Minimal, deadlock-free routing in hypercubic and arbitrary networks

In this paper we consider the problem of creating minimal, deadlock-free routing algorithms, where a routing algorithm is said to be minimal if it uses only shortest paths. In particular we examine the possibility of creating scalable algorithms that use only a constant number of buffers per node. Minimal, scalable, deadlock-free routing algorithms are known for many important networks including meshes, tori, trees and hypercubes. In addition, it is known that a scalable, deadlock -free routing algorithm exists for every network. However, it is unknown whether or not a minimal scalable, deadlock-free routing algorithm exists for every network; and no such algorithm is known for the de Bruijn or shuffle-exchange networks. We present three main results. First, we prove that there is no minimal, scalable, deadlock-free routing algorithm for the hypercube that uses only the standard Ascend (dimension-order) paths. Second, we prove that there exist networks for which no minimal, scalable, deadlock-free routing algorithm is possible. Third, we create minimal, scalable, deadlock- free routing algorithms for the de Bruijn and shuffle-exchange networks. The algorithm for the de Bruijn network appears to be of practical interest, as it uses only four buffers per node. Our results apply to oblivious and adaptive store-and-forward and virtual cut-through routing algorithms, and to oblivious wormhole routing algorithms.