# A fast incremental algorithm for building lattices

@article{Nourine2002AFI,
title={A fast incremental algorithm for building lattices},
author={Lhouari Nourine and Olivier Raynaud},
journal={Journal of Experimental \& Theoretical Artificial Intelligence},
year={2002},
volume={14},
pages={217 - 227}
}
• Published 2002
• Computer Science
• Journal of Experimental & Theoretical Artificial Intelligence
This paper presents an incremental algorithm to compute the covering graph of the lattice generated by a family B of subsets of a totally ordered set X. The implementation of this algorithm has O (((|X| + |B|).|B|).|F|) time complexity, where F is the number of elements in the lattice. This improves the complexity of the previous algorithms which is roughly in O(Min(|X|, |B|)3.|F|). This algorithm may be used in many applications in computer sciences such as the computations of Galois (concept… Expand
42 Citations

#### Topics from this paper

Faster Algorithms for Constructing a Concept (Galois) Lattice
• V. Choi
• Mathematics, Computer Science
• ArXiv
• 2006
This paper presents a fast algorithm for constructing a concept (Galois) lattice of a binary relation, including computing all concepts and their lattice order, which is faster than all other existing algorithms for these problems. Expand
Maximal Antichain Lattice Algorithms for Distributed Computations
A new online incremental algorithm, OLMA, is presented that computes the newly added elements to the lattice without requiring the prior lattice and gets a significant reduction in the space complexity. Expand
Faster Algorithms for Constructing a Galois Lattice, Enumerating All Maximal Bipartite Cliques and Closed Frequent Sets
• Mathematics
• 2006
In this paper, we give a fast algorithm for constructing a Galois lattice of a binary relation. When the binary relation is represented as a bipartite graph, each vertex of the lattice (called aExpand
Lattice Completion Algorithms for Distributed Computations
New algorithms to construct or enumerate the lattice of normal cuts, the smallest lattice that embeds the poset such that all meets and joins are defined, are proposed. Expand
The Use of Associative Concepts for Fast Incremental Concept Formation in Sparse Contexts ?
Formal Concept Analysis (FCA) is interested in the formation of concept lattices from binary relations between objects and attributes, a.k.a. contexts. Many algorithms have been proposed to generateExpand
Enumeration aspects of maximal cliques and bicliques
• Computer Science, Mathematics
• Discret. Appl. Math.
• 2009
The notion of the transition graph T(G) whose vertices are maximal cliques of G and arcs are transitions between cliques is introduced and it is shown that under some specific numbering, the transition graphs has a hamiltonian path for chordal and comparability graphs. Expand
Algorithm of Building Concept Lattice Based on Its Structure
• Computer Science
• 2007
An algorithm is presented to generate all concepts and its lattice order given a formal concept and it can produce all the children and identify all successors from them. Expand
Building Concept (Galois) Lattices from Parts: Generalizing the Incremental Methods
• Computer Science
• ICCS
• 2001
The method is shown to improve a key flaw of the major incremental technique and provides a set of structural results for the case of single object insertions which underlie a new incremental algorithm. Expand
On Independent Sets and Bicliques in Graphs
• Mathematics, Computer Science
• WG
• 2008
It is shown that the maximum number of maximal bicliques in a graph on n vertices is exactly 3 n /3 (up to a polynomial factor). Expand
On Independent Sets and Bicliques in Graphs
• Mathematics, Computer Science
• Algorithmica
• 2010
It is shown that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3), and an exact exponential-time algorithm is used that computes the number of distinct maximal independent sets in a graphs in time O(1.3642n), where n is thenumber of vertices of the input graph. Expand

#### References

SHOWING 1-10 OF 17 REFERENCES
A Fast Algorithm for Building Lattices
• Mathematics, Computer Science
• Inf. Process. Lett.
• 1999
This algorithm can be used to compute the Galois (concept) lattice, the maximal antichains lattice or the Dedekind‐MacNeille completion of a partial order, without increasing time complexity. Expand
Efficient algorithms on distributive lattices
• Computer Science, Mathematics
• Discret. Appl. Math.
• 2001
We present several efficient algorithms on distributive lattices. They are based on a compact representation of the lattice, called the ideal tree. This allows us to exploit regularities in theExpand
INCREMENTAL CONCEPT FORMATION ALGORITHMS BASED ON GALOIS (CONCEPT) LATTICES
• Mathematics, Computer Science
• Comput. Intell.
• 1995
Empirical evidence shows that, on the average, the incremental update of the Galois lattice is done in time proportional to the number of instances previously treated, and the worst‐case analysis of the algorithm also shows linear growth with respect to thenumber of instances. Expand
Tree Structure for Distributive Lattices and its Applications
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 1996
This representation (called ideal tree) can be efficiently computed and compared with respect to time and space complexity and allow us to achieve best running time algorithms for most of the applications in which distributive lattices are involved. Expand
Computing on-line the lattice of maximal antichains of posets
• Mathematics
• 1994
AbstractWe consider the on-line computation of the lattice of maximal antichains of a finite poset $$\tilde P$$ . This on-line computation satisfies what we call the “linear extension hypothesis”:Expand
A lattice conceptual clustering system and its application to browsing retrieval
• Computer Science
• Machine Learning
• 2004
This paper presents GALOIS, a system that automates and applies the theory of concept lattices, and describes a prototype user interface for browsing through the concept lattice of a document-term relation, possibly enriched with a thesaurus of terms. Expand
Comparing performance of algorithms for generating concept lattices
• Computer Science
• J. Exp. Theor. Artif. Intell.
• 2002
Several algorithms that generate the set of all formal concepts and diagram graphs of concept lattices are considered and algorithmic complexity of the algorithms is studied both theoretically and experimentally. Expand
An Introduction to Lattices and Order
• Mathematics, Computer Science
• 1989
This chapter discusses the structure of finite distributive lattices and finite Boolean algebras, and the role of lattices in algebra in this structure. Expand
The Diclique Representation and Decomposition of Binary Relations
The algebraic structure is used to show how dicliques can be coalesced, the relationship between cliques and dicLiques is discussed, and an algorithm for determining cliques from dICliques is described. Expand
Formal Concept Analysis: Mathematical Foundations
• Computer Science
• 1998
From the Publisher: This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science,Expand