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.

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