Sequential clustering with radius and split criteria

  • Nenad Mladenovic
  • Pierre Hansen
  • Jack Brimberg
Original Paper


Sequential clustering aims at determining homogeneous and/or well-separated clusters within a given set of entities, one at a time, until no more such clusters can be found. We consider a bi-criterion sequential clustering problem in which the radius of a cluster (or maximum dissimilarity between an entity chosen as center and any other entity of the cluster) is chosen as a homogeneity criterion and the split of a cluster (or minimum dissimilarity between an entity in the cluster and one outside of it) is chosen as a separation criterion. An O(N 3) algorithm is proposed for determining radii and splits of all efficient clusters, which leads to an O(N 4) algorithm for bi-criterion sequential clustering with radius and split as criteria. This algorithm is illustrated on the well known Ruspini data set.


Clustering Sequential Efficient cluster Radius Split 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Batagelj V, Ferligoj A (1990) Agglomerative Hierarchical Multicriteria clustering using decision rules. In: COMPSTAT 1990. Physica Verlag, Heidelberg, pp 15–20Google Scholar
  2. Carrizosa E, Mladenović N, Todosijević R (2011) Sum-of-squares clustering on networks. Yugosl J Oper Res 21: 157–161CrossRefGoogle Scholar
  3. Chang MS, Tang CY, Lee RCT (1991) A unified approach for solving bottleneck k-bipartition problems. Proceedings of the 19th annual computer science conference, San Antonio, TX, 5–7 March 1991, ACM, pp 39–47Google Scholar
  4. Delattre M, Hansen P (1980) Bicriterion cluster analysis. IEEE Trans Pattern Anal Mach Intell PAMI-2(4): 277–291CrossRefGoogle Scholar
  5. Du Merle O, Hansen P, Jaumard B, Mladenović N (1999) An interior point algorithm for minimum sum of squares clustering problem. SIAM J Sci Stat Comput 21(4): 1485–1505CrossRefGoogle Scholar
  6. Ferligoj A, Batagelj V (1992) Direct multicriteria clustering algorithms. J Classif 9(1): 43–61CrossRefGoogle Scholar
  7. Fortunato S (2010) Community detection in graphs. Phys Rep 486: 75–174CrossRefGoogle Scholar
  8. Gordon AD (1981) Classification. Chapman and Hall, LondonGoogle Scholar
  9. Gower JC, Ross GJS (1969) Minimum spanning trees and single linkage cluster analysis. Appl Stat 18: 54–64CrossRefGoogle Scholar
  10. Guénoche A, Hansen P, Jaumard B (1991) Efficient algorithm for divisive hierarchical clustering with the diameter criterion. J Classif 8(1): 5–30CrossRefGoogle Scholar
  11. Hansen P, Delattre M (1978) Bicriterion cluster analysis as an exploration tool. In: Zionts S (ed) Multiple criteria problem solving. Springer, LondonGoogle Scholar
  12. Hansen P, Jaumard B (1997) Cluster Analysis and Mathematical Programming, series B, pp 191–215Google Scholar
  13. Hansen P, Jaumard B, Mladenović N (1994) How to choose k entities among n. In: Cox I, Hansen P, Julesz B (eds) Partitioning data sets, pp 105–116Google Scholar
  14. Hansen P, Jaumard B, Mladenović N (1998) Minimum sum of squares clustering in a low dimensional space. J Classif 15: 37–56CrossRefGoogle Scholar
  15. Hartigan JA (1975) Clustering algorithms. Wiley, New YorkGoogle Scholar
  16. Jain AK, Dubes RC (1988) Algorithms for clustering data. Prentice Hall, Englewood CliffsGoogle Scholar
  17. Kaufman L, Rousseeuw PJ (2005) Finding groups in data: an introduction to cluster analysis. Wiley, New YorkGoogle Scholar
  18. Mirkin B (1987) Additive clustering and qualitative factor analysis methods for similarity matrices. J Classif 4:7–31 (Erratum 6, 271–272)Google Scholar
  19. Mirkin B (1996) Mathematical classification and clustering. Kluwer, DordrechtCrossRefGoogle Scholar
  20. Prim JC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36: 1389–1401Google Scholar
  21. Romesburg HC (2004) Cluster analysis for researchers. Lulu Press, North CarolinaGoogle Scholar
  22. Rosenstiehl P (1967) L’arbre minimum d’un graphe. In: Rosentiehl P (ed) Théorie des Graphes. Dunod, Paris, pp 357–368Google Scholar
  23. Ruspini EH (1970) Numerical methods for fuzzy clustering. Inf Sci 2: 319–350CrossRefGoogle Scholar
  24. Shepard N, Arabie P (1979) Additive clustering representation of similarities as combinations of discrete overlapping properties. Psychol Rev 86: 87–133CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Nenad Mladenovic
    • 1
  • Pierre Hansen
    • 2
  • Jack Brimberg
    • 3
  1. 1.Brunel UniversityLondonUK
  2. 2.GERADHEC MontrealMontrealCanada
  3. 3.Royal Military CollegeKingstonCanada

Personalised recommendations