Hierarchical optimization of linear constraint processing

We consider the problem of solving a large number of simple systems of constraints. This problem occurs in the context of databases with very large collections of data while the constraints are over a small number of variables. The methodology we develop is based on a hierarchical evaluation of the constraints which are rst simpliied, and replaced by constraints approximating the initial ones. We focus on systems of linear constraints over the reals, which model spatial objects, and consider both geometric and topological approximations, deened with very simple constraints. We show that these constraints can be used either to solve the initial systems, or at least to lter out unsatissable systems. More generally, we consider the manipulation of the spatial objects with rst-order queries. We show how the queries can be evaluated by taking advantage of the approximations. In general, it is undecidable if a query can be completely answered on approximated data. The main contribution of the paper is the development of a set of rewriting rules, that allow the transformation of queries into equivalent ones that make use of the approximation. As usual with optimization problems, an optimal solution is out of reach, but the rules are optimal for individual operations of relational algebra. The eeciency of the technique is important for the range of applications considered, where the probability of ltering from the approximated data is very high. The technique is illustrated on a practical example.