The Subset Partial Order: Computing and Combinatorics

Given a family F of k sets with cardinalities s1, s2, . . . , sk and N = ∑ k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O( ∑ si≤B 2i + N/ logN · ∑ si>B log (2si/B)), where B = log (N/ log N). This implies a simpler proof to the O(N/ log N) bound concluded in [2]. We also give an algorithm that computes the subset graph for any family of sets F . Our algorithm requires O(nk/ log k) time and space on a pointer machine, where n is the number of domain elements. When F is dense, i.e. N = Θ(nk), the algorithm requires O(N/ log N) time and space. We give a construction for a dense family whose subset graph is of size Θ(N/ log N), indicating the optimality of our algorithm for dense families. The subset graph can be dynamically maintained when F undergoes set insertion and deletion in O(nk/ log k) time per update (that is sub-linear in N for the case of dense families). If we assume words of b ≤ k bits, allow bits to be packed in words, and use bitwise operations, the above running time and space requirements can be reduced by a factor of b log (k/b + 1)/ log k and b log (k/b + 1)/ log k respectively.