Embedding hyperpyramids into hypercubes

A &k, d ) hyperpyramid is a level structure of k hypercubes, where the hypercube at level i is of dimension id, and a node at level i 1 is connected to every node in a &dimensional subcube at level i, except for the leaf level k. Hyperpyramids contain pyramids as propy subgraphs. We show that a hyperpyramid P(k, d ) can be embedded in a hypercube with minimal expansion and dilation = d. The congestion is bounded from above by r(2‘ l)/dl and from below by 1 + r(2‘ d)/(kd + 1)1. We also present embeddings of a hyperpyramid &k, d ) together with 2d 2 hyperpyramids &k 1, d ) such that only one hypercube node is unused. The dilation of the embedding is d + 1, with a congestion of O(2‘). A corollary is that a complete mary tree can be embedded in a hypercube with dilation = max(2, [los, “1) and expansion = (2krlo02n1+1 )(n l)/(d+‘ 1).