Wang tilings and distributed orientation on anonymous torus networks (extended abstract)
This paper investigates the orientation problem on anonymous n×n torus networks. In the orientation problem, the goal is to reach an agreement, among all processors, on a consistent channel labelling. This paper shows that, if processors can label their channels arbitrarily, then there is no distributed orientation algorithm. Therefore the orientation problem is studied in various orientation models ℳ that restrict local channel labellings. For many ℳ, there are network labellings that, to the processors, are indistinguishable from orientations. These pseudo-orientations give rise to two problems related to orientation. In the verification problem, processors in an n×n torus distributively verify whether a network labelling in ℳ is an orientation. The verification problem is shown to be equivalent to a problem of tiling the torus with Wang tiles. In the pseudo-orientation problem, the goal is to distributively find a pseudo-orientation. The pseudo-orientation problem is shown to be equivalent to a problem in graph theory.
Unable to display preview. Download preview PDF.
- 1.Attiya, H., Snir, M., Warmuth, M.: Computing on an Anonymous Ring. Journal of the ACM, Vol. 35, No. 4 (1988), 845–875.Google Scholar
- 2.Beame, P.W., Bodlaender, H.L.: Computing on Transitive Networks: The Torus. In B. Monien and R. Cori, editors, Sixth Annual Symposium on Theoretical Aspects of Computer Science (1989), 294–303.Google Scholar
- 3.Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Elsevier Science Pub. Co., 1976.Google Scholar
- 4.Budden, F.J.: The Fascination of Groups. Cambridge University Press, 1972.Google Scholar
- 5.Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman, 1987.Google Scholar
- 6.Harary, F., Wilcox, G.W.: Boolean Operations on Graphs. Math. Scand. 20 (1967), 41–51.Google Scholar
- 7.Kranakis, E., Krizanc, D.: Distributed Computing on Anonymous Hypercube Networks. Proceedings of the 3rd IEEE Symposium on Parallel and Distributed Processing (1991), 722–729.Google Scholar
- 8.Norris, N.: Universal Covers of Edge-Labelled Digraphs: Isomorphism to Depth n-1 Implies Isomorphism to all Depths. Technical Report UCSC-CRL-90-49, University of California at Santa Cruz, 1990.Google Scholar
- 9.Syrotiuk, V.R.: Wang Tilings and Distributed Orientation on Anonymous Torus Networks. Ph.D. Thesis, Department of Computer Science, University of Waterloo, 1992.Google Scholar
- 10.Tel, G.: Network Orientation. Technical Report RUU-CS-91-8, Utrecht University, March 1991.Google Scholar
- 11.Wang, H.: Notes on a Class of Tiling Problems. Fundamenta Mathematicae 82 (1975), 295–305.Google Scholar
- 12.Yamashita, M., Kameda, T.: Computing on an Anonymous Network. Proceedings of the 7th Annual ACM Symposium on Principles of Distributed Computing (1988), 117–130.Google Scholar