Quad-splitting algorithm for a window query on a Hilbert curve

Space-filling curves, particularly, Hilbert curves, have been extensively used to maintain spatial locality of multi-dimensional data in a wide variety of applications. A window query is an important query operation in spatial (image) databases. Given a Hilbert curve, a window query reports its corresponding orders without the need to decode all the points inside this window into the corresponding Hilbert orders. Given a query window of size p times q on a Hilbert curve of size T times T, Chung et al. have proposed an algorithm for decomposing a window into the corresponding Hilbert orders, which needs O(n log T) time, where n = max (p,q). By employing the properties of Hilbert curves, the authors present an efficient algorithm, named as Quad-Splitting, for decomposing a window into the corresponding Hilbert orders on a Hilbert curve without individual sorting and merging steps. Although the proposed algorithm also takes O(n log T) time, it does not perform individual sorting and merging steps which are needed in Chung et al.´s algorithm. Therefore experimental results show that the Quad-Splitting algorithm outperforms Chung et al.´s algorithm.