Efficient approach to embed binary trees in 3-D rectangular arrays

Abstract

The complete binary tree has long been known to support important applications. This paper presents an efficient embedding of a complete binary tree in a 3-D rectangular array. The array hosting the binary tree is called the host, and the binary tree is referred to as the guest. The scheme is modular and high level trees are made of low level trees inductively. It is shown that all PEs (except one) of the host array may be utilised in embedding a k-level complete binary tree. The dilation (maximum length of an edge in the guest graph in terms of the number of edges in the host graph) is kept as low as possible by using building blocks, which allow tree embedding, with dilation of two at most occurring between leaf nodes and their parents where the message traffic is minimum. Upperbounds on propagation delay are reduced from O(2k/2 to O(2k/3) as a result of the use of the third dimension. Results are compared with those dealing with 2-D layouts. As discussed, these bounds are the best known to date for structures with practically implementable dimensionality.