Toward optimal complete exchange on wormhole-routed tori

In this paper, we propose new routing schemes to perform all-to-all personalized communication (or known as complete exchange) in wormhole-routed, one-port tori. On tori of equal size along each dimension, our algorithms use both asymptotically optimal startup and transmission time. The results are characterized by several interesting features: (1) the use of gather-scatter tree to achieve optimality in startup time, (2) enforcement of shortest paths in routing messages to achieve optimality in transmission time, (3) application of network-partitioning techniques to reduce the constant associated with the transmission time, and (4) the dimension-by-dimension and gather-scatter-tree approach to make possible applying the results to nonsquare, any-size tori. In the literature, some algorithms are optimal in only one of startup and transmission costs, while some, although asymptotically optimal in both costs, will incur much larger constants associated with the costs. Numerical analysis and experiment both show that significant improvement can be obtained by our scheme on total communication latency over existing results.