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A fast incremental algorithm for building lattices. (English) Zbl 1022.68027
Summary: This paper presents an incremental algorithm to compute the covering graph of the lattice generated by a family ${\cal B}$ of subsets of a totally ordered set $X$. The implementation of this algorithm has $O(((|X|+|{\cal B}|).|{\cal B}|).|{\cal F}|)$ time complexity, where ${\cal F}$ is the number of elements in the lattice. This improves the complexity of the previous algorithms which is roughly in $O(\text{Min}(|X|,|{\cal B}|)^3.|{\cal F}|)$. This algorithm may be used in many applications in computer sciences such as the computations of Galois (concept) lattice, the maximal antichains lattice or the Dedekind-MacNeille completion of a partial order. All these lattices can be computed incrementally using this algorithm without increasing time complexity.

68P05Data structures
68P15Database theory
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