Algebraic foundations and broadcasting algorithms for wormhole-routed all-port tori

The one-to-all broadcast is the most primary collective communication pattern in a multicomputer network. We consider this problem in a wormhole-routed torus which uses all-port and dimension-ordered routing model. We derive our routing algorithms based on the concept of "span of vector spaces" in linear algebra. For instance, in a 3-D torus, the nodes receiving the broadcast message will be "spanned" from the source node to a line of nodes, to a plane of nodes, and then to a cube of nodes. Our results require at most 2(k-1) steps more than the optimal number of steps for any square k-D torus. Existing results, as compared to ours, can only be applied to tori of very restricted dimensions or sizes, and either rely on an undesirable non-dimension-ordered routing or require more numbers of steps.