Database States Exhibiting Tree Projection Necessity

Abstract

Consider determining whether the join of n relations with relation schemas $\textbf{R}_1, \ldots , \textbf{R}_n$ is non-empty. Abstracting the essence of many query processors, we may form new m intermediate relations, with relation schema $\textbf{S}_1, \ldots , \textbf{S}_m$, then repeatedly use in-place semijoins to obtain an overall pair-wise compatible database state, and finally test that this state is non-empty. Recently, it was proved that if the relation schemas $\textbf{S}_1, \ldots , \textbf{S}_m$ guarantee a correct solution, i.e., for any initial database state, then there must exist an acyclic database schema (tree schema), say over relation schemas $\textbf{R}_1, \ldots , \textbf{R}_n, \textbf{U}_1, \ldots , \textbf{U}_q$ such that each relation schema $\textbf{U}_i$ is a subset of some relation schema $\textbf{S}_j$, $1\le i \le q$, $1\le j \le m$. Such an acyclic database schema is called a tree projection of $\textbf{R}_1, \ldots , \textbf{R}_n$, $\textbf{S}_1, \ldots , \textbf{S}_m$ with respect to (w.r.t.) $\textbf{R}_1, \ldots , \textbf{R}_n$. Suppose such a tree projection does not exist. The proof provides no mechanism, except for exhaustive search, to exhibit an initial database state over which the non-emptiness problem is solved incorrectly. Constructing such a database state is interesting combinatorially and it may also prove useful in testing query processors. We construct such a database state for two classes of cyclic database schemas: Arings and Cliques. Constructing such database states for arbitrary cyclic database schemas remains an open problem.