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

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.