Sharif Digital Repository

The square of a graph G denoted by G2, is the graph with the same vertex set as G and edges linking pairs of vertices at distance at most 2 in G. The chromatic number of the square of the Cartesian product of two cycles was previously determined for some cases. In this paper, we determine the precise value of χ((Cm□Cn)2) for all the remaining cases. We show that for all ordered pairs (m,n) except for (7,11) we have χ(Cm□Cn)2)=γV((Cm□Cn)2)|α((Cm□Cn)2), where α(G) denotes the independent number of G. This settles a conjecture of Sopena and Wu (2010). We also show that the smallest integer k such that χ(Cm□Cn2)≤6 for every m,≥k is 10. This answers a question of Shao and Vesel (2013)  

In this paper, we propose a solution for the problem of capturing an intruder in two popular interconnection topologies namely, the mesh and the torus. A set of agents collaborate to capture a hostile intruder in the network. While the agents can move in the network one hop at a time, the intruder is assumed to be arbitrarily fast i.e. it can traverse any number of nodes contiguously as far as there are no agents in those nodes. Here we consider a new version of the problem where each agent can replicate new agents when needed, i.e. the algorithm starts with a single agent and new agents are created on demand. We define a new class of algorithms for capturing an intruder. In particular, we... 

Several studies have examined the relative performance merits of the torus and hypercube taking into account the channel bandwidth constraints imposed by implementation technology. While the torus has been shown to outperform the hypercube under the constant wiring density constraint, the opposite conclusion has been reached when the constant pin-out constraint is considered. However, all these studies have assumed deterministic routing and have not taken into account the internal hardware cost of routers. This paper re-examines the performance merits of the torus and hypercube using both fully-adaptive and deterministic routing strategies. Moreover, it uses a new cost model which takes into... 

Several studies have examined the relative performance merits of the torus and hypercube taking into account the channel bandwidth constraints imposed by implementation technology. While the torus has been shown to outperform the hypercube under the constant wiring density constraint, the opposite conclusion has been reached when the constant pin-out constraint is considered. However, these studies have assumed a pure uniform traffic pattern and deterministic routing. The 'uniform traffic' assumption is not always justifiable in practice as there are many real-world parallel applications that exhibit non-uniform traffic patterns, which can create unbalanced traffic such as hotspots in the... 

In massively parallel processing systems that incorporate hundreds and thousands of processing nodes, the problem of placing a limited number of resources (e.g. printers, disks, software copies, etc.) over processing nodes is of great importance. In this paper, we investigate resource placement in the three-dimensional torus network which is the underlying interconnection network in most current multicomputer. We show that there exists an irregular distance-1 placement only for tori of the form 2 × 3 i × 6 j, i, j ∈ N. In addition, we present a new formal proof for the necessary and sufficient conditions for the existence of irregular distance-d placement in 3D tori. © 2009 Elsevier B.V. All... 

Modern SoC architectures use NoCs for high-speed inter-IP communication. For NoC architectures, high-performance efficient routing algorithms with low power consumption are essential for real-time applications. NoCs with mesh and torus interconnection topologies are now popular due to their simple structures. A torus NoC is very similar to the mesh NoC, but has rather smaller diameter. For a routing algorithm to be deadlock-free in a torus, at least two virtual channels per physical channel must be used to avoid cyclic channel dependencies due to the warp-around links; however, in a mesh network deadlock freedom can be insured using only one virtual channel. The employed number of virtual... 

NoC is an efficient on-chip communication architecture for SoC architectures. It enables integration of a large number of computational and storage blocks on a single chip. NoCs have tackled the SoCs disadvantages and are scalable. In this paper, we compare two popular NoC topologies, i.e., mesh and torus, in terms of different figures of merit e.g., latency, power consumption, and power/throughput ratio under different routing algorithms and two common traffic models, uniform and hotspot. To the best of our knowledge, this is the first effort in comparing mesh and torus topologies under different routing algorithms and traffic models with respect to their performance and power consumption.... 

