Faster Algorithms for Constructing a Galois Lattice, Enumerating All Maximal Bipartite Cliques and Closed Frequent Sets

In this paper, we give a fast algorithm for constructing a Galois lattice of a binary relation. When the binary relation is represented as a bipartite graph, each vertex of the lattice (called a concept) corresponds to a maximal bipartite clique of the bipartite graph. Thus, our algorithm also enumerates all maximal bipartite cliques. Further, our algorithm can be naturally modified to compute only large concepts that are known as closed frequent sets in data mining. The running time of our algorithm depends on the lattice structure and is faster than all other existing algorithms for these problems. Let B denote the set of all concepts, and L =< B,≺> be the corresponding lattice. For a concept C ∈ B, a descendant D = (ext(D), int(D)) of C is called an upper descendant of C if there exists i ∈ int(D) such that for any descendant E ≺ C with i ∈ int(E), ext(E) ⊆ ext(D). Denote the set of upper descendants of C by UC . For most of concepts, UC consists of all successors of C only. The running time of our algorithm is O( ∑ C∈B ∑ a∈ext(C) |{D ∈ UG(C) : a ∈ ext(D)}|) where G(C) is a predecessor of C.