Abstract
This paper is devoted to the notion of average consensus together with some generalizations involving L p -norms.
We prove that finding one of these consensus dissimilarities out of a profile of dissimilarities is NP-hard for ultrametrics, quasi-utrametrics and proper dissimilarities satisfying the Bertrand and Janowitz k-point inequality. The NP-hardness of finding a consensus dissimilarity for a pyramid (also called an indexed pseudo-hierarchy) is also proved in the case of one of the two possible alternatives for generalized average consensus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BANDELT, H.J. and DRESS, A.W.M. (1989): Weak hierarchies associated with dissimilarity measures: an additive clustering technique. Bulletin of Mathematical Biology, 51 (1), 133–166.
BARTHELEMY, J.-P. and JANOWITZ, M.F. (1991): A formal theory of consensus. SIAM J. Discr. Math., 4, 305–322.
BARTHELEMY, J.-P., LECLERC, B. and MONJARDET, B. (1986): On the use of ordered sets in problems of comparison and consensus of classifications. J. of Classification, 3, 187–224.
BARTHELEMY, J.-P. and MONJARDET, B. (1981): The median procedure in cluster analysis and social choice theory. Math. Soc. Sci., 1, 235–267.
BENZECRI, J.-P. (1973): L’analyse des données, Volume 1: Taxonomie. Dunod, Paris.
BERTRAND, P. and JANOWITZ, M.J. (1999): The k-weak hierarchies: an extension of weak hierarchies. preprint.
BOCK, H.H. (1994): Classification and clustering: Problems for the future. In Diday et al. (eds): New approaches in classification and data analysis. Springer-Verlag, Berlin, 3–24.
CHEPOI, V. and FICHET, B. (2000): le-Approximation via subdominants. Journal of Mathematical psychology, to appear.
CUCUMEL, G. (1990): Construction d’une hiérarchie consensus à l’aide d’une ultramétrique centrale. In Recueil des Textes des Présentations du Colloque sur les Méthodes et Domaines d’Applications de la Statistique 1990, Bureau de la Statistique du Québec, Quebec, 235 - 243.
DIATTA, J. and FICHET, B. (1994): From Asprejan hierarchies and Bandelt-Dress weak hierarchies to quasi-hierarchies. In E. Diday et al. (eds.), New approaches in classification and data analysis, Springer Verlag, Berlin, 111–118.
DIATTA, J. and FICHET, B. (1998): Quasi-ultrametriccs and their 2-balls hyper-graphs. Discrete Math., 192, 87–102.
DIDAY, E. (1984): Orders and overlapping clustering by pyramids. Research Report, 73, INRIA.
DIDAY, E. (1986): Orders and overlapping clusters in pyramids. In De Leeuw et al. (ed.): Multidimensional Data Analysis, DSWO Press, Leiden, 201–234.
FICHET, B. (1984): Sur une extension de la notion de hiórarchie et son üquivalence avec une matrice de Robinson. In Journóes de Statistiques de la Grande Motte.
FICHET, B. (1986): Data analysis: Geometric and algebraic structures. In Y.A Pro- horov and V.U. Sasonov (eds.): First world Congress of the Bernoulli Society Proceedings. V.N.U. Science Press, Utrecht, 123–132.
FICHET, B. (1988): 4-spaces in data analysis. In H.H. Bock (ed.): Classification and Related Methods of Data Analysis. Amsterdam: North Holland, 439–444.
GAREY, M.R. and JOHNSON, D.S. (1979): Computer and intractability: A guide in the theory of NP-completeness. Freeman, New York.
GORDON, A. (1980): On the assessment and comparison of classification. In R. Tomassone (ed.): Analyse des données et informatique, INRIA, Le Chesnay, 149–160.
GORDON, A. (1981): Classification methods for the exploratory analysis of multivariate data. Chapman and Hill, London.
GORDON, A. (1986): Consensus supertrees: the synthesis of rooted trees containing overlapping sets of labelled leaves. J. of Classification, 3, 2, 335–348.
JARDINE, J.P.J., JARDINE, N. and SIBSON, R.S. (1967): The structure and construction of taxonomic hierarchies. Mathematical Biosciences, 1, 171–179.
JOHNSON, S.C. (1967): Hierarchical clustering schemes. Psychometrika, 32, 241–254.
KRIVANEK, M. and MORAVEK, J. (1986): NP-hard problems in hierarchical clustering. Acta Informatica, 23, 311–323.
LAPOINTE, F.-J. and CUCUMEL, G. (1997): The average consensus procedure. Combination of weighted trees containing identical or overlapping sets of taxa. Systematic Biology, 46, 306–312.
MARGUSH, T. and MCMORRIS, F.R. (1981): Consensus n-trees. Bull. Math. Biology, 43, 239–224.
MCMORRIS, F.R. (1985): Axioms for consensus functions defined on undirected phylogenetic trees. Math. Biosciences, 74, 77–80.
RÉGNIER, S. (1965): Sur quelques aspects mathématiques des problèmes de classification automatique. ICC Bull., 4, 175–191. Reprinted in Math. Sc. Hum., 82(1983), 13–29.
WAKABAYASHI, Y. (1986): Aggregation of binary relations: algorithmic and polyhedral investigations, Ph.D. Dissertation, University of Augsburg.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Barthélemy, JP., Brucker, F. (2000). Average Consensus in Numerical Taxonomy and Some Generalizations. In: Gaul, W., Opitz, O., Schader, M. (eds) Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58250-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-58250-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67731-4
Online ISBN: 978-3-642-58250-9
eBook Packages: Springer Book Archive