This Article 
 Bibliographic References 
 Add to: 
Addressing, Routing, and Broadcasting in Hexagonal Mesh Multiprocessors
January 1990 (vol. 39 no. 1)
pp. 10-18

A family of six-regular graphs, called hexagonal meshes or H-meshes, is considered as a multiprocessor interconnection network. Processing nodes on the periphery of an H-mesh are first wrapped around to achieve regularity and homogeneity. The diameter of a wrapped H-mesh is shown to be of O(p/sup 1/2/), where p is the number of nodes in the H-mesh. An elegant, distributed routing scheme is developed for wrapped H-meshes so that each node in an H-mesh can compute shortest paths from itself to any other node with a straightforward algorithm of O(1) using the addresses of the source-destination pair only, i.e. independent of the network's size. This is in sharp contrast with those previously known algorithms that rely on using routing tables. Furthermore, the authors also develop an efficient point-to-point broadcasting algorithm for the H-meshes which is proved to be optimal in the number of required communication steps. The wrapped H-meshes are compared against some other existing multiprocessor interconnection networks, such as hypercubes, trees, and square meshes. The comparison reinforces the attractiveness of the H-mesh architecture.

[1] K. Hwang and J. Ghosh, "Hypernet: A communication-efficient architecture for constructing massively parallel computers,"IEEE Trans. Comput., vol. C-36, pp. 1450-1466, Dec. 1987.
[2] W. D. Hillis,The Connection Machine. Cambridge, MA: MIT Press, 1985.
[3] T. Y. Feng, "A survey of interconnection networks,"IEEE Comput. Mag., pp. 12-27, Dec. 1981.
[4] C. Wu and T. Y. Feng, "On a class of multistage interconnection networks,"IEEE Trans. Comput., vol. C-29, no. 8, pp. 694-702, Aug. 1980.
[5] R. S. Wilkov, "Analysis and design of reliable computer networks,"IEEE Trans. Comput., vol. C-20, no. 3, pp. 670-678, June 1972.
[6] G. A. Anderson and E. D. Jensen, "Computer interconnection networks: Taxonomy, characteristics and examples,"ACM Comput. Surveys, vol. 7, pp. 197-213, Dec. 1975.
[7] W. A. Wulf, E. Cohe, W. Corwin, A. Jones, R. Levin, C. Pierson, and F. Pollack, "HYDRA: The kernel of a multiprocessor operating system,"Commun. ACM, vol. 17, pp. 337-345, June 1974.
[8] D. Mills, "An overview of the distributed computer network," inProc. Nat. Comput. Conf., vol. 45, 1976, pp. 523-531.
[9] K. S. Stevens, S. V. Robison, and A. L. Davis, "The Post Office--Communication support for distributed ensemble architectures," inProc. Six Int. Conf. Distrib. Comput. Syst., 1986, pp. 160-166.
[10] C. L. Seitz, "The Cosmic Cube,"Commun. ACM, pp. 22-33, Jan. 1985.
[11] T. Y. Feng, "Data manipulation functions in parallel processors and their implementations,"IEEE Trans. Comput., vol. C-23, no. 3, pp. 309-318, Mar. 1974.
[12] B. W. Arden and H. Lee, "Analysis of chordal ring network,"IEEE Trans. Comput., vol. C-30, no. 4, pp. 291-294, Apr. 1981.
[13] R. A. Finkel and M. H. Solomon, "Processor interconnection strategies,"IEEE Trans. Comput., vol. C-29, no. 5, pp. 360-370, May 1980.
[14] L. D. Wittie, "Communication structures for large networks of microcomputers,"IEEE Trans. Comput., vol. C-30, no. 4, pp. 264-273, Apr. 1981.
[15] A. J. Martin, "The Torus: An exercise in constructing a processing surface," inProc. 2nd Caltech Conf. VLSI, 1981, pp. 527-537.
[16] M. Watkins, "Connectivity of transitive graphs,"J. Combinatorial Theory, vol. 8, pp. 23-29, 1970.
[17] J. Turner, "Point-symmetric graphs with a prime number of points,"J. Combinatorial Theory, vol. 3, pp. 136-145, 1967.
[18] S. L. Hakimi, "An algorithm for construction of the least vulnerable communication network or the graph with the maximum connectivity,"IEEE Trans. Circuit Theory., vol. CT-16, pp. 229-230, May 1969.
[19] F. T. Boesch and R. E. Thomas, "On graphs of invulnerable communication nets,"IEEE Trans. Circuit Theory, vol. CT-17, no. 5, pp. 183-191, May 1970.
[20] R. Halin, "A theorem on N-connected graphs,"J. Combinatorial Theory, vol. 7, pp. 150-154, 1969.
[21] C. T. Ho and S. L. Johnsson, "Distributed routing algorithms for broadcasting and personalized communication in hypercubes," inProc. Int. Conf. Parallel Processing, Aug. 1986, pp. 640-648.
[22] F. Harary,Graph Theory. Reading, MA: Addison-Wesley, 1969.
[23] E. Horowitz and A. Zorat, "The binary tree as an interconnection network: Applications to multiprocessor systems and VLSI,"IEEE Trans. Comput., vol. C-30, no. 4, pp. 247-253, Apr. 1981.
[24] J. R. Goodman and C. H. Sequin, "Hypertree: A multiprocessor interconnection topology,"IEEE Trans. Comput., vol. C-30, no. 12, pp. 923-933, Dec. 1981.

Index Terms:
addressing; routing; broadcasting; hexagonal mesh multiprocessors; six-regular graphs; H-meshes; multiprocessor interconnection network; hypercubes; trees; square meshes; graph theory; multiprocessor interconnection networks.
M.-S. Chen, K.G. Shin, D.D. Kandlur, "Addressing, Routing, and Broadcasting in Hexagonal Mesh Multiprocessors," IEEE Transactions on Computers, vol. 39, no. 1, pp. 10-18, Jan. 1990, doi:10.1109/12.46277
Usage of this product signifies your acceptance of the Terms of Use.