# Embedding grids into hypercubes

Simulation And Embedding Of Parallel Networks Invited Presentation

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## Abstract

We consider efficient simulations of mesh connected networks by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that has at least as many points as G, called

*the optimal hypercube*for G. In order to minimize simulation time we derive embeddings, i.e. one-to-one mappings of points in G to points in the hypercube, which minimize*dilation*, i.e. the maximum distance in the hypercube between images of adjacent points of G. Our results are:- (1)
There is a dilation 2 embedding of the [m×k] grid into its optimal hypercube, under conditions described in Theorem 2.1.

- (2)
For any k<d, there is a dilation k+1 embedding of a [a

_{1}×a_{2}× ... ×a_{d}] grid into its optimal hypercube, under conditions described in Theorem 3.1. - (3)
A lower bound on dilation in embedding multi-dimensional meshes into their optimal hypercube as described in Theorem 3.2.

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© Springer-Verlag Berlin Heidelberg 1988