This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
A Note on Orthogonal Graphs
May 1993 (vol. 42 no. 5)
pp. 624-630

Orthogonal graphs are natural extensions of the classical binary and b-ary hypercubes b=2/sup l/ and are abstractions of interconnection schemes used for conflict-free orthogonal memory access in multiprocessor design. Based on the type of connection mode, these graphs are classified into two categories: those with disjoint and those with nondisjoint sets of modes. The former class coincides with the class of b-ary b=2/sup l/ hypercubes, and the latter denotes a new class of interconnection. It is shown that orthogonal graphs are Cayley graphs, a certain subgroup of the symmetric (permutation) group. Consequently these graphs are vertex symmetric, but it turns out that they are not edge symmetric. For an interesting subclass of orthogonal graphs with minimally nondisjoint set of modes, the shortest path routing algorithm and an enumeration of node disjoint (parallel) paths are provided. It is shown that while the number of node disjoint paths is equal to the degree, the distribution is not uniform with respect to Hamming distance as in the binary hypercube.

[1] S. B. Akers and B. Krishnamurthy, "A group theoretic model for symmetric interconnection networks,"IEEE Trans. Comput., vol. 38, pp. 555-566, 1989.
[2] H. M. Alnuweiri and V. P. K. Kumar, "A reduced mesh of trees organization for efficient solution to graph problems," inProc. 22nd Annu. Conf. Inform. Sci., Mar. 1988.
[3] H. M. Alnuweiri and V. K. Prasanna Kumar, "Optimal image computations on reduced VLSI architectures,"IEEE Trans. Circuits Syst., Oct. 1989.
[4] S. Bhatia, D. S. Hirschberg, and I. D. Scherson, "Shortest paths in orthogonal graphs," inProc. 29th Annu. Allerton Conf. Commun., Contr., Comput., 1991, pp. 488-497.
[5] L. N. Bhuyan and D. P. Agarwal, "Generalized hypercube and hyperbus structures for computer networks,"IEEE Trans. Comput., vol. 33, pp. 323-333, 1984.
[6] N. Biggs,Algebraic Graph Theory, London, England: Cambridge University Press, 1974.
[7] R. E. Buehrer, H. J. Brundiers, H. Benz, B. Bron, H. Friess, W. Haelg, H. J. Halin, A. Isacson, and M. Tadian "The ETH multiprocessor EMPRESS: A dynamically reconfigurable MIMD system,"IEEE Trans. Comput., vol. C-31, pp. 1035-1044, Nov. 1982.
[8] F. Harary,Graph Theory. Ontario, Canada: Addison-Wesley, 1969.
[9] K. Hwang, P. S. Tseung, and D. Kim, "An orthogonal multiprocessor for large grain scientific computations,"IEEE Trans. Comput., vol. 38, no. 1, pp. 47-61, 1989.
[10] K. Hwang and P. S. Tseung, "An efficient VLSI multiprocessor for signal image processing," inProc. Int. Conf. Comput. Design, Oct. 1985, pp. 1720-1726.
[11] S. Lakshmivarahan and S. K. Dhall, "A new hierarchy of hypercube interconnection schemes for parallel computers,"J. Supercomput., vol. 2, pp. 81-108, 1988.
[12] S. Lakshmivarahan and Sudarshan K. Dhall,Analysis and Design of Parallel Algorithms, McGraw-Hill Pub., 1990.
[13] W. Ledermann,An Introduction to The Theory of Finite Groups. International Science Press, 1949.
[14] I. D. Scherson and Y. Ma, "Vector computations on an orthogonal memory access multiprocessor system," inProc. 8th Symp. Comput. Architecture, May 1987, pp. 28-37.
[15] I. D. Scherson, "A theory for the description and analysis of a class of interconnection networks," CE-S89-002, Dep. Elec. Eng., Princeton Univ., Princeton, NJ, 1989.
[16] I. D. Scherson, "Orthogonal graphs for the construction of a class of interconnection networks,"IEEE Trans. Parallel Distributed Syst., vol. 2, pp. 3-19, 1991.
[17] I. D. Scherson and Y. Ma, "Analysis and applications of an orthogonal access multiprocessor,"J. Parallel Distributed Comput., vol. 7, pp. 232-255, 1989.
[18] H. S. Stone, "Parallel processing with perfect shuffle,"IEEE Trans. Comput., vol. 20, pp. 153-161, 1978.
[19] P. S. Tseung, K. Hwang, and V. P. K. Kumar, "A VLSI multiprocessor architecture for implementing parallel algorithms," inProc. 13th Int. Conf. Parallel Processing, Aug. 1985.

Index Terms:
b-ary hypercubes; interconnection schemes; conflict-free orthogonal memory access; multiprocessor design; connection mode; orthogonal graphs; Cayley graphs; vertex symmetric; shortest path routing algorithm; node disjoint paths; binary hypercube; graph theory; hypercube networks; parallel algorithms.
Citation:
S.V.R. Madabhushi, S. Lakshmivarahan, S.K. Dhall, "A Note on Orthogonal Graphs," IEEE Transactions on Computers, vol. 42, no. 5, pp. 624-630, May 1993, doi:10.1109/12.223683
Usage of this product signifies your acceptance of the Terms of Use.