Current massively parallel systems are often composed of hundreds or thousand of components (such as routers, channels and connectors) that collectively possess failure rates higher than what arise in the ordinary systems. Therefore, these systems are required to he equipped with fault-tolerant mechanisms to ensure that the system will keep running in a degraded made until the failed component in repaired. Normally, the faulty components are coalesced into fault regions which may be classified into two major categories: convex and concave regions. In this paper, we propose the first general solution to calculate the probability of occurrences of common fault patterns in torus networks which... 

Parallel computers, such as multiprocessors system-on-chip (Mp-SoCs), multicomputers and cluster computers, are consisting of hundreds or thousands multiple processing units and components (such as routers, channels and connectors) connected via some interconnection network that collectively may undergo high failure rates. Therefore, these systems are required to be equipped with fault-tolerant mechanisms to ensure that the system will keep running in a degraded mode. Normally, the faulty components are coalesced into fault regions, which are classified into two major categories: convex and concave regions. In this paper, we propose the first solution to calculate the probability of... 

Over the past decade, many fully adaptive routing algorithms have been proposed in the literature, of which Duato's routing algorithm has gained considerable attention for analytical modeling. In this study we propose an analytical model to predict message latency in wormhole routed 2-dimensional torus networks in which fully adaptive routing, based on Linderr-Harden's methodology [10], is employed. This methodology presents a framework in which adaptive routing algorithms can be developed for the k-ary n-cube network. Simulation experiments reveal that the latency results predicted by the proposed analytical model are in good agreement with those provided by simulation experiments. ©... 

While for shallow waters the use of old offshore jackets still seems efficient and justifiable, for deep and ultra-deep waters such platforms cannot be used. During the past few decades the old generation of fixed offshore platforms was succeeded by the new floating platforms and new designs such as FPSOs. A new family of floating offshore platforms has been developed. These should be able to respond to size, weight and space for operating equipment, i.e. they can be constructed to have a wide range of load-bearing capacities. Use is made of the old concept of Life Saving Tubes, which in their simplest form can be a toroidal shaped tyre inner tube. The Torus-shaped idea can be further... 

This paper presents a comprehensive performance model for fully adaptive routing in wormhole-switched Cartesian product networks. Besides the generality of the model which makes it suitable to be used for any product graph, experimental (simulation) results show that the proposed model exhibits high accuracy even in heavy traffic and saturation region, where other models have severe problems to predict the performance of the network. Most popular interconnection network can be defined as a Cartesian product of two or more networks including the mesh, hypercube, and torus networks. Torus and mesh networks are the most popular topologies used in recent supercomputing parallel machines. They... 

Modern SoC architectures use NoCs for high-speed inter-IP communication. For NoC architectures, high-performance efficient routing algorithms with low power consumption are essential for real-time applications. NoCs with mesh and torus interconnection topologies are now popular due to their simple structures. A torus NoC is very similar to the mesh NoC, but has rather smaller diameter. For a routing algorithm to be deadlock-free in a torus, at least two virtual channels per physical channel must be used to avoid cyclic channel dependencies due to the warp-around links; however, in a mesh network deadlock freedom can be insured using only one virtual channel. The employed number of virtual... 

This paper presents a comprehensive performance model for fully adaptive routing in wormhole-switched Cartesian product networks. Besides the generality of the model which makes it suitable to be used for any product graph, experimental (simulation) results show that the proposed model exhibits high accuracy even in heavy traffic and saturation region, where other models have severe problems to predict the performance of the network. Most popular interconnection network can be defined as a Cartesian product of two or more networks including the mesh, hypercube, and torus networks. Torus and mesh networks are the most popular topologies used in recent supercomputing parallel machines. They... 

The Pyramid network is a desirable network topology used as both software data-structure and hardware architecture. In this paper, we propose a general definition for a class of pyramid networks that are based on grid connections between the nodes in each level. Contrary to the conventional pyramid network in which the nodes in each level form a mesh, the connections between these nodes may also be according to other grid-based topologies such as the torus, hypermesh or WK-recursive. Such pyramid networks form a wide class of interconnection networks that possess rich topological properties. We study a number of important properties of these topologies for general-purpose parallel processing... 

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability...