Abstract
We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning forest of a graph with n vertices and m edges that runs in time O(T(m,n)) where T is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine.
Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T are T(m,n)=Ω(m) and T(m,n)=O(itm·α(m,n)), where α is a certain natural inverse of Ackermann’s function.
Even under the assumption that T is super-linear, we show that if the input graph is selected from G n,m, our algorithm runs in linear time w.h.p., regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for G n,m similar to the edge-exposure martingale for G n,p.
Part of this work was supported by Texas Advanced Research Program Grant 003658-0029-1999. Seth Pettie was also supported by an MCD Fellowship.
Chapter PDF
References
N. Alon, J. Spencer. The Probabilistic Method. Wiley, New York, 1992.
O. Borüvka. O jistém problému minimaálním. Moravské Přírodovědecké Společnosti 3, pp. 37–58, 1926. (In Czech).
B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In FOCS’ 97, pp. 22–31, 1997.
B. Chazelle. Car-pooling as a data structuring device: The soft heap. In ESA’ 98(Venice), pp. 35–42, LNCS 1461, Springer, 1998.
B. Chazelle. A minimum spanning tree algorithm with inverse-Ackermann type complexity. NECI Technical Report 99-099, 1999.
K. W. Chong, Y. Han and T. W. Lam. On the parallel time complexity of undirected connectivity and minimum spanning trees. In Proc. SODA, pp. 225–234, 1999.
E. W. Dijkstra. A note on two problems in connexion with graphs. In Numer. Math., 1, pp. 269–271, 1959.
B. Dixon, M. Rauch, R. E. Tarjan. Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Jour. Comput., vol 21, pp. 1184–1192, 1992.
P. Erdös, A. Rényi On the evolution of random graphs. Bull. Inst. Internat. Statist. 38, pp. 343–347, 1961.
M. L. Fredman, R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. In JACM 34, pp. 596–615, 1987.
M. Fredman, D. E. Willard. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. In Proc. FOCS’ 90, pp. 719–725, 1990.
H. N. Gabow, Z. Galil, T. Spencer, R. E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. In Combinatorica 6, pp. 109–122, 1986.
R. L. Graham, P. Hell. On the history of the minimum spanning tree problem. Annals of the History of Computing 7, pp. 43–57, 1985.
V. Jarník. O jistém problému minimaálním. Moravské Přírodovědecké Společnosti 6, pp. 57–63, 1930. (In Czech).
D. R. Karger, P. N. Klein, and R. E. Tarjan. A randomized linear-time algorithm to find minimum spanning trees. JACM, 42:321–328, 1995.
R. M. Karp, R. E. Tarjan. Linear expected-time algorithms for connectivity problems. J. Algorithms 1 (1980), no. 4, pp. 374–393.
L. L. Larmore. An optimal algorithm with unknown time complexity for convex matrix searching. IPL, vol. 36, pp. 147–151, 1990.
S. Pettie, V. Ramachandran. A randomized time-work optimal parallel algorithm for finding a minimum spanning forest Proc. RANDOM’ 99, LNCS 1671, Springer, pp. 233–244, 1999.
S. Pettie, V. Ramachandran. An optimal minimum spanning tree algorithm. Tech Report TR99-17, Univ. of Texas at Austin, 1999.
S. Pettie. Finding minimum spanning trees in O(mα(m, n)) time. Tech Report TR99-23, Univ. of Texas at Austin, 1999.
R. C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, 36:1389–1401, 1957.
R. E. Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets. In JCSS, 18(2), pp 110–127, 1979.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pettie, S., Ramachandran, V. (2000). An Optimal Minimum Spanning Tree Algorithm. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_6
Download citation
DOI: https://doi.org/10.1007/3-540-45022-X_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67715-4
Online ISBN: 978-3-540-45022-1
eBook Packages: Springer Book Archive