A Novel Construction Method for n-Dimensional Hilbert Space-Filling Curves

Abstract

SUMMARYWe develop a novel construction method for n-dimensional Hilbert space-filling curves. The construction method includes four steps: block allocation, Gray permutation, coordinate transformation and recursive construction. We use the tensor product theory to formulate the method. An n-dimensional Hilbert space-filling curve of 2 r elements on each dimension is specified as a permutation which rearranges 2 rn data elements stored in the row major order as in C language or the column major order as in FORTRAN language to the order of traversing an n-dimensional Hilbert space-filling curve. The tensor product formulation of n-dimensional Hilbert space-filling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas of n-dimensional Hilbert space-filling curves. The tensor product formulas are directly translated into computer programs which can be used in various applications. The process of program generation is explained in the paper. key words: block recursive algorithm, tensor product, n-dimensional Hilbert space-filling curve, Gray permutation