Querying multi-dimensional data indexed using the Hilbert space-filling curve

Mapping to one-dimensional values and then using a one-dimensional indexing method has been proposed as a way of indexing multi-dimensional data. Most previous related work uses the Z-Order Curve but more recently the Hilbert Curve has been considered since it has superior clustering properties. Any approach, however, can only be of practical value if there are effective methods for executing range and partial match queries. This paper describes such a method for the Hilbert Curve

Using state diagrams for Hilbert curve mappings

The Hilbert Curve describes a method of mapping between one and n dimensions. Such mappings are of interest in a number of application domains including image processing and, more recently, in the indexing of multi-dimensional data. Relatively little work, however, has been devoted to techniques for mapping in more that 2 dimensions. This paper presents a technique for constructing state diagrams to facilitate mappings and is a specialization of an incomplete generic process described by Bially. Although the storage requirements for state diagrams increase exponentially with the number of dimensions, they are useful in up to about 9 dimensions

Query integrity assurance of location-based services accessing outsourced spatial databases

Abstract. Outsourcing data to third party data providers is becoming a common practice for data owners to avoid the cost of managing and maintaining databases. Meanwhile, due to the popularity of locationbased-services (LBS), the need for spatial data (e.g., gazetteers, vector data) is increasing exponentially. Consequently, we are witnessing a new trend of outsourcing spatial datasets by data collectors. Two main challenges with outsourcing datasets is to keep the data private (from the data provider) and ensure the integrity of the query result (for the clients). Unfortunately, most of the techniques proposed for privacy and integrity do not extend to spatial data in a straightforward manner. Hence, recent studies proposed various techniques to support either privacy or integrity (but not both) on spatial datasets. In this paper, for the first time, we propose a technique that can ensure both privacy and integrity for outsourced spatial data. In particular, we first use a one-way spatial transformation method based on Hilbert curves, which encrypts the spatial data before outsourcing and hence ensures its privacy. Next, by probabilistically replicating a portion of the data and encrypting it with a different encryption key, we devise a technique for the client to audit the trustworthiness of the query results. We show the applicability of our approach for both k-nearest-neighbor and spatial range queries, the building blocks of any LBS application. Finally, we evaluate the validity and performance of our algorithms with real-world datasets.