In this work, we present off-chip communications architectures for line cards to increase the throughput of the currently used memory system. In recent years there is a significant increase in memory bandwidth demand on line cards as a result of higher line rates, an increase in deep packet inspection operations and an unstoppable expansion in lookup tables. As line-rate data and NPU processing power increase, memory access time becomes the main system bottleneck during data store/retrieve operations. The growing demand for memory bandwidth contrasts the notion of indirect interconnect methodologies. Moreover, solutions to the memory bandwidth bottleneck are limited by physical constraints such as area and NPU I/O pins. Therefore, indirect interconnects are replaced with direct, packet-based networks such as mesh, torus or k-ary n-cubes. We investigate multiple k-ary n-cube based interconnects and propose two variations of 2-ary 3-cube interconnect called the 3D-bus and 3D-mesh. All of the k-ary n-cube interconnects include multiple, highly efficient techniques to route, switch, and control packet flows in order to minimize congestion spots and packet loss. We explore the tradeoffs between implementation constraints and performance. We also developed an event-driven, interconnect simulation framework to evaluate the performance of packet-based off-chip k-ary n-cube interconnect architectures for line cards. The simulator uses the state-of-the-art software design techniques to provide the user with a flexible yet robust tool, that can emulate multiple interconnect architectures under non-uniform traffic patterns. Moreover, the simulator offers the user with full control over network parameters, performance enhancing features and simulation time frames that make the platform as identical as possible to the real line card physical and functional properties. By using our network simulator, we reveal the best processor-memory configuration, out of multiple configurations, that achieves optimal performance. Moreover, we explore how network enhancement techniques such as virtual channels and sub-channeling improve network latency and throughput. Our performance results show that k-ary n-cube topologies, and especially our modified version of 2-ary 3-cube interconnect - the 3D-mesh, significantly outperform existing line card interconnects and are able to sustain higher traffic loads. The flow control mechanism proved to extensively reduce hot-spots, load-balance areas of high traffic rate and achieve low transmission failure rate. Moreover, it can scale to adopt more memories and/or processors and as a result to increase the line card's processing power.

On étudie divers problèmes concernant le n-cube. On décrit les exemples connus de (0,2) graphes (bipartis de diamètre 2 ou 3). On présente des constructions de (0,2) graphes. On étudie les (0,2) graphes avec des triangles. On montre comment construire certains des (0,2) graphes comme graphes de Cayley de groupes. On étudie les invariants immédiats du n-cube.

O presente trabalho é uma apresentação de aplicações de estruturas da álgebra de dimensões altas para a teoria de homotopia. Mais precisamente mostramos que existe uma equivalência entre as categorias dos cat$^n$-grupos e a dos $n$-cubos cruzados de grupos, ambas equivalentes a categoria das $n$-categorias estritas internas à categoria de grupos, e uma certa subcategoria da categoria dos $n$-cubos fibrantes, os chamados $n$-cubos de Eilenberg-MacLane. Além disso existe uma equivalência entre uma localização dessa subcategoria e a categoria homotópica dos $(n+1)$-tipos homotópicos, o que sugere a utilidade de usar as estruturas algébricas apresentadas como invariantes topológicas. O teorema central dessa teoria, o teorema generalizado de Seifert-van Kampen, diz que o funtor dos $n$-cubos de fibração aos cat$^n$-grupos usado para mostrar a equivalência mencionada preserva o colimite de certos diagramas e que nesses casos conectividade é preservada, o que permite certas computações. Apresentaremos definições das estruturas algébricas mencionadas além de como calcular certos colimites na categoria de $n$-cubos cruzados de grupos, demonstraremos os teoremas principais da teoria e mostramos como usar esses resultados para generalizar resultados clássicos da topologia algébrica como o teorema de Blakers-Massey, o teorema de Hurewicz e a fórmula de Hopf para homologia de grupos. / The present work is a presentation of applications to homotopy theory of structures in higher dimensional algebra. More precisely we show how the categories of crossed $n$-cubes of groups and of cat$^n$-groups, both equivalent to the category of strict $n$-categories internal to the category of groups, are equivalent to a subcategory of the category of fibrant $n$-cubes, namely the Eilenberg-MacLane $n$-cubes. There is also an equivalence between a localization of the category of Eilenberg-MacLane $n$-cubes and the homotopy category of homotopy $(n+1)$-types, which suggests the usefulness of the presented algebraic structures as topological invariants. The central theorem of this theory, the generalized Seifert-van Kampen theorem, states that the functor from $n$-cube of fibrations to the cat$^n$-groups used to show the aforementioned equivalence preserves the colimit of certain diagrams, and in these cases connectivity is preserved, which permits some computations. We present definitions of the relevant algebraic structures and also how to calculate certain colimits in the category of crossed $n$-cubes of groups, we demonstrate the main theorems of the theory and then we show how to generalize classical results in algebraic topology like the Blakers-Massey theorem, Hurewicz theorem and Hopf\'s formula for the homology of groups.

O presente trabalho é uma apresentação de aplicações de estruturas da álgebra de dimensões altas para a teoria de homotopia. Mais precisamente mostramos que existe uma equivalência entre as categorias dos cat$^n$-grupos e a dos $n$-cubos cruzados de grupos, ambas equivalentes a categoria das $n$-categorias estritas internas à categoria de grupos, e uma certa subcategoria da categoria dos $n$-cubos fibrantes, os chamados $n$-cubos de Eilenberg-MacLane. Além disso existe uma equivalência entre uma localização dessa subcategoria e a categoria homotópica dos $(n+1)$-tipos homotópicos, o que sugere a utilidade de usar as estruturas algébricas apresentadas como invariantes topológicas. O teorema central dessa teoria, o teorema generalizado de Seifert-van Kampen, diz que o funtor dos $n$-cubos de fibração aos cat$^n$-grupos usado para mostrar a equivalência mencionada preserva o colimite de certos diagramas e que nesses casos conectividade é preservada, o que permite certas computações. Apresentaremos definições das estruturas algébricas mencionadas além de como calcular certos colimites na categoria de $n$-cubos cruzados de grupos, demonstraremos os teoremas principais da teoria e mostramos como usar esses resultados para generalizar resultados clássicos da topologia algébrica como o teorema de Blakers-Massey, o teorema de Hurewicz e a fórmula de Hopf para homologia de grupos. / The present work is a presentation of applications to homotopy theory of structures in higher dimensional algebra. More precisely we show how the categories of crossed $n$-cubes of groups and of cat$^n$-groups, both equivalent to the category of strict $n$-categories internal to the category of groups, are equivalent to a subcategory of the category of fibrant $n$-cubes, namely the Eilenberg-MacLane $n$-cubes. There is also an equivalence between a localization of the category of Eilenberg-MacLane $n$-cubes and the homotopy category of homotopy $(n+1)$-types, which suggests the usefulness of the presented algebraic structures as topological invariants. The central theorem of this theory, the generalized Seifert-van Kampen theorem, states that the functor from $n$-cube of fibrations to the cat$^n$-groups used to show the aforementioned equivalence preserves the colimit of certain diagrams, and in these cases connectivity is preserved, which permits some computations. We present definitions of the relevant algebraic structures and also how to calculate certain colimits in the category of crossed $n$-cubes of groups, we demonstrate the main theorems of the theory and then we show how to generalize classical results in algebraic topology like the Blakers-Massey theorem, Hurewicz theorem and Hopf\'s formula for the homology of groups.