“Minimax length links” of a dissimilarity matrix and minimum spanning trees
Article
Received:
Revised:
- 78 Downloads
- 1 Citations
Abstract
A theorem is proved stating that the set of all “minimax links”, defined as links minimizing, over paths, the maximum length of links in any path connecting a pair of objects comprising nodes in an undirected weighted graph, comprise the union of all minimum spanning trees of that graph. This theorem is related to methods of fitting network models to dissimilarity data, particularly a method called “Pathfinder” due to Schvaneveldt and his colleagues, as well as to single linkage clustering, and results concerning the relationship between minimum spanning trees and single linkage hierarchical trees.
Key words
minimum spanning trees hierarchical clustering single linkage network models proximity data undirected weighted graphsPreview
Unable to display preview. Download preview PDF.
References
- Dearholt, D. W., & Schvaneveldt, R. W. (1990). Properties of Pathfinder networks. In R. W. Schvaneveldt (Ed.),Pathfinder associative networks: Studies in knowledge organization (pp. 1–30). Norwood, NJ: Ablex.Google Scholar
- Gower, J. C., & Ross, G. J. S. (1969). Minimum spanning trees and single linkage cluster analysis.Applied Statistics, 18, 54–64.Google Scholar
- Hartigan, J. A. (1975).Clustering algorithms. New York: John Wiley & Sons.Google Scholar
- Hutchinson, J. W. (1981). Network representations of psychological relations. Unpublished doctoral dissertation, Stanford University.Google Scholar
- Hutchinson, J. W. (1989). NETSCAL: A network scaling algorithm for non-symmetric proximity data.Psychometrika, 54, 25–52.Google Scholar
- Johnson, S. C. (1967). Hierarchical clustering schemes.Psychometrika, 32, 241–254.Google Scholar
- Klauer, K. C. (1989). Ordinal representation: Representing proximities by graphs.Psychometrika, 54, 737–750.Google Scholar
- Klauer, K. C., & Carroll, J. D. (1989). A mathematical programming approach to fitting general graphs.Journal of Classification, 6, 247–270.Google Scholar
- Klauer, K. C., & Carroll, J. D. (1991). A comparison of two approaches to fitting directed graphs to nonsymmetric proximity measures.Journal of Classification, 8, 251–268.Google Scholar
- Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem.Proceedings of the American Mathematical Society, 7, 48–50.Google Scholar
- Prim, R. C. (1957). Shortest connection networks and some generalizations.The Bell System Technical Journal, XXXVI, 1389–1401.Google Scholar
- Schvaneveldt, R. W., Dearholt, D. W., & Durso, F. T. (1988). Graph theoretic foundations of Pathfinder networks.Comput. Math. Applic., 15, 337–345.Google Scholar
Copyright information
© The Psychometric Society 1995