Abstract
We consider finite periodic graphs G m defined by nonnegative integer vectors m and directed graphs G whose edges are labeled with integer vector-weights. G m has a vertex (u, x) for each vertex u of G and each nonnegative integer vector x less than or equal to m. G m has an edge from (u, x) to (v, x + z) if and only if G has an edge from u to v with vector weight z.
We analyze the complexity and present algorithms for finding paths and cycles in finite periodic graphs. The present paper shows that path and cycle problems on finite periodic graphs are PSPACE-complete under various restrictions, but solvable in polynomial time if the vector weights of the edges are bounded.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
W. Backes, U. Schwiegelshohn, and L. Thiele. Analysis of free schedule in periodic graphs. Proceedings of the fourth annual ACM Symposium on Parallel Algorithms and Architectures, pages 333–343, 1992.
E. Cohen and N. Megiddo. Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs. In Annual ACM Symposium on Theory of Computing, pages 523–534, 1989.
E. Cohen and N. Megiddo. Recognizing properties of periodic graphs. The Victor Klee Festschrift, Honorary Volume of Applied Geometry and Discrete Mathematics, 1990.
M.R. Garey and D.S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco, 1979.
F. Höfting and E. Wanke. Minimum cost path in periodic graphs. Technical Report 99, Universität-Gesamthochschule-Paderborn, Paderborn, FRG, April 1992.
F. Höfting and E. Wanke. Polynomial algorithms for minimum cost paths in periodic graphs. Proceedings of the fourth annual ACM-SIAM Symposium on Discrete Algorithms, pages 493–499, 1993.
K. Iwano and K. Steiglitz. Testing for cycles in infinite graphs with periodic structure. In Proceedings of Annual ACM Symposium on Theory of Computing '87, pages 46–55, 1987.
R.M. Karp, R.E. Miller, and A. Winograd. The organization of computations for uniform recurrence equations. Journal of the ACM, 14(3):563–590, July 1967.
M. Kodialam and J.B. Orlin. Recognizing strong connectivity in (dynamic) periodic graphs and its relation to integer programming. Proceedings of the second annual ACM-SIAM Symposium on Discrete Algorithms, pages 131–135, 1991.
S.R. Kosaraju and G.F. Sullivan. Detecting cycles in dynamic graphs in polynomial time. In Proceedings of Annual ACM Symposium on Theory of Computing '88, pages 398–406, 1988.
J.B. Orlin. Some problems on dynamic/periodic graphs. In W.R. Pulleyblank, editor, Progress in Combinatorial Optimization, pages 273–293. Academic Press, Orlando, Florida, 1984.
S.K. Rao. Regular iterative algorithms and their implementations on processor arrays. PhD thesis, Department of Electrical Engineering, Stanford University, 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wanke, E. (1993). Paths and cycles in finite periodic graphs. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_66
Download citation
DOI: https://doi.org/10.1007/3-540-57182-5_66
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57182-7
Online ISBN: 978-3-540-47927-7
eBook Packages: Springer Book Archive