All to-all communication with minimum start-up costs in 2D/3D tori and meshes

All-to-all communication patterns occur in many important parallel algorithms. This paper presents new algorithms for all-to-all communication patterns (all-to-all broadcast and all-to-all personalized exchange) for wormhole switched 2D/3D torus- and mesh-connected multiprocessors. The algorithms use message combining to minimize message start-ups at the expense of larger message sizes. The unique feature of these algorithms is that they are the first algorithms that we know of that operate in a bottom-up fashion rather than a recursive, top-down manner. For a 2^d×2^d torus or mesh, the algorithms for all-to-all personalized exchange have time complexity of O(2^3d). An important property of the algorithms is the O(d) time due to message start-ups, compared with O(2^d) for current algorithms. This is particularly important for modern parallel architectures where the start-up cost of message transmissions still dominates, except for very large block sizes. Finally, the 2D algorithms for all-to-all personalized exchange are extended to O(2^4d) algorithms in a 2^d×2^d×2^d3D torus or mesh. These algorithms also retain the important property of O(d) time due to message start-ups