A Fast Bit-Parallel Algorithm for Computing the Subset Partial Order

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 implementations of a variant of a previously proposed algorithm which exploit bit-parallel operations on a RAM with (log N )-bit words. One is shown to have a worst-case complexity of O(N 2 log log N / log 2 N ) operations. This is the first known o(N 2 / log N ) worst case or randomized expected running time for this problem.

On the Complexity of Strongly Connected Components in Directed Hypergraphs

“…Then the transitive reduction of R is defined as in [AGU72]. The subset partial order has been well studied in the literature [YJ93,Pri95,Pri99a,Pri99b,Elm09]. It has been proved in [YJ93,Elm09] that the size of the transitive reduction of the subset partial order can be superlinear in the size of the input (F , D) (defined as |D| + S∈F |S|).…”

The Orthogonal Vectors Conjecture for Branching Programs and Formulas

“…An instructive way of viewing the OV problem is that we have a collection of n sets over [d], and wish to find two disjoint sets among them. The obvious algorithm runs in time O(n 2 • d), and log(n) factors can be shaved [Pri99]. For d < log 2 (n), stronger improvements are possible: there are folklore O(n • 2 d • d)-time and Õ(n + 2 d )-time algorithms (for a reference, see [CST17]).…”

Mining interesting itemsets

“…Analyzing the complexity, the for-loop in Lines 2-3 can be performed in O(n|X|) time using a hash table. For extracting maximal sets, the current fastest algorithm has O(N 2 lg lg N/ lg 2 N ) time complexity [79], where N is the sum of the set cardinalities. However, this approach is only of theoretical interest, since it assumes the worst case distribution.…”