On the Orthographic Dimension of Constraint Databases
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 O 1,..., O n, such that O = O 1×...× O n. 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 efficient optimization mechanism, very easy to use in practical applications.
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- BL98.M. Benedikt and L. Libkin. Safe constraint queries. In Proc. ACM Symp. on Principles of Database Systems, 1998.Google Scholar
- CGK96.J. Chomicki, D.Q. Goldin, and G. Kuper. Variable Independence and Aggregation Closure. In Proc. ACM Symp. on Principles of Database Systems, pages 40–48, 1996.Google Scholar
- GK97.S. Grumbach and G. Kuper. Tractable recursion over geometric data. In International Conference on Constraint Programming, 1997.Google Scholar
- GO97.Jacod E. Goodman and Joseph O’Rourke. Handbook of Discrete and Computational Geometry. CRC Press, 1997.Google Scholar
- GRS98a.S. Grumbach, P. Rigaux, and L. Segoufin. Spatio-Temporal Data Handling with Constraints. In Proc. Intl. Symp. on Geographic Information Systems, 1998.Google Scholar
- GRS98b.S. Grumbach, P. Rigaux, and L. Segoufin. The DEDALE System for Complex Spatial Queries. In Proc. ACM SIGMOD Symp. on the Management of Data, 1998.Google Scholar
- GST94.S. Grumbach, J. Su, and C. Tollu. Linear constraint query languages: Expressive power and complexity. In D. Leivant, editor, Logic and Computational Complexity, Indianapolis, 1994. Springer Verlag. LNCS 960.Google Scholar
- KG94.P. Kanellakis and D. Goldin. Constraint programming and database query languages. In Manuscript, 1994.Google Scholar
- KKR90.P. Kanellakis, G Kuper, and P. Revesz. Constraint query languages. In Proc. 9th ACM Symp. on Principles of Database Systems, pages 299–313, Nashville, 1990.Google Scholar
- KPV95.B. Kuijpers, J. Paredaens, and J. Van den Bussche. Lossless representation of topological spatial data. In M.J. Egenhofer and J. R. Herring, editors, Advances in Spatial Databases, 4th Int. Symp., SSD’95, pages 1–13. Springer, 1995.Google Scholar
- Mai83.D. Maier. The Theory of Relational Databases. Computer Science Press, 1983.Google Scholar
- PVV94.J. Paredaens, J. Van den Bussche, and D. Van Gucht. Towards a theory of spatial database queries. In Proc. 13th ACM Symp. on Principles of Database Systems, pages 279–288, 1994.Google Scholar
- Sch86.A. Schrijver. Theory of Linear and Integer Programming. Wiley, 1986.Google Scholar
- Ull88.J.D. Ullman. Database and Knowledge Base Systems. Computer Science Press, 1988.Google Scholar
- VGV96.L. Vandeurzen, M. Gyssens, and D. Van Gucht. On query languages for linear queries definable with polynomial constraints. In Proc. Second Int. Conf. on Principles and Practice of Constraint Programming, pages 468–481. LNCS 1118, August 1996.Google Scholar
- VMM94.L. Van den Dries, A. Macintyre, and D. Marker. The elementary theory of restricted analytic fields with exponentiation. Annals of Mathematics, 1994.Google Scholar