## In-Close, a Fast Algorithm for Computing Formal Concepts (2009)

Venue: | the Seventeenth International Conference on Conceptual Structures |

Citations: | 12 - 7 self |

### BibTeX

@INPROCEEDINGS{Andrews09in-close,a,

author = {Simon Andrews},

title = {In-Close, a Fast Algorithm for Computing Formal Concepts},

booktitle = {the Seventeenth International Conference on Conceptual Structures},

year = {2009}

}

### OpenURL

### Abstract

Abstract. This paper presents an algorithm, called In-Close, that uses incremental closure and matrix searching to quickly compute all formal concepts in a formal context. In-Close is based, conceptually, on a well known algorithm called Close-By-One. The serial version of a recently published algorithm (Krajca, 2008) was shown to be in the order of 100 times faster than several well-known algorithms, and timings of other algorithms in reviews suggest that none of them are faster than Krajca. This paper compares In-Close to Krajca, discussing computational methods, data requirements and memory considerations. From experiments using several public data sets and random data, this paper shows that In-Close is in the order of 20 times faster than Krajca. In-Close is small, straightforward, requires no matrix pre-processing and is simple to implement. 1

### Citations

1854 |
Data Mining: Concepts and Techniques
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- 2000
(Show Context)
Citation Context ...s that object i does not have attribute j. This form of data is the basis for Formal Concept Analysis (FCA) [9, 20, 18], where it is referred to as the formal context, and is important in data mining =-=[11]-=-. Data analysis applications are numerous, in fields such as marketing, census analysis and bioinformatics. A formal description of formal concepts begins with a set of objects G and a set of attribut... |

809 |
Formal Concept Analysis: Mathematical Foundations
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(Show Context)
Citation Context ...ays that object i has attribute j. A presence of the value 0 (false) in row i, column j, says that object i does not have attribute j. This form of data is the basis for Formal Concept Analysis (FCA) =-=[9, 20, 18]-=-, where it is referred to as the formal context, and is important in data mining [11]. Data analysis applications are numerous, in fields such as marketing, census analysis and bioinformatics. A forma... |

105 | Incremental concept formation algorithms based on galois (concept) lattices
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(Show Context)
Citation Context ... others like it, use a search tree, or trie, to quicky find repeated results. Others use a hash function where the cardinalityof results is used to divide them into groups, thus narrowing the search =-=[10]-=-. Berry’s algorithm [4] avoids searching by pruning the matrix before generating concepts from it - eliminating columns that are strict subsets of other columns and pairing those that are equal. ⎡ ⎤ 0... |

99 |
UCI machine learning repository, http://www.ics.uci.edu/ ∼mlearn/ mlrepository.html
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(Show Context)
Citation Context ...ke’. Two groups of experiments were carried out to compare their performance. One group of experiments was carried out using four publicly available data sets from the UCI Machine Learning Repository =-=[1]-=- and one group using several sets of random data. The UCI data sets were chosen on the basis of suitability in terms of size (thousands of objects, hundreds of attributes) and data type (predominantly... |

92 |
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(Show Context)
Citation Context ...oach to closure, as will be seen later. 2.1 The Lexicographical Order of Concepts Ganter showed how an order of concepts could be used to circumvent the need to explicitly search for repeated results =-=[8]-=-. In mathematics, combinations have a lexicographical order, or cannon, where {1, 2, 3} comes before {1, 2, 4}, for example, and {1, 2, 6} comes before {1, 3}. If concepts are generated in this order ... |

91 | S.: Comparing performance of algorithms for generating concept lattices
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(Show Context)
Citation Context ..., ‘object’ and ‘row’ are substituted for ‘B’, ‘intent’, ‘attribute’ and ‘column’, and vice-versa. Reviews of algorithms that take a variety of approaches to generating formal concepts may be found in =-=[2, 14, 21]-=-. Many of these algorithms perform additional tasks, such as constructing the concept lattice (Hasse diagram), and not all were designed with fast performance as a requirement, nor with the intention ... |

55 | Efficient algorithms for mining closed itemsets and their lattice structure
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(Show Context)
Citation Context ..., ‘object’ and ‘row’ are substituted for ‘B’, ‘intent’, ‘attribute’ and ‘column’, and vice-versa. Reviews of algorithms that take a variety of approaches to generating formal concepts may be found in =-=[2, 14, 21]-=-. Many of these algorithms perform additional tasks, such as constructing the concept lattice (Hasse diagram), and not all were designed with fast performance as a requirement, nor with the intention ... |

53 |
Concept Data Analysis: Theory and Applications
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(Show Context)
Citation Context ...text is that every combination of columns (or rows) can generate a concept. Only a proportion of these will be closed and the number of formal concepts grows exponentially with the size of the matrix =-=[6]-=-. In addition, many different combinations of columns may share the same rows, leading to the same result being generated many times. This leads to a large number of computations and the searching of ... |

41 | Formal concept analysis in information science
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(Show Context)
Citation Context ...ays that object i has attribute j. A presence of the value 0 (false) in row i, column j, says that object i does not have attribute j. This form of data is the basis for Formal Concept Analysis (FCA) =-=[9, 20, 18]-=-, where it is referred to as the formal context, and is important in data mining [11]. Data analysis applications are numerous, in fields such as marketing, census analysis and bioinformatics. A forma... |

23 |
Formal concept analysis as mathematical theory of concepts and concept hierarchies
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(Show Context)
Citation Context ...ays that object i has attribute j. A presence of the value 0 (false) in row i, column j, says that object i does not have attribute j. This form of data is the basis for Formal Concept Analysis (FCA) =-=[9, 20, 18]-=-, where it is referred to as the formal context, and is important in data mining [11]. Data analysis applications are numerous, in fields such as marketing, census analysis and bioinformatics. A forma... |

19 | N.: Reasoning about Sets using Redescription Mining
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(Show Context)
Citation Context ...Boolean factor analysis where they are optimal factors [3]. There is also a strong correlation with algorithms determining the rectangular groups used to simplify Boolean expressions in Karnaugh maps =-=[22]-=-. Elsewhere, algorithms have referred to formal concepts as maximal rectangles [4], although this term has also been used when referring to the largest contiguous empty rectangles in a Boolean matrix ... |

18 |
Learning of Simple Conceptual Graphs from Positive and Negative Examples. PKDD
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(Show Context)
Citation Context ...ata set.) This has led to a variety of interesting approaches to computing formal concepts. This paper presents an algorithm, called In-Close, based conceptually on Kuznetsov’s Close-By-One algorithm =-=[13]-=-, as a contender for being the fastest method to date. 2 Computing Formal Concepts In computing formal concepts, a Boolean matrix is often used to represent the formal context, where rows and columns ... |

12 | Parallel recursive algorithm for FCA
- Krajca, Outrata, et al.
(Show Context)
Citation Context ...empty set smaller than A. 3. The same set, i.e. A ∩ {j} ′ = A. Krajca and In-Close deal with these outcomes in different ways. 2.2 The Krajca Method A recent algorithm by Krajca, Outrata and Vychodil =-=[12, 16, 19]-=-, referred to here as Krajca, was shown to be particularly efficient. They have presented a serial and a parallel version of their algorithms but only the serial version (and results from it) is consi... |

11 | Mining for Empty Rectangles in Large Data Sets
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(Show Context)
Citation Context .... Elsewhere, algorithms have referred to formal concepts as maximal rectangles [4], although this term has also been used when referring to the largest contiguous empty rectangles in a Boolean matrix =-=[7]-=-. Fundamental to the performance of algorithms to generate formal concepts is how they deal with the complexity of the computation. As mentioned in section 1, many values of B can generate the same va... |

