A Closed-Form Algorithm for Converting Hilbert Space-Filling Curve Indices∗

We use the tensor product theory to formulate a closed-form algorithm for converting Hilbert space-filling curve indices of individual points. A twodimensional Hilbert space-filling curve is specified as a permutation which rearranges two-dimensional 2n×2n 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 a twodimensional Hilbert space-filling curve. The closedform algorithm converts the row-major index or the column index of a single point to the index of Hilbert space-filling curve order. The time complexity of the closed-form algorithm is a function of the length of the binary representation of the index and its space complexity is bounded by a constant. In addition, the closed-form tensor product formula can be directly translated into computer programs which can be used in various applications such as image compression. The process of program generation is explained in the paper.