Esteban Stafford

Modeling Toroidal Networks with the Gaussian Integers

In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian integers, the subset of the complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.

Modeling hexagonal constellations with Eisenstein-Jacobi graphs

A set of signal points is called a hexagonal constellation if it is possible to define a metric so that each point has exactly six neighbors at distance 1 from it. As sets of signal points, quotient rings of the ring of Eisenstein-Jacobi integers are considered. For each quotient ring, the corresponding graph is defined. In turn, the distance between two points of a quotient ring is defined as the corresponding graph distance. Under certain restrictions, a quotient ring is a hexagonal constellation with respect to this metric. For the considered hexagonal constellations, some classes of perfect codes are known. Using graphs leads to a new way of constructing these codes based on solving a standard graph-theoretic problem of finding a perfect dominating set. Also, a relation between the proposed metric and the well-known Lee metric is considered.

A first approach to king topologies for on-chip networks

In this paper we propose two new topologies for on-chip networks that we have denoted as king mesh and king torus. These are a higher degree evolution of the classical mesh and torus topologies. In a king network packets can traverse the networks using orthogonal and diagonal movements like the king on a chess board. First we present a topological study addressing distance properties, bisection bandwidth and path diversity as well as a folding scheme. Second we analyze different routing mechanisms. Ranging from minimal distance routings to missrouting techniques which exploit the topological richness of these networks. Finally we make an exhaustive performance evaluation comparing the new king topologies with their classical counterparts. The experimental results show a performance improvement, that allow us to present these new topologies as better alternative to classical topologies.

Perfect Codes over Lipschitz Integers

Cayley graphs over quotients of the quaternion integers are going to be used to define a new metric over four dimensional lattices. We will consider perfect 1-error correcting codes according to this metric space. We will show that, in some cases, these lattices can be represented as two-dimensional constellations, which allow us to state a relation between the Lee metric and this new Lipschitz metric.

Graph-based metrics over QAM constellations

In order to propose a new metric over QAM constellations, diagonal Gaussian graphs defined over quotients of the Gaussian integers are introduced in this paper. Distance properties of the constellations are detailed by means of the vertex-to-vertex distribution of this family of graphs. Moreover, perfect codes for this metric are considered. Finally, notable subgraphs of diagonal Gaussian graphs are studied which leads to relate the proposed metric to other well-known graph-based metrics such as the Lee distance.

Energy efficiency of load balancing for data-parallel applications in heterogeneous systems

The use of heterogeneous systems in supercomputing is on the rise as they improve both performance and energy efficiency. However, the programming of these machines requires considerable effort to get the best results in massively data-parallel applications. Maat is a library that enables OpenCL programmers to efficiently execute single data-parallel kernels using all the available devices on a heterogeneous system. It offers a set of load balancing methods, which perform the data partitioning and distribution among the devices, exploiting more of the performance of the system and consequently reducing execution time. Until now, however, a study of the implications of these on the energy consumption has not been made. Therefore, this paper analyses the energy efficiency of the different load balancing methods compared to a baseline system that uses just a single GPU. To evaluate the impact of the heterogeneity of the system, the GPUs were set to different frequencies. The obtained results show that in all the studied cases there is at least one load balancing method that improves energy efficiency.

Advanced Switching Mechanisms for Forthcoming On-Chip Networks

Many current VLSI on-chip multiprocessors and systems-on-chip employ point-to-point switched interconnection networks. Rings and 2D-meshes are among the most popular interconnection topologies for these increasingly important onchip networks. Nevertheless, rings cannot scale beyond dozens of nodes and meshes are asymmetric. Two of the key features of square 2D-tori are their scalability and symmetry. As higher scalability is demanded by the increasing number of cores (or specialized units) integrated on a chip and symmetry is critical for high-performance and load balancing, we concentrate on 2D-tori. However, most popular deadlock-free routing mechanisms are based on Dimension Order Routing (DOR) which breaks the torus symmetry when managing adversarial traffic patterns. This paper analyzes this problem and its consequences. After that, it proposes a new deadlock-free fully adaptive minimal routing, denoted as σDOR, that preserves tori symmetry under any load. It uses just two virtual channels to avoid DOR-induced asymmetry, the same as in previous competitive proposals. σDOR exhibits better behavior than any of previous solutions as it allows packets to dynamically adapt to local congestion. Experimental results evidence the superior performance of our mechanism, confirming the negative impact of DOR asymmetry.

Quotients of Gaussian graphs and their application to perfect codes

A graph-based model of perfect two-dimensional codes is presented in this work. This model facilitates the study of the metric properties of the codes. Signal spaces are modeled by means of Cayley graphs defined over the Gaussian integers and denoted as Gaussian graphs. Codewords of perfect codes will be represented by vertices of a quotient graph of the Gaussian graph in which the signal space has been defined. It will be shown that any quotient graph of a Gaussian graph is indeed a Gaussian graph. This makes it possible to apply previously known properties of Gaussian graphs to the analysis of perfect codes. To illustrate the modeling power of this graph-based tool, perfect Lee codes will be analyzed in terms of Gaussian graphs and their quotients.

King Topologies for Fault Tolerance

This paper analyzes the robustness of the king networks for fault tolerance. To this aim, a performance evaluation of two well known fault tolerant routing algorithms in king as well as 2d networks is done. Immunet that uses two virtual channels and Immucube, that has a better performance while requiring three virtual channels. Experimental results confirm the excellent behavior, both in performance and scalability, of the king topologies in the presence of failures. Finally, taking advantage of the topological features of king networks, a new fault tolerance routing algorithm for these networks is presented. From a cost/performance point of view this algorithm is a compromise between the two previous algorithms.

Extending OmpSs for OpenCL Kernel Co-Execution in Heterogeneous Systems

Heterogeneous systems have a very high potential performance but present difficulties in their programming. OmpSs is a well known framework for task based parallel applications, which is an interesting tool to simplify the programming of these systems. However, it does not support the co-execution of a single OpenCL kernel instance on several compute devices. To overcome this limitation, this paper presents an extension of the OmpSs framework that solves two main objectives: the automatic division of datasets among several devices and the management of their memory address spaces. To adapt to different kinds of applications, the data division can be performed by the novel HGuided load balancing algorithm or by the well known Static and Dynamic. All this is accomplished with negligible impact on the programming. Experimental results reveal that there is always one load balancing algorithm that improves the performance and energy consumption of the system.