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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Vydané v:International journal of approximate reasoning Ročník 142; s. 370 - 382
Hlavní autori: Gély, Alain, Couceiro, Miguel, Miclet, Laurent, Napoli, Amedeo
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier Inc 01.03.2022
Elsevier
Predmet:
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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Shrnutí: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.
ISSN:0888-613X
1873-4731
DOI:10.1016/j.ijar.2021.12.011