Gossiping on Meshes and Tori- Tradeoff bw startup time and transmission time

Gossiping on Meshes and Tori- Tradeoff bw startup time and transmission time

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IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 9, NO. 6, JUNE 1998 513 Gossiping on Meshes and Tori Ben H.H. Juurlink, Jop F. Sibeyn, Member , IEEE Computer Society , and P.S. Rao, Abstract —Algorithms for performing gossiping on one- and higher-dimensional meshes are presented. As a routing model, the practically important wormhole routing is assumed. We especially focus on the trade-off between the start-up time and the transmission time. For one-dimensional arrays and rings, we give a novel lower bound and an asymptotically optimal gossiping algorithm for all choices of the parameters involved. For two-dimensional meshes and tori, a simple algorithm composed of one- dimensional phases is presented. For an important range of packet and mesh sizes, it gives clear improvements upon previously developed algorithms. The algorithm is analyzed theoretically and the achieved improvements are also convincingly demonstrated by simulations, as well as an implementation on the Paragon. On the Paragon, our algorithm even outperforms the gossiping routine provided in the NX message-passing library. For higher-dimensional meshes, we give algorithms which are based on an interesting generalization of the notion of a diagonal. These algorithms are analyzed theoretically, as well as by simulation. Index Terms —Gossip, global communication, wormhole routing, mesh networks, torus networks. —————————— ±±± F ±±± —————————— 1I NTRODUCTION 1.1 Meshes and Tori One of the most thoroughly investigated interconnection schemes for parallel computers is the n ± n mesh , in which n 2 processing units ( PUs ) are connected by a two- dimensional grid of communication links. A torus is a mesh with wrap-around connections. Their immediate generali- zations are d -dimensional n ± L ± n meshes and tori. Al- though these networks have a large diameter in compari- son to the various hypercubic networks, they are never- theless of great importance due to their simple structure and efficient layout. Numerous parallel machines with mesh and torus topologies have been built, and various algorithmic problems have been analyzed on theoretical models of the mesh. 1.2 Wormhole Routing Traditionally, algorithms for the mesh have been developed using a store-and-forward routing model in which a packet is treated as an atomic unit that can be transferred between two adjacent PUs in unit time. However, many modern parallel architectures employ wormhole routing instead. Briefly, in this model, a packet consists of a number of atomic data units called flits which are routed through the network in a pipelined fashion. As long as there is no con- gestion in the network, the time to send a packet consisting of l flits between two arbitrary PUs is well approximated by t s + l ¿ t l , where t s is the start-up time (the time needed to initiate the message transmission) and t l is the flit-transfer time (the time required for actually transferring the data).
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Gossiping on Meshes and Tori- Tradeoff bw startup time and transmission time

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