Entropy Bounds for Conjunctive Queries with Functional Dependencies

Abstract

We study the problem of finding the worst-case size of the result Q(D) of a fixed conjunctive query Q applied to a database D satisfying given functional dependencies. We provide a characterization of this bound in terms of entropy vectors, and in terms of finite groups. In particular, we show that an upper bound provided by [Gottlob, Lee, Valiant and Valiant, J.ACM, 2012] is tight, and that a correspondence of [Chan and Yeung, ACM TOIT, 2002] is preserved in the presence of functional dependencies. However, tightness of a weaker upper bound provided by Gottlob et al., which would have immediate applications to evaluation of join queries ([Khamis, Ngo, and Suciu, PODS, 2016]) remains open. Our result shows that the problem of computing the worst-case size bound, in the general case, is closely related to difficult problems from information theory.