Carmen Martínez

• PhD
• Professor (Associate) at University of Cantabria

Since today’s HPC and data center systems can comprise hundreds of thousands of servers and beyond, it is crucial to equip them with a network that provides high performance. New topologies proposed to achieve such performance need to be evaluated under different traffic conditions, aiming to closely replicate real-world scenarios. While most optim...

To interconnect their growing number of servers, current supercomputers and data centers are starting to adopt low-diameter networks, such as HyperX, Dragonfly and Dragonfly+. These emergent topologies require balancing the load over their links and finding suitable non-minimal routing mechanisms for them becomes particularly challenging. The Valia...

Understanding the architecture of a processor can be uninteresting and deterring for computer science students, since low-level details of computer architecture are often perceived to lack real-world impact. These courses typically have a strong practical component where students learn the fundamentals of the computer architecture and the handling...

Supercomputers and datacenters comprise hundreds of thousands of servers. Different network topologies have been proposed to attain such a high scalability, from Flattened Butterfly and Dragonfly to the most disruptive Jellyfish, which is based on a random graph. The routing problem on such networks remains a challenge that can be tackled either as...

Fat-trees (FTs) are widely known topologies that, among other advantages, provide full bisection bandwidth. However, many implementations of FTs are made slimmed to cheapen the infrastructure, since most applications do not make use of this full bisection bandwidth. In this paper Extended Generalized Random Folded Clos (XGRFC) interconnection netwo...

Large computer systems, like those in the TOP 500 ranking, comprise about hundreds of thousands cores. Simulating application execution in these systems is very complex and costly. This article explores the option of using application skeletons, together with an analytic simulator, to study the performance of these large systems. With this aim, the...

Big scale, high performance and fault-tolerance, low-cost and graceful expandability are pursued features in current datacenter networks (DCN). Although there have been many proposals for DCNs, most modern installations are equipped with classical folded Clos networks. Recently, regular random topologies, as the Jellyfish, have been proposed for DC...

The interconnection network comprises a significant portion of the cost of large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with the aim of minimizing these costs. However, they fail to consider potential unbalance in the network...

The interconnection network comprises a significant portion of the cost of large parallel computers, both in economic terms and power consumption. Several previous proposals exploit large-radix routers to build scalable low-distance topologies with the aim of minimizing these costs. However, they fail to consider potential unbalance in the network...

In the late years many different interconnection networks have been used with two main tendencies. One is characterized by the use of high-degree routers with long wires while the other uses routers of much smaller degree. The latter rely on two-dimensional mesh and torus topologies with shorter local links. This paper focuses on doubling the degre...

A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It is known that there are infinite such primes. Perfect codes for the Lee-metric were conjectured by Golomb and Welch not to exist, which has been proved for large radii and also for low dimension. The...

In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of the codes in this construction are also perfect. The graphs considered include some well-known graphs such as tori, twisted tori and Kronecker products of two cycles. Therefore, the codes can be...

The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.

Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when used for executing applications that require near-neighbor communications. Nevertheless, they are not so good when dealing with global communications. Hence, typical 3D implementations have evolved to 5D networks, among other reasons, to redu...

Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when used for executing applications that require near-neighbor communications. Nevertheless, they are not so good when dealing with global communications. Hence, typical 3D implementations have evolved to 5D networks, among other reasons, to redu...

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes h...

A complete family of Cayley graphs of degree four, denoted as L-networks, is considered in this paper. L-networks are 2D mesh-based topologies with wrap-around connections. L-networks constitute a graph-based model which englobe many previously proposed 2D interconnection networks. Some of them have been extensively used in the industry as the unde...

Twisted torus topologies have been proposed as an alternative to toroidal rectangular networks, improving distance parameters and providing network symmetry. However, twisting is apparently less amenable to task mapping algorithms of real life applications. In this paper we make an analytical study of different mapping and concentration techniques...

This work attempts to compare size and cost of two network topologies proposed for large-radix routers: concentrated torus and dragonflies. We study and compare the scalability, cost and fault tolerance of each network. On average, we found that a concentrated torus can be a cost-efficient option for middle-range networks.

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...

Many current parallel computers are built around a torus interconnection network. Machines from Cray, HP, and IBM, among others, make use of this topology. In terms of topological advantages, square (2D) or cubic (3D) tori would be the topologies of choice. However, for different practical reasons, 2D and 3D tori with different number of nodes per...

