Embedding of Binary Trees into Hypercubes

The authors present a mathematical model of parallel computing in a hypercubical parallel computer. This is based on embedding binary trees or forests into the n-dimensional hypercube. They consider three different models corresponding to three different computing situations. First, they assume that the processing time at each level of the binary tree is arbitrary, and develop the corresponding mathematical model of an embedding of a binary tree into the hypercube. Then they assume that the processing time at each level of the binary tree is the same for all processors involved at that level, and for this they develop the mathematical model of a loop embedding of a binary tree into the hypercube. The most general case is that in which only certain neighboring levels are active. Here they assume for simplicity that only the processors corresponding to two neighboring levels are active at the same time, and correspondingly they develop the mathematical model of a level embedding of a binary tree into the hypercube to cover this case. Both loop embeddings and level embeddings allow the authors to use the same processor several times during the execution of a program.