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Embedding hyperpyramids into hypercubes

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2 Author(s)
Ho, C.-T. ; IBM Research Division, Almaden Research Center, 650 Hany Road, San Jose, California 95120, USA ; Johnsson, S.L.

A ̂P(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 d-dimensional subcube at level i, except for the leaf level k. Hyperpyramids contain pyramids as proper 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 ⌈(2d - 1)/d⌉ and from below by 1 + ⌈(2d - d)/(kd + 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(2d). A corollary is that a complete n-ary tree can be embedded in a hypercube with dilation = max(2,⌈log2 n⌉) and expansion = equation (n - 1)/(nk+1 - 1).

Note: The Institute of Electrical and Electronics Engineers, Incorporated is distributing this Article with permission of the International Business Machines Corporation (IBM) who is the exclusive owner. The recipient of this Article may not assign, sublicense, lease, rent or otherwise transfer, reproduce, prepare derivative works, publicly display or perform, or distribute the Article.  

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IBM Journal of Research and Development  (Volume:38 ,  Issue: 1 )