Abstract
We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m logβ(m, n)=k 2); for planar graphs this bound can be improved toO(n+k 2). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k 2 n+n logn,k 2+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k 2).
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A. Aggarwal, L. J. Guibas, J. Saxe, and P. W. Shor,A linear time algorithm for computing the Voronoi diagram of a convex polygon, Discrete and Comput. Geom. 4, 1989, 591–604.
A. Aggarwal, H. Imai, N. Katoh, and S. Suri,Finding k points with minimum diameter and related problems, J. Algorithms, to appear.
A. Aggarwal and J. Wein,Computational Geometry, MIT LCS Research Seminar Series 3, 1988.
M. Blum, R. W. Floyd, V. R. Pratt, R. L. Rivest, and R. E. Tarjan,Time bounds for selection, J. Comput. Syst. Sci. 7, 1972, 448–461.
H. Booth and J. Westbrook,Linear algorithms for analysis of minimum spanning and shortest path trees in planar graphs, Tech. Rep. TR-763, Department of Computer Science, Yale University, Feb. 1990.
R. N. Burns and C. E. Haff,A ranking problem in graphs, 5th Southeast Conf. Combinatorics, Graph Theory and Computing 19, 1974, 461–470.
P. M. Camerini, L. Fratta, and F. Maffioli,The k shortest spanning trees of a graph, Int. Rep. 73-10, IEEE-LCE Politechnico di Milano, Italy, 1974.
D. Cheriton and R. E. Tarjan,Finding minimum spanning trees, SIAM J. Comput. 5, 1976, 310–313.
L. P. Chew and S. Fortune,Sorting helps for Voronoi diagrams, 13th Symp. Mathematical Progr., Japan, 1988.
D. Eppstein,Offline algorithms for dynamic minimum spanning tree problems, 2nd Worksh. Algorithms and Data Structures, Springer Verlag LNCS 519, 1991, 392–399.
D. Eppstein, Z. Galil, R. Giancarlo, and G. F. Italiano,Sparse dynamic programming, 1st ACM-SIAM Symp. Discrete Algorithms, San Francisco, 1990. 513–522.
D. Eppstein, G. F. Italiano, R. Tamassia, R. E. Tarjan, J. Westbrook, and M. Yung,Maintenance of a minimum spanning forest in a dynamic planar graph, 1st ACM-SIAM Symp. Discrete Algorithms, 1990, 1–11.
G. N. Frederickson,Data structures for on-line updating of minimum spanning trees, SIAM J. Comput. 14(4), 1985, 781–798.
G. N. Frederickson,Ambivalent data structures for dynamic 2-edge-connectivity and k smallest spanning trees, 32nd IEEE Conf. Foundations of Computer Science, 1991, to appear.
H. N. Gabow,Two algorithms for generating weighted spanning trees in order, SIAM J. Comput. 6, 1977, 139–150.
H. N. Gabow, Z. Galil, T. H. Spencer, and R. E. Tarjan,Efficient algorithms for minimum spanning trees on directed and undirected graphs, Combinatorica 6, 1986, 109–122.
H. N. Gabow and M. Stallman,Efficient algorithms for graphic matroid intersection and parity, 12th Int. Conf. Automata, Languages, and Programming, Springer-Verlag LNCS 194, 1985, 210–220.
N. Katoh, T. Ibaraki, and H. Mine,An algorithm for finding k minimum spanning trees, SIAM J. Comput. 10, 1981, 247–255.
E. W. Mayr and C. G. Plaxton,On the spanning trees of weighted graphs, manuscript, 1990.
N. Sarnak and R. E. Tarjan,Planar point location using persistent search trees, C. ACM 29(7), 1986, 669–679.
R. E. Tarjan,Applications of path compression on balanced trees, J. ACM 26, 1979, 690–715.
P. van Emde Boas,Preserving order in a forest in less than logarithmic time, 16th IEEE Symp. Found. Comput. Sci., 1975, and Info. Proc. Lett. 6, 1977, 80–82.