Abstract
We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total sum of vertex costs at most \(c\sqrt {\Sigma _{v \in V(G)} (cost(v))^2 }\) and that this bound is optimal to within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results. We describe applications of our separator theorems to graph embedding problems, to graph pebbling, and to multi-commodity flow problems.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for graphs with an excluded minor and its applications. Proceedings of the 22nd Symp. on Theory of Computing, pages 293–299, 1990.
S. Bhatt, F. Chung, T. Leighton, and A. Rosenberg. Optimal simulations of tree machines. In Proc. 27th IEEE Symposium on Foundations of Computer Science (FOCS), pages 274–282, 1986.
S. N. Bhatt, F. R. K. Chung, J. W. Hong, F. T. Leighton, and A. L. Rosenberg. Optimal simulations by butterfly networks. In Proc. 20th ACM Symposium on Theory of Computing (STOC), pages 192–204, 1988.
S.N. Bhatt and F.T. Leighton. A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences, 28:300–343, 1984.
Krzystof Diks, Hristo N. Djidjev, Ondrej Sykora, and Imrich Vrto. Edge separators of planar and outerplanar graphs with applications. J. Algorithms, 14:258–279, 1993.
Hristo N. Djidjev. A separator theorem. Compt. rend. Acad. bulg. Sci., 34:643–645, 1981.
Hristo N. Djidjev and Shankar Venkatesan. Reduced constants for simple cycle graph separation. Acta Informatica, 1995. in print.
G.N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16:1004–1022, 1987.
Hillel Gazit and Gary L. Miller. Planar separators and the Euclidean norm. In SIGAL 90, Lecture Notes in Computer Science, vol. 450, pages 338–347. SpringerVerlag, Berlin, Heidelberg, New York, Tokio, 1990.
John R. Gilbert, Joan P. Hutchinson, and Robert E. Tarjan. A separator theorem for graphs of bounded genus. J. Algorithms, 5:391–407, 1984.
John R. Gilbert, Donald J. Rose, and Anders Edenbrandt. A separator theorem for chordal graphs. SIAM Journal on Algebraic and Discrete Methods, pages 306–313, 1984.
John R. Gilbert and Robert E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377–404, 1987.
Michael T. Goodrich. Planar separators and parallel polygon triangulation. Proceedings of 24th Symp. on Theory of Computing, pages 507–516, 1992.
Samir Khuller, Balaji Raghavachari, and Neal Young. Designing multi-commodity flow trees. Information Processing Letters, 50:49–55, 1994.
P. Klein, S. Rao, M. Rauch, and S. Subramanian. Faster shortest-path algorithms for planar graphs. In 26th ACM Symp. Theory of Computing, pages 27–37, 1994.
Leighton, Makedon, Plotkin, Stein, Tardos, and Tragoudas. Fast approximation algorithms for multicommodity flow problems. Journal of Computer and System Sciences, 50:228–243, 1995.
F. Thomas Leighton and Satish Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In Proceedings of the 29th IEEE Symposium on the Foundations of Computer Science, pages 422–431, 1988.
C.E. Leiserson. Area efficient VLSI computation. In Foundations of Computing. MIT Press, Cambridge, MA, 1983.
Richard J. Lipton, D. J. Rose, and Robert E. Tarjan. Generalized nested dissection. SIAM J. Numer. Anal., 16:346–358, 1979.
Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math, 36:177–189, 1979.
Richard J. Lipton and Robert E. Tarjan. Applications of a planar separator theorem. SIAM Journal on Computing, 9:615–627, 1980.
Gary L. Miller.Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Sciences, pages 265–279, 1986.
Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis. A unified geometric approach to graph separators. Proceedings of the 32nd FOCS, pages 538–547, 1991.
Gary L. Miller and William Thurston. Separators in two and three dimensions. Proceedings of the 22nd Symp. on Theory of Computing, pages 300–309, 1990.
B. Monien and H. Sudborough. Comparing interconnection networks. In Symposium on Mathematical Foundations of Computer Science, volume 320, pages 138–153, 1988.
N. Pippenger. Advances in pebbling. In Annual International Colloquium on Automata, Languages and Programming, pages 407–417, 1982.
P. D. Seymour and R. Thomas. Call routing and the ratcatcher. Combinatorica, 14:217–241, 1994.
Shang-Hua Teng. Points, spheres, and separators: A unified geometric approach to graph partitioning. Ph.D.-Thesis CMU-CS-91-184, Carnegie Mellon University, Pittsburgh, 1991.
H. Venkateswaran and M. Tompa. A new pebble game that characterizes parallel complexity classes. SIAM Journal on Computing, 18:533–549, 1989.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Djidjev, H.N. (1997). Weighted graph separators and their applications. In: Burkard, R., Woeginger, G. (eds) Algorithms — ESA '97. ESA 1997. Lecture Notes in Computer Science, vol 1284. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63397-9_11
Download citation
DOI: https://doi.org/10.1007/3-540-63397-9_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63397-6
Online ISBN: 978-3-540-69536-3
eBook Packages: Springer Book Archive