Polynomial-Delay and Polynomial-Space Algorithms for Mining Closed Sequences, Graphs, and Pictures in Accessible Set Systems

Abstract

In this paper, we study efficient closed pattern mining in a general framework of set systems, which are families of subsets ordered by set-inclusion with a certain structure, proposed by Boley, Horváth, Poigné, Wrobel (PKDD'07 and MLG'07). By modeling semi-structured data such as sequences, graphs, and pictures in a set system, we systematically study efficient mining of closed patterns. For a class of accessible set systems with a tree-like structure, we present an efficient depth-first search algorithm that finds all closed sets in accessible set systems without duplicates in polynomial-delay and polynomial-space w.r.t. the total input size using efficient oracles for the membership test and the closure computation for the pattern class. From the above results, we show that the closed pattern mining problems are efficiently solvable both in time and space for the following classes: convex hulls, picture patterns in 2-D planes, maximal bi-cliques, closed relational graphs, closed patterns for rigid motifs with wildcards.

Para Miner: a generic pattern mining algorithm for multi-core architectures

“…In order to provide a generic definition for closed patterns we extend the work of Boley et al (2010) and (Arimura and Uno, 2009). We rely on a homogeneous representation of the patterns and of the dataset using sets.…”

“…Arimura and Uno (2009) have shown that it is possible to build polynomial-space enumeration strategies if the set system that consists in the ground set and the set of patterns satisfies the property of accessibility.…”

“…Regarding genericity Uno (2009), andBoley et al (2010) have recently proposed to study the problem of closed pattern enumeration as the problem of enumerating sets in a set system. The definition of a pattern relies on a simple predicate called selection criterion and their theoretical framework can capture a broad range of pattern mining problems such as frequent itemsets, connected relational graphs, rigid sequences and others, with strong complexity guarantees (output-polynomial in time, polynomial in space).…”

“…Unlike other dataset reduction operators of the litterature, the EL-reduction is independent of the pattern mining problem considered. In order to define this operator, we restricted the framework proposed by Arimura and Uno (2009) and Boley et al (2010) to a slightly more specialized class of pattern mining problems where the patterns have to occur in a dataset. We also impose some limitations on the selection criterion, expressed through a new property called decomposability that it has to satisfy.…”

Closed Patterns and Abstraction Beyond Lattices

“…First, various mining problems have been investigated that come down to searching for closed motifs which cannot be considered, strictly speaking, as support-closed motifs, such as convex hulls of subsets of a given set of points, or sequential motifs with wildcards. Solving the problem then means defining and building the corresponding closure operator [5]. To characterize such closure operators, the authors make use of a wellknown theorem stating that in a finite lattice T there is a one-to-one correspondence between the families closed under the meet operator and the closure operators on T .…”