Abstract
Nodes in the hexagonal mesh and torus network are placed at the vertices of a regular triangular tessellation, so that each node has up to six neighbors. The routing algorithm for the Hexagonal Torus is very complicated, and it is an open problem by now. Hexagonal mesh and torus are known to belong to the class of Cayley digraphs. In this paper, we use Cayley-formulations for the hexagonal torus, along with some result on subgraphs and Coset graphs, to develop the optimal routing algorithm for the Hexagonal Torus, and then we draw conclusions to the network diameter of the Hexagonal Torus.
This research is supported by the Natural Science Foundation of Guangdong Province, China (No. 04020130).
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References
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)
Chen, M.S., Shin, K.G., Kandlur, D.D.: Addressing, Routing and Broadcasting in Hexagonal Mesh Multiprocessors. IEEE Trans. Computers 39(1), 10–18 (1990)
Dolter, J.W., Ramanathan, P., Shin, K.G.: Performance Analysis of Virtual Cut-Through Switching in HARTS: A Hexagonal Mesh Multicomputer. IEEE Trans. Computers 40(6), 669–680 (1991)
Decayeux, C., Seme, D.: 3D hexagonal network: modeling, topological properties, addressing scheme, and optimal routing algorithm. IEEE Trans. on Parallel and Distrib. Sys. 16(9), 875–884 (2005)
García, F., Solano, J., Stojmenovic, I., Stojmenovic, M.: Higher dimensional hexagonal networks. Journal of Parallel and Distributed Computing 63(11), 1164–1172 (2003)
Heydemann, M.: Cayley Graphs and Interconnection Networks. In: Graph Symmetry: Algebraic Methods and Applications, pp. 167–224 (1997)
He, M.X., Xiao, W.J.: A Unified Addressing Schema for Hexagonal and Honeycomb Networks with Isomorphic Cayley Graphs. In: IMSCCS 2006. Proc. of 1st Int. Multi-Symp. of Computer and Computational Sciences, Hangzhou, China, vol. 1, pp. 363–368 (2006)
Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypecubes. Morgan Kaufmann, San Francisco (1992)
Nocetti, F.G., Stojmenovic, I., Zhang, J.Y.: Addressing and Routing in Hexagonal Networks with Applications for Tracking Mobile Users and Connection Rerouting in Cellular Networks. IEEE Trans. Parallel and Distributed Systems 13(9), 963–971 (2002)
Parhami, B.: Introduction to Parallel Processing: Algorithms and Architectures, Plenum (1999)
Robic, B., Silc, J.: High performance Computing on a Honeycomb Arcbhitecture. In: Proc. Int. ACPC Parallel Computation Conf. (1993)
Stojmenovic, I.: Honeycomb Networks: Topological Properties and Communication Algorithms. IEEE Trans. Parallel and Distributed systems 8(10), 1036–1042 (1997)
Shin, K.G.: HARTS: A Distributed Real-Time Architectrue. Computer 24(5), 25–35 (1991)
Tosic, R., Masulovic, D., Stojmenovic, I., et al.: Enumeration of Polyhex Hydrocarbons up to h=17. J. Chemical Infromation and Computer Sciences 35, 181–187 (1995)
Xiao, W.J., Parhami, B.: Some Mathematical Properties of Cayley Digraphs with Applications to Interconnection Network Design. Int. J. Computer Mathematics 82, 521–528 (2005)
Xiao, W.J., Parhami, B.: Further Mathematical Properties of Cayley Digraphs Applied to Hexagonal and Honeycomb Meshes. Discrete Applied Mathematics (to appear, 2007)
Xiao, W.J., Parhami, B.: Hexagonal and Pruned Torus Networks as Cayley Graphs. In: Proc. International Conf. on Communications in Computing, Las Vegas, June 21-24, 2004, pp. 107–112 (2004)
Xiao, W.J., Parhami, B.: Structural Properties of Cayley Digraphs with Applications to Mesh and Pruned Torus Interconnection Networks. Int. J. of Computer and System Sciences, Special Issue on Network-Based Computing (to appear, 2007)
Xiao, W.J., Parhami, B.: A Group Consruction Method with Applications to Deriving Pruned Interconnection Networks. IEEE Trans. on Parallel and Distrib. Sys. 18(5), 637–643 (2007)
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Zhang, Z., Xiao, W., He, M. (2007). Optimal Routing Algorithm and Diameter in Hexagonal Torus Networks. In: Xu, M., Zhan, Y., Cao, J., Liu, Y. (eds) Advanced Parallel Processing Technologies. APPT 2007. Lecture Notes in Computer Science, vol 4847. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76837-1_28
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DOI: https://doi.org/10.1007/978-3-540-76837-1_28
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