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+ LlogmJ-l. We show how to embed a 2+I -I-node complete binary tree T(q) into a (m + 1)2111+1_ node wrap-around butterfly B1l1(m +1) with a dilation of 4, and how to embed T(q) into a (m + 2)2+_node wrap-around butterfly B 1l1 (m +2) with an optimal dilation of 2. We also present an embedding of a wrap-around butterfly B 1l1 (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) witb a dilation of 8 (resp. 5).