Asymptotic properties of keys and functional dependencies in random databases

Abstract

Practical database applications give the impression that sets of constraints are rather small and that large sets are unusual and are caused by bad design decisions. Theoretical investigations, however, show that minimal constraint sets are potentially very large. Their size can be estimated to be exponential in terms of the number of attributes. The gap between observation in practice and theory results in the rejection of theoretical results. However, practice is related to average cases and is not related to worst cases. The theory used until now considered the worst-case complexity. This paper aims to develop a theory for the average-case complexity. Several probabilistic models and asymptotics of corresponding probabilities are investigated for random databases formed by independent random tuples with a common discrete distribution. Poisson approximations are studied for the distributions of some characteristics for such databases where the number of tuples is sufficiently large. We intend to prove that the exponential complexity of key sets and sets of functional dependencies is rather unusual and almost all minimal keys in a relation have a length which depends mainly on the size of the relation.