Embedding Complete Binary Trees into Star Networks

Abs t rac t . Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedding is the maximum distance between f(u), f(v) taken over all edges (u, v) of G. Low dilation embeddings of binary trees into star graphs correspond to efficient simulations of parallel algorithms that use the binary tree topology, on parallel computers interconnected with star networks. First, we give a construction of embeddings of dilation 1 of complete binary trees into n-dimensional star graphs. These trees are subgraphs of star graphs. Their height is fl(n log n), which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 28 and 26 + 1, for 8 > 1, into star graphs are then given. The use of larger dilation allows embeddings of trees of greater height into star graphs. For example, the difference of the heights of the trees embedded with dilation 2 and 1 is greater than n/2. All these constructions can be modified to yield embeddings of dilation 1, and 26, for ~ > 1, of complete binary trees into pancake graphs. Our results show that massively parallel computers interconnected with star networks are well suited for efficient simulations of parallel algorithms with complete binary tree topology.