10 | Performances of Galois Sub-Hierarchy-Building Algorithms
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(Show Context)
Citation Context ..., ‘object’ and ‘row’ are substituted for ‘B’, ‘intent’, ‘attribute’ and ‘column’, and vice-versa. Reviews of algorithms that take a variety of approaches to generating formal concepts may be found in =-=[2, 14, 21]-=-. Many of these algorithms perform additional tasks, such as constructing the concept lattice (Hasse diagram), and not all were designed with fast performance as a requirement, nor with the intention ... |

10 |
Fast concept analysis. In: Working with Conceptual Structures
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(Show Context)
Citation Context ... is the closed B. As As are generated, a common method is to search previous results and discard repeats. A key to good peformance is how efficiently previous results are searched. Lindig’s algorithm =-=[15]-=-, and others like it, use a search tree, or trie, to quicky find repeated results. Others use a hash function where the cardinalityof results is used to divide them into groups, thus narrowing the se... |

9 | J.F.: Towards fault-tolerant formal concept analysis
- Pensa
(Show Context)
Citation Context ... small errors in the data set, and not generating them is therefore sensible. It is acknowledged that this is not always the case and that useful results may be lost by ‘over-pruning’ the computation =-=[17]-=-. Similar approaches to reducing computation have been explored that discard results with small |B| or small |B × A|, or use some other constraint (‘powers of 2’ rectangles in Karnaugh maps, for examp... |

7 | Actionability and Formal Concepts: A Data Mining Perspective
- Boulicaut, Besson
- 2008
(Show Context)
Citation Context ... j. A formal concept is represented by an extent, A[r] (an ordered list of objects), and an intent, B[r] (an ordered list of attributes), where r is the concept number (index). For example, if B[r] = =-=(3, 5, 7)-=-, B[r][2] = 7. For the purposes of the following pseudocode, A[r] and B[r] will be treated as sets of objects and attributes, respectively, where convenient. Thus, B[r] ∪ {j} appends attribute j to B[... |

4 |
A local approach to concept generation
- Berry, Bordat, et al.
- 2007
(Show Context)
Citation Context ...correlation with algorithms determining the rectangular groups used to simplify Boolean expressions in Karnaugh maps [22]. Elsewhere, algorithms have referred to formal concepts as maximal rectangles =-=[4]-=-, although this term has also been used when referring to the largest contiguous empty rectangles in a Boolean matrix [7]. Fundamental to the performance of algorithms to generate formal concepts is h... |

3 |
Optimal decompositions of matrices with grades
- Belohlavek
(Show Context)
Citation Context ...rent areas where formal concepts are important: in data mining [21], where formal concepts may be referred to as closed frequent itemsets and in Boolean factor analysis where they are optimal factors =-=[3]-=-. There is also a strong correlation with algorithms determining the rectangular groups used to simplify Boolean expressions in Karnaugh maps [22]. Elsewhere, algorithms have referred to formal concep... |

3 |
Vychodil V. Fast algorithm for computing maximal rectangles from object-attribute relational data (submitted
- Outrata
(Show Context)
Citation Context ...empty set smaller than A. 3. The same set, i.e. A ∩ {j} ′ = A. Krajca and In-Close deal with these outcomes in different ways. 2.2 The Krajca Method A recent algorithm by Krajca, Outrata and Vychodil =-=[12, 16, 19]-=-, referred to here as Krajca, was shown to be particularly efficient. They have presented a serial and a parallel version of their algorithms but only the serial version (and results from it) is consi... |

3 |
A new algorithm for computing formal concepts
- Vychodil
(Show Context)
Citation Context ...empty set smaller than A. 3. The same set, i.e. A ∩ {j} ′ = A. Krajca and In-Close deal with these outcomes in different ways. 2.2 The Krajca Method A recent algorithm by Krajca, Outrata and Vychodil =-=[12, 16, 19]-=-, referred to here as Krajca, was shown to be particularly efficient. They have presented a serial and a parallel version of their algorithms but only the serial version (and results from it) is consi... |