Laws for Rewriting Queries Containing Division Operators

Relational division, also known as small divide, is derived operator of the relational algebra that realizes many-to-one set containment test, where a set is represented as a group of tuples: Small divide discovers which sets in a dividend relation contain all elements of the set stored in a divisor relation. The great divide operator extends small divide by realizing many-to-many set containment tests. It is also similar to the set containment join operator for schemas that are not in first normal form. Neither small nor great divide has been implemented in commercial relational database systems although the operators solve important problems and many efficient algorithms for them exist. We present algebraic laws that allow rewriting expressions containing small or great divide, illustrate their importance for query optimization, and discuss the use of great divide for frequent itemset discovery, an important data mining primitive. A recent theoretic result shows that small divide must be implemented by special purpose algorithms and not be simulated by pure relational algebra expressions to achieve efficiency. Consequently, an efficient implementation requires that the optimizer treats small divide as a first-class operator and possesses powerful algebraic laws for query rewriting.