Recognition of Robinsonian dissimilarities

This paper is devoted to some selected topics relating Combinatorial Optimization and Hierarchical Classification. It is oriented toward extensions of the standard classification schemes (the hierarchies): pyramids, quasi-hierarchies, circular clustering, rigid clustering and others. Bijection theorems between these models and dissimilarity models allow to state some clustering problems as optimization problems. Within the galaxy of optimization we have especially discussed the following: NP-completeness results and search for polynomial instances; problems solved in a polynomial time (e.g. subdominant theory); design, analysis and applications of algorithms. In contrast with the orientation to "new" clustering problems, the last part discusses some standard algorithmic approaches.

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“…It is stated in Chepoi and Fichet (1997). An algorithm proving assertion (ii) can be found in Osswald (2003), see also Quillot (1984).…”

Section: Proposition 4 Rig-rec(g) Can Be Solved In: (I) O(n 3 ) Whementioning

confidence: 92%

“…Assertion (iii) comes from Chepoi and Fichet (1997) and assertion (iv) from Osswald (2003b). Despite the fact that the definition of an ultrametric involves three element subsets, the complexity of its recognition is O(n 2 ).…”

Section: Dissimilarity Recognition Problemsmentioning

confidence: 96%

Combinatorial optimization and hierarchical classifications

Barthélemy

Brucker

Osswald

2004

4OR

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“…Assertion (i) can be inferred from Duchet (1979). It is stated in Chepoï and Fichet (1997). An algorithm proving assertion (ii) can be found in Osswald (2003a), see also Quilliot (1984).…”

Section: Proposition 4 Rig-rec(g) Can Be Solved Inmentioning

confidence: 96%

Combinatorial optimisation and hierarchical classifications

2007

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“…For this, they build up the hypergraph of all balls of d and test using the PQ-tree algorithm if this hypergraph is an interval hypergraph. A simple divide-and-conquer O(n 3 )-time algorithm for the same recognition problem has been designed in [20]. Barthelemy and Brucker [9] established that the problem of an optimal l p -approximation of a dissimilarity by a particular Robinsonian dissimilarity (namely, strongly Robinsonian dissimilarity) is NP-hard for p < ∞.…”

Section: Related Workmentioning

confidence: 99%

Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

Chepoi

Seston

2009

Algorithmica

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The classical seriation problem consists in finding a permutation of the rows and the columns of the distance (or, more generally, dissimilarity) matrix d on a finite set X so that small values should be concentrated around the main diagonal as close as possible, whereas large values should fall as far from it as possible. This goal is best achieved by considering the Robinson property: a distance d R on X is Robinsonian if its matrix can be symmetrically permuted so that its elements do not decrease when moving away from the main diagonal along any row or column. If the distance d fails to satisfy the Robinson property, then we are lead to the problem of finding a reordering of d which is as close as possible to a Robinsonian distance.In this paper, we present a factor 16 approximation algorithm for the following NP-hard fitting problem: given a finite set X and a dissimilarity d on X, we wish to find a Robinsonian dissimilarity d R on X minimizing the l ∞ -error d − d R ∞ = max x,y∈X {|d(x, y) − d R (x, y)|} between d and d R .

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Recognition of Robinsonian dissimilarities

Abstract: Robinsonian dissimilarities, Order compatible with a dissimilarity, Divide-and-conquer algorithm,

Cited by 31 publications

References 8 publications

Combinatorial optimization and hierarchical classifications

Combinatorial optimization and hierarchical classifications

Combinatorial optimisation and hierarchical classifications

Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

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