# Linear Orderings of Random Geometric Graphs

• Josep Díaz
• Mathew D. Penrose
• Jordi Petit
• María Serna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1665)

## Abstract

In random geometric graphs, vertices are randomly distributed on [0, 1]2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.

## Keywords

Linear Ordering Random Graph Layout Problem Geometric Graph Grid Graph
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.

## References

1. 1.
N. Alon, J. H. Spencer, and P. Erd?os. The probabilistic method. Wiley-Interscience, New York, 1992. 292
2. 2.
M. J. Appel and R. P. Russo. The connectivity of a graph on uniform points in [0, 1]d. Technical Report #275, Department of Statistics and Actuarial Science, University of Iowa, 1996. 294Google Scholar
3. 3.
J.E. Atkins, E.G. Boman, and B. Hendrickson. A spectral algorithm for seriation and the consecutive ones problem. SIAM J. Comput., 28(1):297–310, 1999. 291
4. 4.
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. 291
5. 5.
R. Boppana. Eigenvalues and graph bisection: An average case analysis. In Proc. on Foundations of Computer Science, pages 280–285, 1987. 292Google Scholar
6. 6.
T. Bui, S. Chaudhuri, T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. Combinatorica, 7:171–191, 1987. 292
7. 7.
Josep Díaz, Mathew D. Penrose, Jordi Petit, and María Serna. Layout problems on lattice graphs. In T. Asano, H. Imai, D.T. Lee, S. Nakano, and T. Tokuyama, editors, Computing and Combinatorics-5th Annual International Conference, COCOON’99, Tokyo, Japan, July 26-28, 1999, number 1627 in Lecture Notes in Computer Science. Springer-Verlag, Berlin, July 1999. 301Google Scholar
8. 8.
J. Díaz, J. Petit, M. Serna, and L. Trevisan. Approximating layout problems on random sparse graphs. Technical Report LSI 98-44-R, Universitat Polit`ecnica de Catalunya, 1998. 292Google Scholar
9. 9.
M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979. 293Google Scholar
10. 10.
F. Gavril. Some NP-complete problems on graphs. In Proc. 11th. Conference on Information Sciences and Systems, pages 91–95, John Hopkins Univ., Baltimore, 1977. 293Google Scholar
11. 11.
D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon. Optimization by simulated annealing: an experimental evaluation; part I, graph partitioning. Operations Research, 37(6):865–892, November 1989. 292
12. 12.
K. Lang and S. Rao. Finding Near-Optimal Cuts: An Empirical Evaluation. In Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 212–221, 1993. 292Google Scholar
13. 13.
T. Lengauer. Black-white pebbles and graph separation. Acta Informatica, 16:465–475, 1981. 293
14. 14.
G. Mitchison and R. Durbin. Optimal numberings of an n×n array. SIAM Journal on Discrete Mathematics, 7(4):571–582, 1986. 291
15. 15.
C. H. Papadimitriou and M. Sideri. The bisection width of grid graphs. In First ACM-SIAM Symposium on Discrete Algorithms, pages 405–410, San Francisco, 1990. 295Google Scholar
16. 16.
M. D. Penrose. The longest edge of the random minimal spanning tree. The Annals of Applied Probability, 7:340–361, 1997. 292
17. 17.
J. Petit. Combining Spectral Sequencing with Simulated Annealing for the MinLA Problem: Sequential and Parallel Heuristics. Technical Report LSI-97-46-R, Departament de Llenguatges i Sistemes Informàtics, Universitat Polit`ecnica de Catalunya, 1997. 291Google Scholar
18. 18.
J. Petit. Approximation Heuristics and Benchmarkings for the MinLA Problem. In R. Battiti and A. Bertossi, editors, Alex’ 98-Building bridges between theory and applications, http://www.lsi.upc.es/?jpetit/MinLA/Benchmark, February 1998. Universit`a di Trento. 291, 292
19. 19.
J. S. Turner. On the probable performance of heuristics for bandwidth minimization. SIAM Journal on Computing, 15:561–580, 1986. 291, 292

## Authors and Affiliations

• Josep Díaz
• 1
• Mathew D. Penrose
• 2
• Jordi Petit
• 1
• María Serna
• 1
1. 1.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain
2. 2.Department of Mathematical SciencesUniversity of DurhamDurhamEngland