8 research outputs found
Executing algorithms with hypercube topology on torus multicomputers
Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost-per-node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, an embedding of hypercubes onto toruses of any given dimension is proposed. This novel embedding is called xor embedding. The paper presents a set of performance figures for both the standard and the xor embeddings and shows that the latter outperforms the former for any torus. In addition, it is proven that for a one-dimensional torus (a ring) the xor embedding is optimal in the sense that it minimizes the execution time of a class of parallel algorithms with hypercube topology. This class of algorithms is frequently found in real applications, such as FFT and some class of sorting algorithms.Peer Reviewe
Executing Algorithms with Hypercube Topology on Torus Multicomputers
Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost per node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, ..
Executing algorithms with hypercube topology on torus multicomputers
Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost-per-node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, an embedding of hypercubes onto toruses of any given dimension is proposed. This novel embedding is called xor embedding. The paper presents a set of performance figures for both the standard and the xor embeddings and shows that the latter outperforms the former for any torus. In addition, it is proven that for a one-dimensional torus (a ring) the xor embedding is optimal in the sense that it minimizes the execution time of a class of parallel algorithms with hypercube topology. This class of algorithms is frequently found in real applications, such as FFT and some class of sorting algorithms.Peer Reviewe
Executing Algorithms with Hypercube Topology on Torus Multicomputers
Many parallel algorithms use hypercubes as the communication topology among their processes. When such algorithms are executed on hypercube multicomputers the communication cost is kept minimum since processes can be allocated to processors in such a way that only communication between neighbor processors is required. However, the scalability of hypercube multicomputers is constrained by the fact that the interconnection cost per node increases with the total number of nodes. From scalability point of view, meshes and toruses are more interesting classes of interconnection topologies. This paper focuses on the execution of algorithms with hypercube communication topology on multicomputers with mesh or torus interconnection topologies. The proposed approach is based on looking at different embeddings of hypercube graphs onto mesh or torus graphs. The paper concentrates on toruses since an already known embedding, which is called standard embedding, is optimal for meshes. In this paper, an embedding of hypercubes onto toruses of any given dimension is proposed. This novel embedding is called xor embedding. The paper presents a set of performance figures for both the standard and the xor embeddings and shows that the latter outperforms the former for any torus. In addition, it is proven that for a one-dimensional torus (a ring) the xor embedding is optimal in the sense that it minimizes the execution time of a class of parallel algorithms with hypercube topology. This class of algorithms is frequently found in real applications, such as FFT and some class of sorting algorithms. 1
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Resource placement, data rearrangement, and Hamiltonian cycles in torus networks
Many parallel machines, both commercial and experimental, have been/are being designed with toroidal interconnection networks. For a given number of nodes, the torus has a relatively larger diameter, but better cost/performance tradeoffs, such as higher channel bandwidth, and lower node degree, when compared to the hypercube. Thus, the torus is becoming a popular topology for the interconnection network of a high performance parallel computers.
In a multicomputer, the resources, such as I/O devices or software packages, are distributed over the networks. The first part of the thesis investigates efficient methods of distributing resources in a torus network. Three classes of placement methods are studied. They are (1) distant-t placement problem: in this case, any non-resource node is at a distance of at most t from some resource nodes, (2) j-adjacency problem: here, a non-resource node is adjacent to at least j resource nodes, and (3) generalized placement problem: a non-resource node must be a distance of at most t from at least j resource nodes.
This resource placement technique can be applied to allocating spare processors to provide fault-tolerance in the case of the processor failures. Some efficient
spare processor placement methods and reconfiguration schemes in the case of processor failures are also described.
In a torus based parallel system, some algorithms give best performance if the data are distributed to processors numbered in Cartesian order; in some other cases, it is better to distribute the data to processors numbered in Gray code order. Since the placement patterns may be changed dynamically, it is essential to find efficient methods of rearranging the data from Gray code order to Cartesian order and vice versa. In the second part of the thesis, some efficient methods for data transfer from Cartesian order to radix order and vice versa are developed.
The last part of the thesis gives results on generating edge disjoint Hamiltonian cycles in k-ary n-cubes, hypercubes, and 2D tori. These edge disjoint cycles are quite useful for many communication algorithms
Una contribució al càlcul de valors i vectors propis i a l'anàlisi de l'escalabilitat
El càlcul de valors i vectors propis és un nucli computacional que forma part de diverses aplicacions de tipus científic i tècnic que requereixen una potència de càlcul molt gran. Aquestes aplicacions no poden resoldre's en sistemes monoprocessadors perquè aquests sistemes no proporcionen la potència de càlcul suficient per resoldre el problema amb un temps raonable. Una solució possible a aquest problema és la utilització de sistemes paral·lels.El contingut d'aquest treball pot dividir-se en quatre parts ben diferenciades; en les tres primeres parts dels valors i vectors propis en sistemes multicomputadors amb diferents topologies: hipercub, malla i torus; en l'última part del treball es proposa una metodologia d'anàlisis de l'escalabilitat de sistemes paral·lels.- En la primera part del treball es proposen un conjunt d'algorismes paral·lels per hipercubs: BR segmentat, alfa-optimal i Grau-4. Tots aquests algorismes es basen en l'algorisme Block Recursive proposat a [42]. Els nous algorismes proposats tenen la capacitat d'utilitzar de forma més eficient el potencial paral·lelisme de comunicacions que ofereix una arquitectura multiple-port amb els que s'aconsegueix una reducció del cost de la comunicació considerable respecte al cost de comunicació de l'algorisme original.- En la segona part del treball es proposa un nou algorisme amb una topologia de comunicació en malla bidimensional (2D). Aquest algorimse l'hem anomenat algorisme 2D. Es veurà que aquest nou algorisme aconsegueix reduir el cost total considerablement respecte als algorismes que han estat proposats per altres autors per malles i torus.- En la tercera part, s'estudia l'eficiència de l'algorisme BR-segmentat (algorisme amb una topologia de comunicació en hipercub proposat en la primera part de la tesi) un cop mapejat en un multicomputador amb una topologia en malla o en torus. A l'hora de realitzar el mapeig s'ha aplicat i ampliat una metodologia desenvolupada en el grup de treball que ens permet realitzar el mapeig de forma eficient i sistemàtic d'una topologia en hipercub a una topologia en malla o torus. El cost de la comunicació del nou algorisme es compara amb el cost de l'algorisme 2D proposat en la segona part del treball.- Finalment, en l'última part d'aquest treball es proposa una metodologia d'anàlisi de l'escalabilitat de sistemes paral·lels orientada a l'usuari final del sistema. S'utilitza l'algorisme 2D mapejat en una línia per mostrar un exemple d'aplicació de la metodologia