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Mapping with Space Filling Surfaces
September 2007 (vol. 18 no. 9)
pp. 1258-1269
The use of space filling curves for proximity-improving mappings is well known and has found many useful applications in parallel computing. Such curves permit a linear array to be mapped onto a 2(respectively, 3)D structure such that points distance d apart in the linear array are distance O(d^1/2) (O(d^1/3)) apart in the 2(3)D array and vice-versa. We extend the concept of space filling curves to space filling surfaces and show how these surfaces lead to mappings from 2D to 3D so that points at distance d^1/2 on the 2D surface are mapped to points at distance O(d^1/3) in the 3D volume. Three classes of surfaces, associated respectively with the Peano curve, Sierpinski carpet, and the Hilbert curve, are presented. A methodology for using these surfaces to map from 2D to 3D is developed. These results permit efficient execution of 2D computations on processors interconnected in a 3D grid. The space filling surfaces proposed by us are the first such fractal objects to be formally defined and are thus also of intrinsic interest in the context of fractal geometry.

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Index Terms:
Fractals, Hilbert curve, parallel computing, Peano curve, Sierpinski carpet, space filling curves, space filling surfaces
Masood Ahmed, Shahid Bokhari, "Mapping with Space Filling Surfaces," IEEE Transactions on Parallel and Distributed Systems, vol. 18, no. 9, pp. 1258-1269, Sept. 2007, doi:10.1109/TPDS.2007.1049
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