Abstract
Minimal generators (or mingen) constitute a remarkable part of the closure space landscape since they are the antipodes of the closures, i.e., minimal sets in the underlying equivalence relation over the powerset of the ground set. As such, they appear in both theoretical and practical problem settings related to closures that stem from fields as diverging as graph theory, database design and data mining. In FCA, though, they have been almost ignored, a fact that has motivated our long-term study of the underlying structures under different perspectives. This paper is a two-fold contribution to the study of mingen families associated to a context or, equivalently, a closure space. On the one hand, it sheds light on the evolution of the family upon increases in the context attribute set (e.g., for purposes of interactive data exploration). On the other hand, it proposes a novel method for computing the mingen family that, although based on incremental lattice construction, is intended to be run in a batch mode. Theoretical and empirical evidence witnessing the potential of our approach is provided.
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References
Barbut, M., Monjardet, B.: Ordre et Classification: Algèbre et combinatoire. Hachette (1970)
Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North Holland, Amsterdam (1989)
Ganter, B.: Two basic algorithms in concept analysis. Preprint 831, Technische Hochschule, Darmstadt (1984)
Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Heidelberg (1999)
Godin, R., Missaoui, R., Alaoui, H.: Incremental Concept Formation Algorithms Based on Galois (Concept) Lattices. Computational Intelligence 11(2), 246–267 (1995)
Guigues, J.L., Duquenne, V.: Familles minimales d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sciences Humaines 95, 5–18 (1986)
Maier, D.: The theory of Relational Databases. Computer Science Press, Rockville (1983)
Pasquier, N.: Extraction de bases pour les règles d’association à partir des itemsets fermés fréquents. In: Proceedings of the 18th INFORSID 2000, Lyon, France, pp. 56–77 (2000)
Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Discovering frequent closed itemsets for association rules. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 398–416. Springer, Heidelberg (1999)
Rouane, M.H., Nehme, K., Valtchev, P., Godin, R.: On-line maintenance of iceberg concept lattices. In: Contributions to the 12th ICCS, Huntsville (AL), p. 14. Shaker Verlag, Aachen (2004)
Stumme, G., Taouil, R., Bastide, Y., Pasquier, N., Lakhal, L.: Computing Iceberg Concept Lattices with Titanic. Data and Knowledge Engineering 42(2), 189–222 (2002)
Valtchev, P., Duquenne, V.: Implication-based methods for the merge of factor concept lattices (32 p.). Submitted to Discrete Applied Mathematics
Valtchev, P., Grosser, D., Roume, C., Rouane Hacene, M.: GALICIA: an open platform for lattices. In: Ganter, B., de Moor, A. (eds.) Using Conceptual Structures: Contributions to 11th Intl. Conference on Conceptual Structures (ICCS 2003), Aachen (DE), pp. 241–254. Shaker Verlag, Aachen (2003)
Valtchev, P., Rouane Hacene, M., Missaoui, R.: A generic scheme for the design of efficient on-line algorithms for lattices. In: de Moor, A., Lex, W., Ganter, B. (eds.) ICCS 2003. LNCS, vol. 2746, pp. 282–295. Springer, Heidelberg (2003)
Valtchev, P., Missaoui, R.: Building concept (Galois) lattices from parts: generalizing the incremental methods. In: Delugach, H.S., Stumme, G. (eds.) ICCS 2001. LNCS (LNAI), vol. 2120, pp. 290–303. Springer, Heidelberg (2001)
Valtchev, P., Missaoui, R., Godin, R.: Formal Concept Analysis for Knowledge Discovery and Data Mining: The New Challenges. In: Eklund, P. (ed.) ICFCA 2004. LNCS (LNAI), vol. 2961, pp. 352–371. Springer, Heidelberg (2004)
Valtchev, P., Missaoui, R., Godin, R., Meridji, M.: Generating Frequent Itemsets Incrementally: Two Novel Approaches Based on Galois Lattice Theory. Journal of Experimental & Theoretical Artificial Intelligence 14(2-3), 115–142 (2002)
Wille, R.: Restructuring lattice theory: An approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered sets, pp. 445–470. Reidel, Dordrecht (1982)
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Nehmé, K., Valtchev, P., Rouane, M.H., Godin, R. (2005). On Computing the Minimal Generator Family for Concept Lattices and Icebergs. In: Ganter, B., Godin, R. (eds) Formal Concept Analysis. ICFCA 2005. Lecture Notes in Computer Science(), vol 3403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32262-7_13
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DOI: https://doi.org/10.1007/978-3-540-32262-7_13
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