Linear Constraint Databases Research Articles

A representation independent language for planar spatial databases with Euclidean distance

Linear constraint databases and query languages are appropriate for spatial database applications. Not only is the data model suitable for representing a large portion of spatial data such as in GIS systems, but there also exist efficient algorithms for the core operations in the query languages. An important limitation of linear constraints, however, is that they cannot model constructs such as Euclidean distance; extending such languages to include such constructs, without obtaining the full power of polynomial constraints has proven to be quite difficult. One approach to this problem, by Kuijpers, Kuper, Paredaens, and Vandeurzen, used the notion of Euclidean constructions with ruler and compass as the basis for a first order query language. While their language had the desired expressive power, the semantics are not really natural, due to its use of an ad hoc encoding. In this paper, we define a language over a similar class of databases, with more natural semantics. We show that this language captures a natural subclass, the representation independent queries of the first order language of Kuijpers, Kuper, Paredaens, and Vandeurzen.

Uniform generation in spatial constraint databases and applications

We study the efficient approximation of queries in linear constraint databases using sampling techniques. We define the notion of an almost uniform generator for a generalized relation and extend the classical generator of Dyer, Frieze and Kannan for convex sets to the union and the projection of relations. For the intersection and the difference, we give sufficient conditions for the existence of such generators. We show how such generators give relative estimations of the volume and approximations of generalized relations as the composition of convex hulls obtained from the samples.

Variable independence in constraint databases

In this paper, we study constraint databases with variable independence conditions (vics). Such databases occur naturally in the context of temporal and spatiotemporal database applications. Using computational geometry techniques, we show that variable independence is decidable for linear constraint databases. We also present a set of rules for inferring vics in relational algebra expressions. Using vics, we define a subset of relational algebra that is closed under restricted aggregation.

Aggregate Operators in Constraint Query Languages

We investigate the problem of how to extend constraint query languages with aggregate operators. We deal with standard relational aggregation, and also with aggregates specific to spatial data, such as volume. We study several approaches, including the addition of a new class of approximate aggregate operators which allow an error tolerance in the computation. We show how techniques of M. Karpinski and A. Macintyre (in “Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht,” Springer Lecture Notes on Computer Science 1261, pp. 162–173, Springer-Verlag, Berlin, 1997) and P. Koiran (in “FOCS '95,” pp. 134–141) based on VC-dimension can be used to give languages with approximation operators, but also show that these languages have a number of shortcomings. We then give a set of results showing that it is impossible to get constraint-based languages that admit definable aggregation operators, both for exact operators and for approximate ones. These results are quite robust, in that they show that closure under aggregation is problematic even when the class of functions permitted in constraints is expanded. This motivates a different approach to the aggregation problem. We introduce a language FO+Poly+Sum, which permits standard discrete aggregation operators to be applied to the outputs of range-restricted constraint queries. We show that this language has a number of attractive closure and expressivity properties, and that it can compute volumes of linear-constraint databases.

Queries with arithmetical constraints

In this paper, we study the expressive power and the complexity of first-order logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains ( N , Z , Q , and R ), and with various arithmetical operations (⩽, +, ×, etc.). We first consider the data complexity of first-order queries. We prove in particular that linear queries can be evaluated in AC 0 over finite integer databases, and in NC 1 over linear constraint databases. This improves previously known bounds. We also show that over all domains, enough arithmetic lead to arithmetical queries, therefore, showing the frontiers of constraints for database purposes. We then tackle the problem of the expressive power, with the definability of the parity and the connectivity, which are the most classical examples of queries not expressible in first-order logic over finite structures. We prove that these two queries are first-order definable in the presence of (enough) arithmetic. Nevertheless, we show that they are not definable with constraints of interest for constraint databases such as linear constraints for instance. Finally, we developed reduction techniques for queries over constraint databases, that allow us to draw conclusions with respect to their undefinability in various constraint query languages.

Toward Practical Query Evaluation for Constraint Databases

Linear constraint databases (LCDBs) extend relational databases to include linear arithmetic constraints in both relations and queries. A LCDB can also be viewed as a powerful extension of linear programming (LP) where the system of constraints is generalized to a database containing constraints and the objective function is generalized to a relational query containing constraints. Our major concern is query optimization in LCDBs. Traditional database approaches are not adequate for combination with LP technology. Instead, we propose a new query optimization approach, based on statistical estimations and iterated trials of potentially better evaluation plans. The resulting algorithms are not only effective on LCDBs, but also applicable to existing query languages. A number of specific constraint algebra algorithms are also developed: select-project-join for two relations, constraint sort-join and constraint multi-join.

The LyriC language

We propose a novel data model and its language for querying object-oriented databases where objects may hold spatial, temporal or constraint data, conceptually represented by linear equality and inequality constraints. The proposed LyriC language is designed to provide a uniform and flexible framework for diverse application realms such as (1) constraint-based design in two-, three-, or higher-dimensional space, (2) large-scale optimization and analysis, based mostly on linear programming techniques, and (3) spatial and geographic databases. LyriC extends flat constraint query languages, especially those for linear constraint databases, to structurally complex objects. The extension is based on the object-oriented paradigm, where constraints are treated as first-class objects that are organized in classes. The query language is an extension of the language XSQL, and is built around the idea of extended path expressions. Path expressions in a query traverse nested structures in one sweep. Constraints are used in a query to filter stored constraints and to create new constraint objects.