Listing closed sets of strongly accessible set systems with applications to data mining

Abstract

We study the problem of listing all closed sets of a closure operator a that is a partial function on the power set of some finite ground set E, i.e., sigma : F -> F with F subset of P(E). A very simple divide-and-conquer algorithm is analyzed that correctly solves this problem if and only if the domain of the closure operator is a strongly accessible set system. Strong accessibility is a strict relaxation of greedoids as well as of independence systems. This algorithm turns out to have delay O (vertical bar E vertical bar (T-F + T-sigma + vertical bar E vertical bar)) and space O (vertical bar E vertical bar + S-F + S-sigma), where T-F, S-F, T-sigma, and S-sigma are the time and space complexities of checking membership in F and computing a, respectively. In contrast, we show that the problem becomes intractable for accessible set systems. We relate our results to the data mining problem of listing all support-closed patterns of a dataset and show that there is a corresponding closure operator for all datasets if and only if the set system satisfies a certain confluence property

Closed Patterns and Abstraction Beyond Lattices

“…More generally, in data mining extensions are denoted as supports, and an element x of a pattern language is said support closed with respect to O whenever for any element y > x we have that ext(y) ⊂ ext(x) [6]. In other words, a support-closed element x is a maximal element of the equivalence class associated to its support ext(x).…”

“…We are then interested in which subsets F of T have support-closures with respect to any O. We connect here to the seminal result of M. Boley and co-authors [6] on confluent systems. To avoid confusion, up sets and down sets of T starting from an element x will be denoted respectively as T x and T x wherease the notations F t and F t will be used for the up sets and down sets of the subset F .…”

“…In [6], the lattice T is a powerset 2 X and a confluent system S is similar to the latter definition of confluences* except that ⊥ = ∅ belongs to S but x ∪ y is only required to belong to F when x ⊇ t and y ⊇ t for any t! = ∅.…”

“…= ∅. Proposition 7 is a straightforward adaptation of the theorem of [6] when T = 2 X , confluent systems replaces confluences*, and which prohibits to have any attribute common to all objects in O. A useful result is the following:…”

“…This means that we can use the same kind of algorithm that specializes a closed pattern, computes the support of the new pattern and closes it in the same way we do in the lattice mining case. In their paper [6] M. Boley and coauthors state in particular the necessary and sufficient conditions that the family F of a set system (F, X) has to fulfill in order to guarantee that whatever the dataset O of objects is 2 , there exists a closure operator to compute support-closed patterns. These conditions on set systems defines the property of confluence that requires a kind of local union closure on F : given three elements t, t 1 , t 2 non empty elements of F , if t 1 and t 2 are greater than or equal to t then necessarily t 1 ∪ t 2 belongs to F .…”