Recognition and top-down generation of β-acyclic database schemes

Abstract

Database schemes can be viewed as hypergraphs with individual relation schemes corresponding to the edges of a hypergraph. Under this setting, a new class of "acyclic" database schemes was recently introduced and was shown to have a claim to a number of desirable properties. However, unlike the case of ordinary undirected graphs, there are several unequivalent notions of acyclicity of hypergraphs. Of special interest among these are α-, β-, and γ-, degrees of acyclicity, each characterizing an equivalence class of desirable properties for database schemes, represented as hypergraphs. In this paper, two complementary approaches to designing β-acyclic database schemes have been presented. For the first part, a new notion called "independent cycle" is introduced. Based on this, a criterion for β-acyclicity is developed and is shown equivalent to the existing definitions of β-acyclicity. From this and the concept of the dual of a hypergraph, an efficient algorithm for testing β-acyclicity is developed. As for the second part, a procedure is evolved for top-down generation of β-acyclic schemes and its correctness is established. Finally, extensions and applications of ideas are described.