Abstract
We study some combinatorial and algorithmic problems associated with an arbitrary motion of input points in space. The motivation for such an investigation comes from two different sources:computer modeling andsensitivity analysis. In modeling, the dynamics enters the picture since geometric objects often model physical entities whose positions can change over time. In sensitivity analysis, the motion of the input points might represent uncertainties in the precise location of objects.
The main results of the paper deal with state transitions in the minimum spanning tree when one or more of the input points move arbitrarily in space. In particular, questions of the following form are addressed: (i) How many different minimum spanning trees can arise if one point moves while the others remain fixed? (ii) When does the minimum spanning tree change its topology if all points are allowed to move arbitrarily?
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T. Asano, B. Bhattacharya, M. Keil, and F. F. Yao. Clustering algorithms based on minimum and maximum spanning trees.Proceedings of the 4th Annual Symposium on Computational Geometry, 1988, pp. 252–257.
M. J. Atallah. Some dynamic computational geometry problems.Journal of Computational and Applied Mathematics, Vol. 11, 1985, pp. 1117–1181.
B. Chazelle, R. Cole, F. P. Preparata, and C. Yap. New upper bounds for neighbor searching.Information and Control, Vol. 68, 1986, pp. 105–124.
L. P. Chew and R. L. Drysdale. Voronoi diagrams based on convex distance functions.Proceedings of the 1st ACM Symposium on Computational Geometry, 1985, pp. 235–244.
W. Cunningham. Optimal attack and reinforcement of a network.Journal of the Association for Computing Machinery, Vol. 32, No. 3, 1985, pp. 549–561.
M. E. Dyer. On a multidimensional search technique and its application to the Euclidean one-center problem.SIAM Journal on Computing, Vol. 15, 1986, pp. 725–738.
H. Edelsbrunner.Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, 1987.
G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm.SIAM Journal on Computing, Vol. 18, 1989, pp. 30–55.
G. Georgakopoulos and C. H. Papadimitriou. The 1-Steiner tree problem.Journal of Algorithms, Vol. 8, 1987, pp. 122–130.
D. Gusfield. Bounds for the parametric spanning tree problem.Proceedings of the Humbolt Conference on Graph Theory, Combinatorics and Computing, 1979, pp. 173–183.
D. Gusfield. Sensitivity analysis for combinatorial optimization. Ph.D. thesis, University of California, Berkeley, 1980.
D. Gusfield. Parametric combinatorial optimization and a problem of program module distribution.Journal of the Association for Computing Machinery, Vol. 30, No. 3, 1983, pp. 551–563.
D. Gusfield and R. Irving. Parametric stable marriage and minimum cuts.Information Processing Letters, Vol. 30, 1989, pp. 255–259.
D. Gusfield and C. Martel. A fast algorithm for the generalized parametric minimum cut problem and applications. Technical Report CSE-89-21, University of California, Davis, 1989.
D. Harel and R. E. Tarjan. Fast algorithms for finding nearest common ancestors.SIAM Journal on Computing, Vol. 13, No. 2, 1984, pp. 338–355.
C. Monma, M. Paterson, S. Suri, and F. Yao. Computing Euclidean maximum spanning trees.Algorithmica, Vol. 5, 1990, pp. 407–419.
J. C. Picard and M. Queyranne. Selected applications of minimum cuts in a network.INFOR—Canadian Journal of Operations Research and Information Processing, Vol. 20, 1982, pp. 394–422.
J. C. Picard and H. Ratliff. A cut approach to the rectilinear distance facility location problem.Operations Research, Vol. 26, 1978, pp. 422–433.
D. D. Sleater and R. E. Tarjan. A data structure for dynamic trees.Journal of Computer and Systems Sciences, Vol. 26, 1983, pp. 362–391.
A. C. C. Yao. On constructing minimum spanning trees ink-dimensional space and related problems.SIAM Journal on Computing, Vol. 11, No. 4, 1982, pp. 721–736.
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Monma, C., Suri, S. Transitions in geometric minimum spanning trees. Discrete Comput Geom 8, 265–293 (1992). https://doi.org/10.1007/BF02293049
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DOI: https://doi.org/10.1007/BF02293049