Embedding Complete Binary Trees into Butter y Networks

In this paper we present embeddings of complete binary trees into butter y networks with or without wrap-around connections. Let m be an even integer and q = m+ blogmc 1. We show how to embed a 2q+1 1-node complete binary tree T (q) into a (m + 1)2m+1node wrap-around butter y Bw(m+1) with a dilation of 4; and how to embed T (q) into a (m+2)2m+2-node wrap-around butter y Bw(m+2) with an optimal dilation of 2: We also present an embedding of a wrap-around butter y Bw(m) into a (m+ 1)2m-node no wrap-around butter y B(m) with a dilation of 3. Using this embedding we show that T (q) can be embedded into a no wrap-around butter y B(m+1) (resp. B(m+2)) with a dilation of 8 (resp. 5).