A simple algorithm to generate the minimal separators and the maximal cliques of a chordal graph

We present a simple unified algorithmic process which uses either LexBFS or MCS on a chordal graph to generate the minimal separators and the maximal cliques in linear time in a single pass

Organizing the atoms of the clique separator decomposition into an atom tree

International audienceWe define an atom tree of a graph as a generalization of a clique tree: its nodes are the atoms obtained by clique minimal separator decomposition, and its edges correspond to the clique minimal separators of the graph.Given a graph GG, we compute an atom tree by using a clique tree of a minimal triangulation HH of GG. Computing an atom tree with such a clique tree as input can be done in O(min(nm,m+nf))O(min(nm,m+nf)), where ff is the number of fill edges added by the triangulation. When both a minimal triangulation and the clique minimal separators of GG are provided, we compute an atom tree of GG in O(m+f)O(m+f) time, which is in O(n2)O(n2) time.We give an O(nm)O(nm) time algorithm, based on MCS, which combines in a single pass the 3 steps involved in building an atom tree: computing a minimal triangulation, constructing a clique tree, and constructing the corresponding atom tree.Finally, we present a process which uses a traversal of a clique tree of a minimal triangulation to determine the clique minimal separators and build the corresponding atom tree in O(n(n+t))O(n(n+t)) time, where tt is the number of 2-pairs of HH (tt is at most View the MathML sourcem¯−f, where View the MathML sourcem¯ is the number of edges of the complement graph); to complete this, we also give an algorithm which computes a minimal triangulation in View the MathML sourceO(n(n+m¯)) time, thus providing an approach to compute the decomposition in View the MathML sourceO(n(n+m¯)) time

Efficient clique decomposition of a graph into its atom graph

We explain how to organize the atoms resulting from clique minimal separator decomposition into a metagraph which we call the atom graph, and give an efficient recursive algorithm to compute this graph at no extra cost than computing the atoms

An Introduction to Clique Minimal Separator Decomposition

International audienceThis paper is a review which presents and explains the decomposition of graphs by clique minimal separators. The pace is leisurely, we give many examples and figures. Easy algorithms are provided to implement this decomposition. The historical and theoretical background is given, as well as sketches of proofs of the structural results involved

Impact of the distance choice on clustering gene expression data using graph decompositions

The study of gene interactions is an important research area in biology and grouping genes with similar expression profiles to clusters is a first step towards a better understanding of their functional relationships. In Kaba et al. 2007, a new clustering approach was presented, using gene interaction graphs to model this data, and decomposing the graphs by means of clique minimal separators. A clique separator is a clique whose removal increases the number of connected components of the graph; the decomposition is obtained by repeatedly copying a clique separator into the components it defines, until only subgraphs with no clique separators are left: these subgraphs will be our clusters. The advantage of our approach is that this decomposition can be computed efficiently, is unique, and yields overlapping clusters. For that, the similarity between each pair of genes is estimated by a distance function, then a family of gene interaction graphs is constructed by choosing several thresholds, where an edge is added between two genes if their distance is below the threshold. Hereby, both the choice of the distance function and of the threshold influences the construction of the gene interaction graphs. In Kaba et al. 2007, several criteria are developed to select thresholds in an appropriate way. Here we discuss the impact of the choice of the distance function; our results suggest that this choice does not effect the final decomposition of the gene interaction graphs into clusters

Clique separator decomposition in less than nm

We address the problem of computing the atoms of the decomposition by clique minimal separators of a graph G (also called the maximal prime subgraphs) when a minimal triangulation H of G is given as part of the input. We present a new algorithmic technique based on the clique tree of H. We introduce a new graph parameter, m0, which is the number of edges belonging to no minimal separator of H. We give an algorithm which runs in O(nm0) time, which improves the current O(nm) time for this problem. Another version of our algorithm runs in O(n(n+t)) time, where t is the number of 2-pairs of H. We show that our technique computes the atoms in O(n2) time for several graph classes, including the graphs with bounded treewidth, which improves the current O(n3) time for dense graphs by a factor of n

