Efficient submesh permutations in wormhole-routed meshes

This paper studies how to concurrently permute related logical or physical submeshes in a d-dimensional n x ⋯ x n physical mesh via wormhole and dimension-ordered routing. Our objective is to minimize the congestion for realizing the permutations, while maximizing the number and dimensionality of permuted submeshes. We show that for d ≤ 2α - β, concurrent independent permutations of n^β related physical submeshes, each of α dimensions, can be performed in two routing steps without congestion. If the permuted submeshes are logical ones, they can be permuted in one, instead of two, routing step. In addition, any shift operation along any axis of the logical mesh can be performed in the physical mesh without congestion. We also show that if all nodes know the permutation function, any permutation within a submesh of dimensions [2(d - 1)/3] can be realized in three routing steps without congestion.