Compact Hilbert Indices for Multi-Dimensional Data

Abstract

Space-filling curves, particularly Hilbert curves, have proven to be a powerful paradigm for maintaining spatial groupings of multi-dimensional data in a variety of application areas including database systems,data structures and distributed information systems. One significant limitation in the standard definition of Hilbert curves is the requirement that the grid size (i.e. the cardinality) in each dimension be the same. In the real world, not all dimensions are of equal size and the work-around of padding all dimensions to the size of the largest dimension wastes memory and disk space, while increasing the time spent manipulating and communicating these "inflated" values. In this paper we define a new compact Hilbert index which, maintains all the advantages of the standard Hilbert curve and permits dimension cardinalities of varying sizes. This index can be used in any application that would have previously relied on Hilbert curves but, in the case of unequal side lengths, provides a more memory efficient representation. This is particularly important in distributed applications (parallel, P2P and grid), in which not only is memory space saved but communication volume reduced