Décomposition par séparateurs minimaux complets et applications

We worked on clique minimal separator decomposition. In order to compute this decomposition on a graph G we need to compute the minimal separators of its triangulation H. In this context, the first efforts were on finding a clique minimal separators in a chordal graph. We defined a structure called atom tree inspired from the clique tree to compute and represent the final products of the decomposition, called atoms. The purpose of this thesis was to apply this technique on biological data. While we were manipulating this data using Galois lattices, we noticed that the clique minimal separator decomposition allows a divide and conquer approach on Galois lattices. One biological application of this thesis was the detection of fused genes which are important evolutionary events. Using algorithms we produced in the course of along our work we implemented a program called MosaicFinder that allows an efficient detection of this fusion event and their pooling. Another biological application was the extraction of genes of interest using expression level data. The atom tree structure allowed us to have a good visualization of the data and to be able to compute large datasets.Nous avons utilisé la décomposition par séparateurs minimaux complets. Pour décomposer un graphe G, il est nécessaire de trouver les séparateurs minimaux dans le graphe triangulé H correspondant. Dans ce contexte, nos premiers efforts se sont tournés vers la détection de séparateurs minimaux dans un graphe triangulé. Nous avons défini une structure, que nous avons nommée 'atom tree'. Cette dernière est inspirée du 'clique tree' et permet d'obtenir et de représenter les atomes qui sont les produits de la décomposition. Lors de la manipulation de données à l'aide de treillis de Galois, nous avons remarqué que la décomposition par séparateurs minimaux permettait une approche de type `Diviser pour régner' pour les treillis de Galois. La détection des gènes fusionnés, qui est une étape importante pour la compréhension de l'évolution des espèces, nous a permis d'appliquer nos algorithmes de détection de séparateurs minimaux complets, qui nous a permis de détecter et regrouper de manière efficace les gènes fusionnés. Une autre application biologique fut la détection de familles de gènes d'intérêts à partir de données de niveaux d'expression de gènes. La structure de `l'atom tree' nous a permis d'avoir un bon outils de visualisation et de gérer des volumes de données importantes

Décomposition par séparateurs minimaux complets et applications

Nous avons utilisé la décomposition par séparateurs minimaux complets. Pour décomposer un graphe G, il est nécessaire de trouver les séparateurs minimaux dans le graphe triangulé H correspondant. Dans ce contexte, nos premiers efforts se sont tournés vers la détection de séparateurs minimaux dans un graphe triangulé. Nous avons défini une structure, que nous avons nommée 'atom tree'. Cette dernière est inspirée du 'clique tree' et permet d'obtenir et de représenter les atomes qui sont les produits de la décomposition. Lors de la manipulation de données à l'aide de treillis de Galois, nous avons remarqué que la décomposition par séparateurs minimaux permettait une approche de type Diviser pour régner' pour les treillis de Galois. La détection des gènes fusionnés, qui est une étape importante pour la compréhension de l'évolution des espèces, nous a permis d'appliquer nos algorithmes de détection de séparateurs minimaux complets, qui nous a permis de détecter et regrouper de manière efficace les gènes fusionnés. Une autre application biologique fut la détection de familles de gènes d'intérêts à partir de données de niveaux d'expression de gènes. La structure de l'atom tree' nous a permis d'avoir un bon outils de visualisation et de gérer des volumes de données importantes.We worked on clique minimal separator decomposition. In order to compute this decomposition on a graph G we need to compute the minimal separators of its triangulation H. In this context, the first efforts were on finding a clique minimal separators in a chordal graph. We defined a structure called atom tree inspired from the clique tree to compute and represent the final products of the decomposition, called atoms. The purpose of this thesis was to apply this technique on biological data. While we were manipulating this data using Galois lattices, we noticed that the clique minimal separator decomposition allows a divide and conquer approach on Galois lattices. One biological application of this thesis was the detection of fused genes which are important evolutionary events. Using algorithms we produced in the course of along our work we implemented a program called MosaicFinder that allows an efficient detection of this fusion event and their pooling. Another biological application was the extraction of genes of interest using expression level data. The atom tree structure allowed us to have a good visualization of the data and to be able to compute large datasets.CLERMONT FD-Bib.électronique (631139902) / SudocSudocFranceF