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A Simple SubQuadratic Algorithm for Computing the Subset Partial Order
, 1995
"... A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal ..."
Abstract

Cited by 4 (2 self)
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A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been presented that compute the partial order (and thereby the minimal and maximal sets, i.e., extremal sets) in worstcase time O(N 2 = log N ). This paper develops a simple algorithm that uses only simple data structures, and gives a simple analysis that establishes the above worstcase bound on its running time. The algorithm exploits a variation on lexicographic order that may be of independent interest. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. Pritchard [4] presented algorithms for finding those sets in F that have no subset in F . Starting from a naive O(N 2 ) algorithm 1 , an algorithm was obtained that had worstcase complexity O...
Faster Concept Analysis
"... Abstract. We introduce a simple but efficient, multistage algorithm for constructing concept lattices (MCA). A concept lattice can be obtained as the closure system generated from attribute concepts (dually, object concepts). There are two strategies to use this as the basis of an algorithm: (a) for ..."
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Cited by 3 (2 self)
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Abstract. We introduce a simple but efficient, multistage algorithm for constructing concept lattices (MCA). A concept lattice can be obtained as the closure system generated from attribute concepts (dually, object concepts). There are two strategies to use this as the basis of an algorithm: (a) forming intersections by joining one attribute concept at a time, and (b) repeatedly forming pairwise intersections starting from the attribute concepts. A straightforward translation of (b) to an algorithm suggests that pairwise intersection be performed among all cumulated concepts. MCA is parsimonious in forming the pairwise intersections: it only performs such operations among the newly formed concepts from the previous stage, instead of cumulatively. We show that this parsimonious multistage strategy is complete: it generates all concepts. To make this strategy really work, one must overcome the need to eliminate duplicates (and potentially save time even further), since concepts generated at a later stage may have already appeared in one of the earlier stages. As considered in several other algorithms in the literature [5], we achieve this by an auxiliary search tree which keeps all existing concepts as paths from the root to a flagged node or a leaf. The depth of the search tree is bounded by the total number of attributes, and hence the time complexity for concept lookup is bounded by the logarithm of the total number of concepts. For constructing lattice diagrams, we adapt a subquadratic algorithm of Pritchard [9] for computing subset partial orders to constructing the Hasse diagrams. Instead of the standard expected quadratic complexity, the Pritchard approach achieves a worstcase time O(N 2 /log N). Our experimental results showed significant improvements in speed for a variety of input profiles against three leading algorithms considered in the comprehensive comparative study [5]: Bordat, Chein, and Norris. 1
A Fast Bitwise Algorithm for Computing the Subset Partial Order
, 1995
"... A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worstcase time O(N 2 = log N ..."
Abstract

Cited by 1 (1 self)
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A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be defined as the sum of the cardinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worstcase time O(N 2 = log N ). This paper gives a variant implementation of a previously proposed algorithm which is shown to have a worstcase complexity of O(N 2 (log log N) 2 = log 2 N) operations on a RAM with \Theta(log N) bit words. This is the first known o(N 2 = log N) worstcase running time. 1 Introduction Given is a collection F = fS 1 ; : : : ; S k g, where each S i is a set over the same domain D. Define the size of the collection to be N = P i jS i j. In [5] we presented algorithms for finding those sets in F that have no subset in F , and obtained a fast algorithm for the important special case when all sets in F are small. A particular implementation was later shown [6] to have worstcas...
Functional Graphical Models
, 2003
"... Functional models are frequently used in computer vision and photogrammetry, as they enable the mathematical formulation of several problems such as pose computation and more generally the parameter estimation problem. However, the structural properties of such models have only seldom been studie ..."
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Functional models are frequently used in computer vision and photogrammetry, as they enable the mathematical formulation of several problems such as pose computation and more generally the parameter estimation problem. However, the structural properties of such models have only seldom been studied. This contribution is dedicated to the analysis of such properties. We propose a new formalism that enables the analysis and design of functional models.
The Subset Partial Order: . . .
"... Given a family F of k sets with cardinalities s1, s2,..., sk and N = ∑k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O ( ∑ si≤B 2s ∑ ..."
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Given a family F of k sets with cardinalities s1, s2,..., sk and N = ∑k i=1 si, we show that the size of the partial order graph induced by the subset relation (called the subset graph) is O ( ∑ si≤B 2s ∑