Performances of Galois Sub-hierarchy-building Algorithms

LNAI est une "Sublibrary de LNCS"International audienceThe Galois Sub-hierarchy (GSH) is a polynomial-size repre- sentation of a concept lattice which has been applied to several fields, such as software engineering and linguistics. In this paper, we analyze the performances, in terms of computation time, of three GSH-building algorithms with very different algorithmic strategies: Ares, Ceres and Pluton. We use Java and C++ as imple- mentation languages and Galicia as our development platform. Our results show that implementations in C++ are significantly faster, and that in most cases Pluton is the best algorithm

Recognizing Chordal-Bipartite Probe Graphs

A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have been studied previously on other families of graphs, such as interval graphs and chordal graphs. In this paper, we give a characterization of chordal-bipartite probe graphs, in the case of a fixed given partition of the vertices into probes and nonprobes. Our results are obtained by solving first the more general case without assuming that N is a stable set, and then this can be applied to the more specific case. Our characterization uses an edge elimination ordering which also implies a polynomial time recognition algorithm for the class. This research was conducted in the context of a France-Israel Binational project, while the French team visited Haifa in March 2007

Data Mining (une approche par les graphes)

CLERMONT FD-BCIU Sci.et Tech. (630142101) / SudocSudocFranceF

Data Mining : using a graph approach

We propose a new approach, which uses graphs, for two data analysis problems. Phylogeny: We use the threshold family of graphs associated with a dissimilarity to define a new distance, more general than an additive tree distance, which we call triangulated distance. We introduce the new concept of maximal sub-triangulation, consistent with the fact that phylogenetic values tend to be too low. We then propose an algorithm ajusting data to a triangulated distance, by sub-triangulating associated graphs. Formal Concept Analysis: We encode a finite binary relation R and its Concept Lattice L(R) by an undirected co-bipartite graph G(R). We show that there is a one-to-one correspondence between the elements of L(R) and the minimal separators of G(R), and a one-to-one correspondence between the maximal paths in L(R) and the minimal triangulations of G(R). Thus, algorithmic processes for G(R) leads us to correspondind processes for L(R). For instance, polynomial-sized lattices can be obtained from L(R), by embeding G(R) in a weakley triangulated graph. Next, we show that there is a domination partial ordering on maximal clique modules of G(R). This domination is inherited from an element to another during the traversal of a maximal path of L(R). We deal dynamically with domination relationships using a data structure we called domination table. We use this table for two algorithmic problems: - Maintaining a Galois sub-hierarchy, in the context of inheritance for object-oriented languages; - Efficient Generation of a Concept Lattice.Nous abordons, par une modélisation à base de graphes, deux problèmes de structuration de données. Phylogénie: Nous utilisons la famille de graphes associée à une dissimilarité pour définir la notion nouvelle de distance triangulée, plus générale qu'une distance additive d'arbre. Nous proposons un algorithme d'ajustement de données à une distance triangulée par triangulation des graphes associés. Nous introduisons pour cela le concept nouveau de sous-triangulation maximale, afin de prendre en compte la sous-évaluation intrinsèque des données phylogénétiques. Nous procédons ensuite à une étude théorique complémentaire. Analyse Formelle de Concepts: Nous codons une relation binaire R et son treillis des concepts L(R) par un graphe non orienté co-biparti G(R). Nous montrons que les éléments de L(R) sont en bijection avec les séparateurs minimaux de G(R), et que les chaînes maximales de L(R) sont en bijection avec les triangulations minimales de G(R). Des procédés algorithmiques appliqués à G(R) trouvent ainsi leurs correspondants dans L(R). En particulier, des treillis de taille polynomiale peuvent être obtenus à partir de L(R), par plongement de G(R) dans un graphe faiblement triangulé. Nous mettons ensuite en évidence un ordre de domination sur les modules complets maximaux de G(R), domination qui s'hérite quand on parcourt une chaîne maximale de L(R). Une structure de données, la table de domination, gère dynamiquement les relations de domination. Nous utilisons cette table pour deux applications algorithmiques: - Mise à jour d'une sous-hiérarchie de Galois matérialisant une hiérarchie d'héritage orienté-objet; - Génération efficace d'un treillis des concepts

Concepts can't afford to stammer

International audienceGenerating concepts defined by a binary relation between a set P of properties and a set O of objects is one of the important current problems encountered in Data Mining. We present a new algorithmic process which generates each concept exactly once, using graph-theoretic results. We present two associated algorithms, both with a good worst-time complexity analysis, which make them competitive with the best existing algorithms. This process has a time complexity of O(|P|.m) per maximal chain of the concept lattice, where m denotes the size of the complement of the relation, and uses a data structure which is of small polynomial size. Our algorithms can be used to compute the edges of the lattice as well as to generate only frequent sets

Representing a Concept Lattice By a Graph

Concept lattices (also called Galois lattices) are an ordering of the maximal rectangles dened by a binary relation. In this paper, we present a new relationship between lattices and graphs: given a binary relation R, we dene an underlying graph GR , and establish a one-to-one correspondence between the set of elements of the concept lattice of R and the set of minimal separators of GR

Maintaining Class Membership Information

Galois lattices (or concept lattices), which are lattices built on a binary relation, are now used in many fields, such as Data Mining and hierarchy organization, but may be of exponential size. In this paper, we propose a decomposition of a Galois sub-hierarchy which is of small size but contains useful inheritance information. We show how to efficiently maintain this information when an element is added to or removed from the relation, using a dynamic domination table which describes the underlying graph with which we encode the lattice

Representing a Concept Lattice By a Graph

In this paper, we present a new relationship between concept lattices and graphs. Given a binary relation R, we define an underlying graph GR , and establish a one-to-one correspondence between the set of elements of the concept lattice of R and the set of minimal separators of GR . We explain ho

Obtaining and Maintaining Polynomial-Sized Concept Lattices

We use our definition of an underlying co-bipartite graph which encodes a given binary relation to propose a new approach to defining a sub-relation or incrementally maintaining a relation which will define only a polynomial number of concepts

Generalized domination in closure systems

In the context of extracting maximal item sets and association rules from a binary data base, the graph-theoretic notion of domination was recently used to characterize the neighborhood of a concept in the corresponding lattice. In this paper, we show that the notion of domination can in fact be extended to any closure operator on a finite universe and be efficiently encoded into propositional Horn functions. This generalization enables us to endow notions and algorithms related to Formal Concept Analysis with Horn minimization and minimal covers of functional dependencies in Relational Databases.