On the orthographic dimension of constraint databases

Abstract

One of the most important advantages of constraint databases is their ability to represent and to manipulate data in arbitrary dimension within a uniform framework. Although the complexity of querying such databases by standard means such as first-order queries has been shown to be tractable for reasonable constraints (e.g. polynomial), it depends badly (roughly speaking exponentially) upon the dimension of the data. A precise analysis of the trade-off between the dimension of the input data and the complexity of the queries reveals that the complexity strongly depends upon the use the input makes of its dimensions. We introduce the concept of orthographic dimension, which, for a convex object O, corresponds to the dimension of the (component) objects O1,…, On, such that O = O1 ×· · ·×On. We study properties of databases with bounded orthographic dimension in a general setting of o-minimal structures, and provide a syntactic characterization of first-order orthographic dimension preserving queries. The main result of the paper concerns linear constraint databases. We prove that orthographic dimension preserving Boolean combination of conjunctive queries can be evaluated independently of the global dimension, with operators limited to the orthographic dimension, in parallel on the components. This results in an extremely e_cient optimization mechanism, very easy to use in practical applications.