A study of algorithms relating distributive lattices, median graphs, and Formal Concept Analysis

In this paper, we study structures such as distributive lattices, distributive semilattices, and median graphs from an algorithmic point of view. Such structures are very useful in classification and phylogeny for representing lineage relationships for example. A distributive lattice can be consider...

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Published in:	International journal of approximate reasoning Vol. 142; pp. 370 - 382
Main Authors:	Gély, Alain, Couceiro, Miguel, Miclet, Laurent, Napoli, Amedeo
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.03.2022 Elsevier
Subjects:	Combinatorics Computer Science Data Structures and Algorithms Discrete Mathematics Distributivity Embedding Formal Concept Analysis Lattice Mathematics Median graph Formal Concept Analysis Embedding Distributivity Median graph Lattice Median Graph
ISSN:	0888-613X, 1873-4731
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Abstract	In this paper, we study structures such as distributive lattices, distributive semilattices, and median graphs from an algorithmic point of view. Such structures are very useful in classification and phylogeny for representing lineage relationships for example. A distributive lattice can be considered as a median graph while a distributive ∨-semilattice can be considered as a median graph provided that some conditions holding on triple of elements are satisfied. Starting from a lattice structure with different representations, we study the problem of building a median graph from such structures. We make precise and propose algorithms for checking how a lattice can be distributive and can be a median graph. Then, we adapt the problem to semilattices as a lattice where the bottom element is removed is a ∨-semilattice. We also state the problem in terms of Formal Concept Analysis and the representation of a lattice as a formal context, i.e., a binary table. Moreover, we also propose as input a system of implications such as the Duquenne-Guigues basis of a lattice, and we study how to compute such a basis for a distributive semilattice. In the paper, we provide algorithms and examples which illustrate the difficulties related to these different classification tasks. In particular, the minimality of the output lattices is a condition which is hard to ensure and which cannot be always achieved.
AbstractList	In this paper, we study structures such as distributive lattices, distributive semilattices, and median graphs from an algorithmic point of view. Such structures are very useful in classification and phylogeny for representing lineage relationships for example. A distributive lattice can be considered as a median graph while a distributive ∨-semilattice can be considered as a median graph provided that some conditions holding on triple of elements are satisfied. Starting from a lattice structure with different representations, we study the problem of building a median graph from such structures. We make precise and propose algorithms for checking how a lattice can be distributive and can be a median graph. Then, we adapt the problem to semilattices as a lattice where the bottom element is removed is a ∨-semilattice. We also state the problem in terms of Formal Concept Analysis and the representation of a lattice as a formal context, i.e., a binary table. Moreover, we also propose as input a system of implications such as the Duquenne-Guigues basis of a lattice, and we study how to compute such a basis for a distributive semilattice. In the paper, we provide algorithms and examples which illustrate the difficulties related to these different classification tasks. In particular, the minimality of the output lattices is a condition which is hard to ensure and which cannot be always achieved.
Author	Couceiro, Miguel Miclet, Laurent Napoli, Amedeo Gély, Alain
Author_xml	– sequence: 1 givenname: Alain orcidid: 0000-0002-3820-6616 surname: Gély fullname: Gély, Alain email: alain.gely@loria.fr organization: Université de Lorraine, CNRS, LORIA, F-57000 Metz, France – sequence: 2 givenname: Miguel orcidid: 0000-0003-2316-7623 surname: Couceiro fullname: Couceiro, Miguel email: miguel.couceiro@loria.fr organization: Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France – sequence: 3 givenname: Laurent surname: Miclet fullname: Miclet, Laurent email: laurent.miclet@gmail.com organization: Université de Rennes, CNRS, IRISA, Rue de Kérampont, 22300 Lannion, France – sequence: 4 givenname: Amedeo surname: Napoli fullname: Napoli, Amedeo email: amedeo.napoli@loria.fr organization: Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
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Keywords	Formal Concept Analysis Embedding Distributivity Median graph Lattice Median Graph
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SubjectTerms	Combinatorics Computer Science Data Structures and Algorithms Discrete Mathematics Distributivity Embedding Formal Concept Analysis Lattice Mathematics Median graph
Title	A study of algorithms relating distributive lattices, median graphs, and Formal Concept Analysis
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