Tensor Product Formulation for Hilbert Space-Filling Curves

We present a tensor product formulation for Hilbert space-filling curves. Both recursive and iterative formulas are expressed in the paper. We view a Hilbert space-filling curve as a permutation which maps twodimensional n n data elements stored in the row major or column major order to the order of traversing a Hilbert space-filling curve. The tensor product formula of Hilbert space-filling curves uses several permutation operations: stride permutation, radix-2 Gray permutation, transposition, and anti-diagonal transposition. The iterative tensor product formula can be manipulated to obtain the inverse Hilbert permutation. Also, the formulas are directly translated into computer programs which can be used in various applications including R-tree indexing, image processing, and process allocation, etc.