Minimum Spanning Tree

A minimum spanning tree (MST) connects all the vertices of a weighted graph by existing edges whose total weight is minimum. It has been the object of intensive study due to its importance in combinatorial optimization. In this paper, we discuss the formulation of the MST problem, present some of the most commonly used algorithms for its construction, and discuss several of its applications in statistics, probability, and data analysis. Keywords: algorithms; multivariate data analysis; optimization; statistical applications

[1]  J. Steele.  Growth Rates of Euclidean Minimal Spanning Trees With Power Weighted Edges , 1988 .

[2]  Bernard Chazelle,et al.  A minimum spanning tree algorithm with inverse-Ackermann type complexity , 2000, JACM.

[3]  J. Friedman,et al.  Multivariate generalizations of the Wald--Wolfowitz and Smirnov two-sample tests , 1979 .

[4]  J. Friedman,et al.  Graphics for the Multivariate Two-Sample Problem , 1981 .

[5]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[6]  Charles T. Zahn,et al.  Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters , 1971, IEEE Transactions on Computers.

[7]  Alfred O. Hero,et al.  Image registration with minimum spanning tree algorithm , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[8]  Robert E. Tarjan,et al.  Efficient algorithms for finding minimum spanning trees in undirected and directed graphs , 1986, Comb..

[9]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[10]  Mauro Leoncini,et al.  On the Symmetric Range Assignment Problem in Wireless Ad Hoc Networks , 2002, IFIP TCS.

[11]  J. Friedman,et al.  Graph-Theoretic Measures of Multivariate Association and Prediction , 1983 .

[12]  Ronald L. Graham,et al.  On the History of the Minimum Spanning Tree Problem , 1985, Annals of the History of Computing.

[13]  N. Henze,et al.  On the multivariate runs test , 1999 .

[14]  D. Bertsimas,et al.  The Minimum Spanning Tree Constant in Geometrical Probability and Under the Independent Model: A Unified Approach , 1992 .

[15]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  Byron J. T. Morgan,et al.  Identifying influential observations in hierarchical cluster analysis , 1995 .

[17]  Alfred O. Hero,et al.  Geodesic entropic graphs for dimension and entropy estimation in manifold learning , 2004, IEEE Transactions on Signal Processing.

[18]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[19]  Godfried T. Toussaint,et al.  The relative neighbourhood graph of a finite planar set , 1980, Pattern Recognit..

[20]  J. Gower,et al.  Minimum Spanning Trees and Single Linkage Cluster Analysis , 1969 .

[21]  J Gasteiger,et al.  Checking the projection display of multivariate data with colored graphs. , 1997, Journal of molecular graphics & modelling.