Regular Article
Embeddings among Meshes and Tori

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Abstract

Given a d-dimensional mesh or torus G and a c-dimensional mesh or torus H of the same size, we study the problem of embedding G in H to minimize the dilation cost. We construct embeddings for increasing dimension cases (d < c) in which the shapes of G and H satisfy the condition of expansion, and for lowering dimension cases (d >c) in which the shapes of G and H satisfy the condition of reduction. We then use these results to construct embeddings for the cases in which G and H are square. The embeddings for square meshes and square tori are optimal for increasing dimension cases in which c is divisible by d, and optimal to within a constant multiplicative factor for fixed values of d and c for all lowering dimension cases.

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