Abstract
In this paper, a parallel algorithm is given that, given a graph G=(V, E), decides whether G is a series parallel graph, and if so, builds a decomposition tree for G of series and parallel composition rules. The algorithm uses O(log¦E¦log*¦E¦) time and O(¦E¦) operations on an EREW PRAM, andO(log¦E¦) time and O(¦E¦) operations on a CRCW PRAM (note that if G is a simple series parallel graph, then ¦E¦=O(¦V¦)). With the same time and processor resources, a tree-decomposition of width at most two can be built of a given series parallel graph, and hence, very efficient parallel algorithms can be found for a large number of graph problems on series parallel graphs, including many well known problems, e.g., all problems that can be stated in monadic second order logic. The results hold for undirected series parallel graphs graphs, as well as for directed series parallel graphs.
This research was partially supported by the Foundation for Computer Science (S.I.O.N) of the Netherlands Organisation for Scientific Research (N.W.O.).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese. An algebraic theory of graph reduction. J. ACM, 40:1134–1164, 1993.
M. W. Bern, E. L. Lawler, and A. L. Wong. Linear time computation of optimal subgraphs of decomposable graphs. J. Algorithms, 8:216–235, 1987.
H. L. Bodlaender and B. de Fluiter. Reduction algorithms for graphs with small treewidth. Technical Report UU-CS-1995-37, Department of Computer Science, Utrecht University, Utrecht, 1995.
H. L. Bodlaender and T. Hagerup. Parallel algorithms with optimal speedup for bounded treewidth. In Z. Fülöp and F. Gécseg, editors, Proceedings 22nd International Colloquium on Automata, Languages and Programming, pages 268–279, Berlin, 1995. Springer-Verlag, Lecture Notes in Computer Science 944.
R. B. Borie, R. G. Parker, and C. A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7:555–581, 1992.
R. J. Duffin. Topology of series-parallel graphs. J. Math. Anal. Appl., 10:303–318, 1965.
D. Eppstein. Parallel recognition of series parallel graphs. Information and Computation, 98:41–55, 1992.
X. He and Y. Yesha. Parallel recognition and decomposition of two terminal series parallel graphs. Information and Computation, 75:15–38, 1987.
T. Kikuno, N. Yoshida, and Y. Kakuda. A linear algorithm for the domination number of a series-parallel graph. Disc. Appl. Math., 5:299–311, 1983.
K. Takamizawa, T. Nishizeki, and N. Saito. Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM, 29:623–641, 1982.
J. Valdes, R. E. Tarjan, and E. L. Lawler. The recognition of series parallel digraphs. SIAM J. Comput., 11:298–313, 1982.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodlaender, H.L., de Fluiter, B. (1996). Parallel algorithms for series parallel graphs. In: Diaz, J., Serna, M. (eds) Algorithms — ESA '96. ESA 1996. Lecture Notes in Computer Science, vol 1136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61680-2_62
Download citation
DOI: https://doi.org/10.1007/3-540-61680-2_62
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61680-1
Online ISBN: 978-3-540-70667-0
eBook Packages: Springer Book Archive