The design and analysis of spatial data structures
The design and analysis of spatial data structures
The Grid File: An Adaptable, Symmetric Multikey File Structure
ACM Transactions on Database Systems (TODS)
An Efficient Data Structure for Lattice Operations
SIAM Journal on Computing
Multidimensional binary search trees used for associative searching
Communications of the ACM
Realm-based spatial data types: the ROSE algebra
The VLDB Journal — The International Journal on Very Large Data Bases
Tree-Based Access Methods for Spatial Databases: Implementation and Performance Evaluation
IEEE Transactions on Knowledge and Data Engineering
COSIT '97 Proceedings of the International Conference on Spatial Information Theory: A Theoretical Basis for GIS
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Formal methods based on the mathematical theory of partially ordered sets (i.e., posets) have been used in the database field for the modelling of spatial data since many years. In particular, the use of the lattice completion (or normal completion) of a poset has been shown by Kainz, Egenhofer and Greasley [13] to be a fundamental technique to build meaningful representations of spatial subdivisions. In fact, they proved that the new elements introduced by the normal completion process can (and have to) be interpreted as being the intersection of poset elements. This is fundamental, from a mathematical point of view, since it means that the lattice resulting from the normal completion is the closure of the given poset with respect to the intersection operation. In this paper we precisely clarify the limitations for the use of lattices as models for spatial subdivisions, by proving sufficient and necessary conditions. Our result gives therefore a sound theoretical basis for the use of lattices built on simplicial complexes as a data model for spatial databases.