Embedding complete binary trees into butterfly networks

The authors 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. The authors show how to embed a 2^q+1-1-node complete binary tree T(q) into a (m+1)2^m+1-node wrap-around butterfly B_w(m+1) with a dilation of 4, and how to embed T(q) into a (m+2)2^m+2-node wrap-around butterfly B_w(m+2) with an optimal dilation of 2. They also present an embedding of a wrap-around butterfly B_w (m) into a (m+1)2^m-node no-wrap-around butterfly B(m) with a dilation of 3. Using this embedding it is shown that T(q) can be embedded into a no-wrap butterfly B(m+1) [resp. B (m+2)] with a dilation of 8 (resp. 5)