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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
Citation:
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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