Narrow Results

Hypermeshes have been given much attention as a versatile interconnection network of parallel computers. A hypermesh is obtained from a mesh by replacing each linear connection with a hyperedge. In this paper, we show how to embed a butterfly or multiple copies of a butterfly into a hypermesh. First, a butterfly B(s) of (s + 1)2^s nodes is embedded into a 2^s × X hypermesh where X = 2^{⌊log₂ s ⌋+ 1}. Second, the butterfly B(s) is embedded into a square hypermesh. Third, multiple copies of the butterfly B(s) are embedded into a hypermesh of variable aspect ratio. The efficiency of these embeddings is measured by alignment cost, congestion, and expansion. The alignment cost of all of these embeddings is optimal. The congestion of the first and third embedding is optimal. The expansion of the first and third embedding is one if s = 2^k - 1 for some integer k, otherwise, less than two. The expansion of the second embedding is 2 + ∊ (s) where ∊(s) = (2log(s + 1) + 2)/(s + 1).

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The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from the Courant algebroid appearing in generalized complex geometry. In this paper, we integrate such a Lie 2-algebra to a strict Lie 2-group in the finite-dimensional case.