Space filling curves and their use in the design of geometric data structures

Abstract

We are given a two-dimensional square grid of size N×N, where N:=2n andn≥0. A space filling curve (SFC) is a numbering of the cells of this grid with numbers from c+1 to c+N 2, for some c≥0. We call a SFC recursive (RSFC) if it can be recursively divided into four square RSFCs of equal size. Examples of well-known RSFCs include the Hilbert curve, the z-curve, and the Gray code. We prove several useful and interesting combinatorial properties of recursive and general SFCs. In addition, we describe grammars for generating RSFCs. For certain optimality criteria, we propose optimal RSFCs. In addition, for an optimality criterion that is important in the design of geometric data structures, we propose a RSFC that outperforms the previously known RSFCs in the worst-case.