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

Abstract

We develop a novel construction method forn-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. Ann-dimensional Hilbert space-filling curve of 2r elements on each dimension is specified as a permutation which rearranges 2rn 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 ann-dimensional Hilbert space-filling curve. The tensor product formulation ofn-dimensional Hilbert space-filling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas ofn-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.