Mapping with Space Filling Surfaces

Abstract

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 2D (respectively, 3D) structure such that points that are distance d apart in the linear array are distance O (d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ) (O(d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sup> )) apart in the 2D (3D) 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> on the 2D surface are mapped to points at distance O(d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sup> ) 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.