Some Results on the Representative Instance in Relational Databases

Abstract

Recently, the representative instance has been proposed as a generalized concept of the pure universal relation. Let ${\bf R} = \{ \langle R_1 ,F_1 \rangle , \cdots , \langle R_n ,F_n \rangle \}$ be a database scheme, where each $R_i $ is a set of attributes and $F_i $ is a set of functional dependencies over $R_i $. ${\bf R}$ is said to be consistent if the representative instance of every database $I = \{ r_1 , \cdots ,r_n \}$ of ${\bf R}$ satisfies $F (= F_1 \cup \cdots \cup F_n )$, that is, if whenever each relation $r_i $ satisfies its own functional dependencies $F_i $, the representative instance satisfies all the functional dependencies F. In this paper, we present the following two results, which are generalizations of the previous results by [Sag2]. (1) It can be determined in $O(n|F|\|F\|)$ time whether ${\bf R}$ is consistent, where $|F|$ is the number of functional dependencies in F and $||F||$ is the size of the description of F. (A polynomial time algorithm for determining whether ${\bf R}$ is consistent is presented, independently, in [GY].) (2) Suppose that ${\bf R}$ is consistent. Given a subset V of $R_1 \cup \cdots \cup R_n $, we can construct in $O(n|F|\|F\|)$ time a relational expression such that (a) its value is the total projection of the representative instance onto V for every database of ${\bf R}$, (b) it consists of projection, extension join, and union, and (c) it contains neither a redundant union nor a redundant join.