EMBEDDING COMPLETE BINARY-TREES INTO BUTTERFLY NETWORKS

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