Abstract

A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worst-case time O(N 2 = log N ). This paper gives a variant implementation of a previously proposed algorithm which is shown to have a worst-case complexity of O(N 2 (log log N) 2 = log 2 N) operations on a RAM with \Theta(log N) bit words. This is the first known o(N 2 = log N) worst-case running time. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. In [5] we presented algorithms for finding those sets in F that have no subset in F , and obtained a fast algorithm for the important special case when all sets in F are small. A particular implementation was later shown [6] to have worst-cas...

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