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...

In this paper we consider perfect codes over two dimensional QAM-type constellations of any cardinal. Such constellations are going to be modeled by L-graphs, which are the two-dimensional family of multidimensional circulants, defined. We show that Gaussian graphs, Lee graphs and the Kronecker product of two cycles are included in this family. The...

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...

The search for perfect error-correcting codes has received intense interest since the seminal work by Hamming. Decades ago, Golomb and Welch studied perfect codes for the Lee metric in multidimensional torus constellations. In this work, we focus our attention on a new class of four-dimensional signal spaces which include tori as subcases. Our cons...

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,...

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. Finall...

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 betwe...

An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant gra...

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 relatio...

Many parallel computers use Tori interconnection networks. Machines from Cray, HP and IBM, among others, exploit these topologies. In order to maintain full network symmetry, 2D and 3D Tori must have the same number of nodes (k) per dimension resulting in square or cubic topologies. Nevertheless, for practical reasons, computer engineers have desig...

In this paper we present perfect codes for two-dimensional constellations derived from generalized Gaussian graphs, a family of graphs built over quotient rings of Gaussian integers. Using the generalized Gaussian graphs distance, we solve the problem of finding t-dominating sets and, then, we build new perfect codes over these graphs. The well-kno...

This paper explores the suitability of dense circulant graphs of degree four for the design of on-chip interconnection networks. Networks based on these graphs reduce the Torus diameter in a factor √2, which translates into significant performance gains for unicast traffic. In addition, they are clearly superior to Tori when managing collective com...

The basis for designing error-correcting codes for two dimensional signal sets is considered in this paper. Both, algebraic and graph-theoretical approaches are employed in this research for establishing the fundamentals of these codes. We give a solution to the t-dominating set problem in a subfamily of degree four circulant graphs which directly...

In this paper we present algorithms for finding a shortest path between two vertices of any weighted undirected and directed circulant graph with two jumps. Our shortest path algorithm only requires O(log N) arithmetic steps and the total bit complexity is O(log3 N), where N is the number of the graph’s vertices. This method has been derived from s...

Circulant graphs have been deeply studied in technical literature. Midimew networks are a class of distance-related optimal circulant graphs of degree four which have applications in network engineering and coding theory. In this research, a new layout for Midimew networks which keeps the maximum link length under the value √5 is presented, conside...

In this paper we present the first polynomial time deterministic algorithm to compute the shortest path between two vertices of a cir-culant graph of degree four. Our spectacular algorithm only requires O(log 3 N) bit operations, where N is the number of the vertices and it is based on shortest vector problems in a special class of lattices for L 1...

Circulant graphs have been deeply studied in technical literature. Midimew networks are a class of distance-related optimal circulant graphs of degree four which have applications in network engineering and coding theory. In a previous work, a new layout for Midimew networks which keeps the maximum link length under 5 has been presented. The most i...

The class of dense circulant graphs of degree four with optimal distance-related properties is analyzed in this paper. An algebraic study of this class is done. Two geometric characterizations are given, one in the plane and other in the space. Both characterizations facilitate the analysis of their topological properties and corroborate their suit...

In this paper we present a distance-hereditary decomposition of optimal chordal rings of 2k² nodes into a set of rings of 2k nodes, where k is the diameter. All the rings belonging to this set have the same length and their diameter corresponds to the diameter of the chordal ring in which they are embedded. The members of this embedded s...

We present in this paper some of the topolog-ical properties of an interesting class of Cir-culant graphs whose nodes are labeled by a subset of the Gaussian integers. Such graphs and the problems we solve on them have direct applications to the design of interconnection networks and, in addition, they can be consid-ered in the design of perfect er...

Gaussian Graphs are introduced and analyzed in this paper. The set of vertices of these Cayley graphs are quotient rings of Gaussian integers. The distance related properties of these graphs are studied and the problem of finding perfect t-dominating sets over them is solved.

In this paper we characterize symmetric L-graphs, which are either Kronecker products of two cycles or Gaussian graphs. Vertex symmetric networks have the property that the commu-nication load is uniformly distributed on all the vertices so that there is no point of congestion. A stronger notion of symmetry, edge symmetry, requires that every edge...