Abstract
We present an optimal deterministic O(n)-work parallel algorithm for finding a minimum spanning tree on an n-vertex planar graph. The algorithm runs in \(O(\log n)\) time on a CRCW PRAM and in \(O(\log n\log ^*n)\) time on an EREW PRAM. Our results hold for any sparse graph that is closed under taking of minors, as well as for a class of graphs with non-bounded genus.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows. Prentice-Hall, Englewood (1993)
Awerbuch, B., Shiloach, Y.: New connectivity and MSF algorithms for shuffle-exchange network and PRAM. IEEE Trans. Comput. 36(10), 1258–1263 (1987)
Bodlaender, H., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput. 27(6), 1725–1746 (1998)
Boruvka, O.: O jistém problému minimálním. Práca Moravské Přírodovědecké Společnosti 3, 37–58 (1926)
Chazelle, B.: A minimum spanning tree algorithm with inverse-Ackermann type complexity. J. ACM 47(6), 1028–1047 (2000)
Cheriton, D., Tarjan, R.E.: Finding minimum spanning trees. SIAM J. Comput. 5, 724–742 (1976)
Chin, F.Y., Lam, J., Chen, I.N.: Efficient parallel algorithms for some graph problems. Commun. ACM 25(9), 659–665 (1982)
Chong, K.W.: Finding minimum spanning trees on the EREW PRAM. In: Proceedings of the International Computer Symposium—ICS’96, pp. 7–14. Taiwan (1996)
Chong, K.W., Lam, T.W.: Finding connected components in \(O(\log n\log \log n)\) time on the EREW PRAM. J. Algorithms 18, 378–402 (1995)
Chong, K.W., He, Y., Lam, T.W.: Concurrent threads and optimal parallel minimum spanning trees algorithm. J. ACM 48(2), 297–323 (2001)
Cole, R., Klein, P.N., Tarjan, R.E.: Finding minimum spanning forests in logarithmic time and linear work using random sampling. In: Proceedings of the 8th ACM symposium on Parallel Algorithms and Architectures (ACM), pp. 243–250 (1996)
Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Inf. Control 70, 32–53 (1986)
Cole, R., Vishkin, U.: Approximate and exact parallel scheduling with applications to list, tree and graph problems. In: Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, pp. 478–491. IEEE (1986)
Fredman, M., Willard, D.E.: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. J. Comput. Syst. Sci. 48, 533–551 (1994)
Gabow, H., Galil, Z., Spencer, T., Tarjan, R.E.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2), 109–122 (1986)
Goldberg, A., Plotkin, S., Shannon, G.: Parallel symmetry-breaking in sparse graphs. SIAM J. Discrete Math. 1, 434–446 (1988)
Hagerup, T.: Optimal parallel algorithms on planar graphs. Inf. Comput. 84, 71–96 (1990)
Hagerup, T.: Optimal Parallel Computation of Minimum Spanning Forests in Planar Graphs, Technical Report 11/1990. Universität des Saarlandes, May 1990
Hagerup, T., Chrobak, M., Diks, K.: Optimal parallel 5-colouring of planar graphs. SIAM J. Comput. 18(2), 288–300 (1989)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
JáJá, J.: An Introduction to Parallel Algorithms. Addison-Wesley, Reding (1992)
Johnson, D.B., Metaxas, P.: Connected components in \(O(\log ^{3/2}|V|)\) parallel time for the CREW PRAM. In: Proceeings of 32nd IEEE Symposium on Foundations of Computer Science, pp. 688–695, IEEE (1991)
Johnson, D.B., Metaxas, P.: A parallel algorithm for computing minimum spanning trees. J. Algorithms 19, 383–401 (1995)
Karger, D.R.: Approximating, verifying, and constructing minimum spanning trees. Unpublished manuscript (1992)
Karger, D.R., Klein, P.N., Tarjan, R.E.: A randomized linear-time algorithm to find minimum spanning trees. J. ACM 42(2), 321–328 (1995)
Karp, R., Ramachandran, V.: Parallel Algorithms for Shared-Memory Machines. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, pp. 869–941. Elsevier, Amsterdam (1990)
Pettie, S., Ramachandran, V.: An optimal minimum spanning tree algorithm. J. ACM 49(1), 16–34 (2002)
Pettie, S., Ramachandran, V.: A randomized time-work optimal parallel algorithm for finding a minimum spanning forest. SIAM J. Comput. 31(6), 1879–1895 (2002)
Zaroliagis, C.D.: Simple and work-efficient parallel algorithms for the minimum spanning tree problem. Parallel Process. Lett. 7(1), 25–37 (1997)
Acknowledgements
The last author is indebted to his mentor Paul Spirakis, who taught him by example to be a scientist and who uniquely affected the shaping of his career.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chong, K.W., Zaroliagis, C. (2015). An Optimal Parallel Algorithm for Minimum Spanning Trees in Planar Graphs. In: Zaroliagis, C., Pantziou, G., Kontogiannis, S. (eds) Algorithms, Probability, Networks, and Games. Lecture Notes in Computer Science(), vol 9295. Springer, Cham. https://doi.org/10.1007/978-3-319-24024-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-24024-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24023-7
Online ISBN: 978-3-319-24024-4
eBook Packages: Computer ScienceComputer Science (R0)