A Uniform Framework for Collection Types

We present a new algebra for complex database objects based on monoids and monoid homomorphisms. The object types supported in our algebra can be any nested composition of collection types, which include lists, sets, multisets, vectors, and matrices. We deene a new calculus equivalent to this algebra, called monoid comprehensions, that captures operations involving diverse collection types in declarative form. We present a normalization algorithm that reduces any expression in our algebra to a canonical form which, when evaluated, generates very few intermediate data structures. This algorithm generalizes some well-known algebraic optimization techniques and heuristics used in relational query optimization. In addition, we demonstrate the modeling power of this language by capturing physical storage structures and algorithms, such as merge join, hash join, and partitioned hash join. Finally, we extend this algebra by incorporating object-oriented features, such as object identity, and destructive updates.