On Computing the Subset Graph of a Collection of Sets

Let a given collection of sets have size N measured by the sum of the cardinalities. Yellin and Jutla presented an algorithm which constructed the partial order induced by the subset relation (a “subset graph”) in a claimed O(N²/log N) operations over a dictionary ADT, and exhibited a collection whose subset graph had Θ(N²/log² N) edges. This paper first establishes a matching upper bound on the number of edges in a subset graph. It also presents a finer analysis of the algorithm, which confirms the claimed upper bound and shows it to be tight. A simple implementation requiring O(1) bit-parallel operations per ADT operation is presented, along with a variant of the algorithm with an implementation requiring O(N²/log N) RAM operations.