A Novel Construction Method for n-Dimensional Hilbert Space-Filling Curves
We 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-tilling 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.
n-dimensional hilbert space-filling curve gray permutation tensor product block recursive algorithm