Efficient Generation of Combinatorial Families

In this paper, we propose a unifying framework for efficient generation of combinatorial objects by giving recursive definition of an abstract combinatorial class. The definition can be instantiated to an array of specific combinatorial classes by specifying the framework parameters appropriately. Our framework defines a rooted spanning tree over a representative graph of that abstract combinatorial class where exhaustive generation can be done by a depth-first traversal. As an illustration, we show an instantiation for the combinatorial class of combinations as well as give a novel constant time generation algorithm for them.