words and give a partial description of the pairs of exponents which parameterize the existence of words both highly repetitive and repetition-free. Finally, we use results and techniques stemming from those problems to solve a question on graph colouring: we introduce a repetition threshold adapted from the thresholds we know for words, and give its value for two classes of graphs, namely, trees and subdivision graphs.

Malgré la hiérarchie mémoire utilisée dans les ordinateurs modernes, il convient toujours d'optimiser l'utilisation des registres du processeur et des mémoires locales gérées de manières logicielles (mémoires locales) présentes dans beaucoup de systèmes embarqués, de processeurs graphiques (GPUs) et de multiprocesseurs. Lors de la compilation, d'un code source vers un langage machine, deux optimisations de la mémoire revêtent une importance capitale : l'allocation de registres et l'allocation de mémoires locales. Dans ce manuscrit de thèse nous nous intéressons à des approches découplées, qui traitent séparément les problèmes d'allocation et d'assignation, permettant d'améliorer les allocations de registres et de mémoires locales. Dans la première partie de la thèse, nous nous penchons sur le problème de l'allocation de registres. Tout d'abord, nous proposons dans le contexte des compilateurs-juste-à-temps, une allocation de registres fractionnées (split register allocation). Avec cette approche l'allocation de registres est effectuée en deux étapes: une faite durant la phase de compilation statique et l'autre pendant la phase de compilation dynamique. Ce qui permet de réduire le temps d'exécution des programmes avec un impact négligeable sur le temps de compilation. Ensuite Nous introduisons une allocation de registres incrémentale qui permet de résoudre d'une manière quasi-optimale le problème d'allocation. Cette méthode est pseudo-polynomiale alors que le problème d'allocation est NP-complet même à l'intérieur d'un « basic block ». Dans la deuxième partie de la thèse nous nous intéressons au problème de l'allocation de mémoires locales. Au vu des dernières avancées dans le domaine de l'allocation de registres, nous étudions dans quelle mesure le problème d'allocation pourrait être séparé de celui de l'assignation dans le contexte des mémoires locales. Dans un premier temps nous validons expérimentalement que les problèmes d'allocation et d'assignation peuvent être résolus séparément. Ensuite, nous procédons à une étude plus théorique d'une approche découplée de l'allocation de mémoires locales. Cela permet d'introduire de nouveaux résultats sur le « submarine-building problem », une variante du « ship-building problem », que nous avons défini. L'un de ces résultats met en évidence pour la première fois une différence de complexité (P vs. NP-complet) entre les graphes d'intervalles et les graphes d'intervalles unitaires. Dans la troisième partie de la thèse nous proposons une nouvelle heuristique, appelée « clustering allocator » fondée sur la construction de sous-graphes stables d'un graphe d'interférence, permettant de découpler aussi bien le problème d'allocation pour les registres que pour les mémoires locales. Cette nouvelle heuristique se veut le pont qui permettra de réconcilier les problèmes d'allocations de registres et de mémoires locales. / Despite the benefit of the memory hierarchy, it is still essential, in order to reduce accesses to higher levels of memory, to have an efficient usage of registers and local memories (also called scratchpad memories) present in most embedded processors, graphical processors (GPUs) and network processors. During the compilation, from a source language to an executable code, there are two optimizations that are of utmost importance: the register allocation and the local memory allocation. In this thesis's report we are interested in decoupled approaches, solving separately the allocation and assignment problems, that helps to improve the quality of the register and local memory allocations. In the first part of this thesis we are interested in two aspects of the register allocation problem: the improvements of the just-in-time (JIT) register allocation and the spill minimization problem. We introduce the split register allocation which leverages the decoupled approach to improve register allocation in the context of JIT compilation. We experimentally validate the effectiveness of split register allocation and its portability with respect to register count variations, relying on annotations whose impact on the bytecode size is negligible. We introduce a new decoupled approach, called iterated-optimal allocation, which focus on the spill minimization problem. The iterated-optimal allocation algorithm achieves results close to optimal while offering pseudo-polynomial guarantees for SSA programs and fast allocations on general programs. In the second part of this thesis, we study how a decoupled local memory allocation can be proposed in light of recent progresses in register allocation. We first validate our intuition for decoupled approach to local memory allocation. Then, we study the local memory allocation in a more theoretical way setting the junction between local memory allocation for linearized programs and weighted interval graph coloring. We design and analyze a new variant of the ship-building problem called the submarine-building problem. We show that this problem is NP-complete on interval graphs, while it is solvable in linear time for proper interval graphs, equivalent to unit interval graphs. The submarine-building problem is the first problem that is known to be NP-complete on interval graphs, while it is solvable in linear time for unit interval graphs. In the third part of this thesis, we propose a heuristic-based solution, the clustering allocator, which decouples the local memory allocation problem and aims to minimize the allocation cost. The clustering allocator while devised for local memory allocation, it appears to be a very good solution to the register allocation problem. After many years of separation, this new algorithm seems to be a bridge to reconcile the local memory allocation and the register allocation problems.