PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Listing Closed Sets of Strongly Accessible Set Systems with Applications to Data Mining
Tamas Horvath, Axel Poigne and Stefan Wrobel
Theoretical Computer Science Volume 411, Number 3, pp. 691-700, 2010.

Abstract

We study the problem of listing all closed sets of a closure operator $\sigma$ that is a partial function on the power set of some finite ground set E, i.e., $\sigma \subseteq \mathcal{F}$ with $\mathcal{F} \subseteq \mathcal{P}(E)$. 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 $O(|E| (T_\mathcal{F}+T_\sigma+|E|))$ and space $O(|E|+S_\mathcal{F}+S_\sigma)$, where $T_\mathcal{F}$, $T_\sigma$ , $S_\mathcal{F}$ , and $S_\sigma$ are the time and space complexities of checking membership in $\mathcal{F}$ and computing $\sigma$, 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.

EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Theory & Algorithms
ID Code:6143
Deposited By:Mario Boley
Deposited On:08 March 2010