Size Bounds for Factorised Representations of Query Results

We study two succinct representation systems for relational data based on relational algebra expressions with unions, Cartesian products, and singleton relations: f-representations, which employ algebraic factorisation using distributivity of product over union, and d-representations, which are f-representations where further succinctness is brought by explicit sharing of repeated subexpressions. In particular we study such representations for results of conjunctive queries. We derive tight asymptotic bounds for representation sizes and present algorithms to compute representations within these bounds. We compare the succinctness of f-representations and d-representations for results of equi-join queries, and relate them to fractional edge covers and fractional hypertree decompositions of the query hypergraph. Recent work showed that f-representations can significantly boost the performance of query evaluation in centralised and distributed settings and of machine learning tasks.