Abstract
We present a method for decomposing a hypergraph with certain regularities into smaller hypergraphs, in a “direct product”-like fashion. By applying this to the set of all canonical covers of a given set of functional dependencies, we obtain more efficient methods for solving several optimization problems in database design. These include finding one or all “optimal” covers w.r.t. different criteria, which can help to synthesize better decompositions, and to reduce the cost of constraint checking. As a central step we investigate how the hypergraph of all canonical covers can be computed efficiently. Our results suggest that decomposed representations of this hypergraph are usually small and can be obtained rather quickly, even if the number of canonical covers is huge.
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References
Armstrong, W.W.: Dependency structures of data base relationships. In: IFIP Congress, pp. 580–583 (1974)
Berge, C.: Hypergraphs: Combinatorics of Finite Sets. Elsevier Science Pub. Co. (1989)
Biskup, J., Dayal, U., Bernstein, P.A.: Synthesizing independent database schemas. In: SIGMOD Conference, pp. 143–151 (1979)
Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)
Gottlob, G.: On the size of nonredundant FD-covers. Inf. Process. Lett. 24(6), 355–360 (1987)
Gottlob, G., Pichler, R., Wei, F.: Tractable database design through bounded treewidth. In: Vansummeren, S. (ed.) PODS, pp. 124–133. ACM (2006)
Habib, M., de Montgolfier, F., Paul, C.: A simple linear-time modular decomposition algorithm for graphs, using order extension. In: Hagerup, T., Katajainen, J. (eds.) SWAT, Lecture Notes in Computer Science, vol. 3111, pp. 187–198. Springer (2004)
Köehler, H.: Finding faithful Boyce–Codd normal form decompositions. In: Cheng, S.W., Poon, C.K. (eds.) AAIM, Lecture Notes in Computer Science, vol. 4041, pp. 102–113. Springer (2006)
Köhler, H.: Autonomous sets—a method for hypergraph decomposition with applications in database theory. In: Hartmann, S., Kern-Isberner, G. (eds.) FoIKS, Lecture Notes in Computer Science, vol. 4932, pp. 78–95. Springer (2008)
Lechtenbörger, J.: Computing unique canonical covers for simple FDs via transitive reduction. Inf. Process. Lett. 92(4), 169–174 (2004)
Levene, M., Loizou, G.: A Guided Tour of Relational Databases and Beyond. Springer (1999)
Lucchesi, C.L., Osborn, S.L.: Candidate keys for relations. J. Comput. Syst. Sci. 17(2), 270–279 (1978)
Maier, D.: Minimum covers in the relational database model. J. ACM 27(4), 664–674 (1980)
Maier, D.: The Theory of Relational Databases. Computer Science Press (1983)
Mannila, H., Räihä, K.J.: The Design of Relational Databases. Addison-Wesley (1987)
Osborn, S.L.: Testing for existence of a covering Boyce–Codd normal form. Inf. Process. Lett. 8(1), 11–14 (1979)
Saiedian, H., Spencer, T.: An efficient algorithm to compute the candidate keys of a relational database schema. Comput. J. 39(2), 124–132 (1996)
Zaniolo, C.: A new normal form for the design of relational database schemata. ACM Trans. Database Syst. 7(3), 489–499 (1982). doi:10.1145/319732.319749
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Köhler, H. Autonomous sets for the hypergraph of all canonical covers. Ann Math Artif Intell 63, 257–285 (2011). https://doi.org/10.1007/s10472-012-9276-z
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DOI: https://doi.org/10.1007/s10472-012-9276-z