Efficient algorithms on distributive lattices

Habib, Michel; Medina, Raoul; Nourine, Lhouari; Steiner, George

doi:10.1016/s0166-218x(00)00258-4

Cited by 25 publications

(19 citation statements)

References 11 publications

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“…Even then the naïve algorithm enumerating each set S ∈ F and testing whether S is ρ-closed is inefficient because |F | can be exponential in |ρ(F )|. There are several results on efficient enumeration of ρ-closed sets for the case that the underlying set system is finite and closed under intersection (see, e.g., [2,3]). Among others, formal concept analysis [7] and closed frequent itemset mining (see, e.g., [6]) provide some representative applications of this case.…”

Section: The General Problemmentioning

confidence: 99%

Efficient Closed Pattern Mining in Strongly Accessible Set Systems (Extended Abstract)

Boley

Horváth

Poigné

et al.

Knowledge Discovery in Databases: PKDD 2007

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Abstract. Many problems in data mining can be viewed as a special case of the problem of enumerating the closed elements of an independence system with respect to some specific closure operator. Motivated by real-world applications, e.g., in track mining, we consider a generalization of this problem to strongly accessible set systems and arbitrary closure operators. For this more general problem setting, the closed sets can be enumerated with polynomial delay if deciding membership in the set system and computing the closure operator can be solved in polynomial time. We discuss potential applications in graph mining.

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Section: The General Problemmentioning

confidence: 99%

Efficient Closed Pattern Mining in Strongly Accessible Set Systems (Extended Abstract)

Boley

Horváth

Poigné

et al.

Knowledge Discovery in Databases: PKDD 2007

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“…The second representation is the ideal tree data structure for distributive lattices due to Habib and Nourine [9,10], described in Section 4. This representation requires only O(n log n) bits of space and computes meets and joins in O(m) time, which may be as fast as O(log n) or as slow as O(n).…”

Section: Ideal Tree (Tree(l))mentioning

confidence: 99%

“…The ideal tree is a very clean and elegant way to manipulate distributive lattices; see also [9]. We use their ideas extensively in this paper.…”

Section: Ideal Treesmentioning

confidence: 99%

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Time and Space Efficient Representations of Distributive Lattices

Munro

Sinnamon

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a space-efficient data structure using O(n log n) bits that represents a distributive lattice on n elements and supports finding meets and joins in O(log n) time. Our data structure extends the ideal tree structure of Habib and Nourine which occupies O(n log n) bits of space and requires O(m) time to compute a meet or join, where m depends on the specific lattice and may be as large as n − 1. We also give an encoding of a distributive lattice using 10 7 n + O(log n) bits, which is very close to the information theoretic lower bound. This encoding can be created or decompressed in O(n log n) time.

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“…We are well aware of the fact that, in lattice theory, ideals have a more specific meaning. We stick to the term ideal because of the literature on the ideal lattice of a poset we draw upon [11]. Let us further denote the set of ideals of the poset P as IðP ; 6 P Þ or as IðP Þ for short, and the set of filters as FðP ; 6 P Þ or FðP Þ.…”

Section: Preliminariesmentioning

confidence: 99%