On the complexity of join dependencies

• Authors: Marc Gyssens
• Venue: ACM Transactions on Database Systems (TODS)
• Year: 1986

In [10] a method is proposed for decomposing join dependencies (jds) in a relational database using the notion of a hinge. This method was subsequently studied in [11] and [12]. We show how the technique of decomposition can be used to make integrity checking more efficient. It turns out that it is important to find a decomposition that minimizes the number of edges of its largest element. We show that the decompositions obtained with the method described in [10] are optimal in this respect. This minimality criterion leads to the definition of the degree of cyclicity, which allows us to classify jds and leads to the notion of n-cyclicity, of which acyclicity is a special case for n = 2. We then show that, for a fixed value of n (which may be greater than 2). integrity checking can be performed in polynomial time provided we restrict ourselves to n-cyclic jds. Finally, we generalize a well-known characterization for acyclic jds by proving that n-cyclicity is equivalent to “n-wise consistency implies global consistency.” As a consequence, consistency checking can be performed in polynomial time if we restrict ourselves to n-cyclic jds, for a tired value of n, not necessarily equal to 2.