This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Class-Congruence Property and Two-Phase Routing of Borel Cayley Graphs
December 1995 (vol. 44 no. 12)
pp. 1462-1468
ASCII Text
x 
 
Bruce W. Arden, K. Wendy Tang, "Class-Congruence Property and Two-Phase Routing of Borel Cayley Graphs," IEEE Transactions on Computers, vol. 44, no. 12, pp. 1462-1468, December, 1995.
 
 
BibTex
x
 
@article{ 10.1109/12.477252,
author = {Bruce W. Arden and K. Wendy Tang},
title = {Class-Congruence Property and Two-Phase Routing of Borel Cayley Graphs},
journal ={IEEE Transactions on Computers},
volume = {44},
number = {12},
issn = {0018-9340},
year = {1995},
pages = {1462-1468},
doi = {http://doi.ieeecomputersociety.org/10.1109/12.477252},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Computers
TI - Class-Congruence Property and Two-Phase Routing of Borel Cayley Graphs
IS - 12
SN - 0018-9340
SP1462
EP1468
EPD - 1462-1468
A1 - Bruce W. Arden,
A1 - K. Wendy Tang,
PY - 1995
KW - Generalized chordal ring
KW - interconnection network
KW - parallel computer.
VL - 44
JA - IEEE Transactions on Computers
ER -
 

Abstract—Dense, symmetric graphs are useful interconnection models for multicomputer systems. Borel Cayley graphs, the densest degree-4 graphs for a range of diameters [1], are attractive candidates. However, the group-theoretic representation of these graphs makes the development of efficient routing algorithms difficult. In earlier reports, we showed that all degree-4 Borel Cayley graphs have generalized chordal ring (GCR) and chordal ring (CR) representations [2], [3]. In this paper, we present the class-congruence property and use this property to develop the two-phase routing algorithm for Borel Cayley graphs in a special GCR representation. The algorithm requires a small space complexity of O(p+k) for n=p×k nodes. Although suboptimal, the algorithm finds paths with length bounded by 2D, where D is the diameter. Furthermore, our computer implementation of the algorithm on networks with 1,081 and 15,657 nodes shows that the average path length is on the order of the diameter. The performance of the algorithm is compared with that of existing optimal and suboptimal algorithms.

[1] D.V. Chudnovsky,G.V. Chudnovsky,, and M.M. Denneau,“Regular graphs with small diameter as models for interconnectionnetworks,” Tech. Report RC 13484(60281), IBM Research Division, Feb. 1988.
[2] B.W. Arden and K.W. Tang,“Representation and routing of Cayley graphs,” IEEE Trans. Comm., vol. 39, no. 11, pp. 1,533-1,537, Nov. 1991.
[3] K.W. Tang and B.W. Arden,“Representations for Borel Cayley graphs,” SIAM J. on Descrete Math. vol. 6, no. 4. pp. 655-676, Nov. 1993.
[4] T.Y. Feng,“A survey of interconnection networks,” Computer, vol. 14, pp. 12-27, Dec. 1981.
[5] G.H. Barnes,“The ILLIAC IV computer,” IEEE Trans. Computers, vol. 17, no. 8, pp. 746-757, Aug. 1968.
[6] J.P. Hayes,“Architecture of a hypercube supercomputer,” IEEE Proc. 1986 Int’l Conf. Parallel Processing, pp. 653-660, 1986.
[7] B.W. Arden and H. Lee,“Analysis of Chordal ring network,” IEEE Trans. Computers, vol. 30, no. 4, pp. 291-295, Apr. 1981.
[8] F.P. Preparata and J. Vuillemin, “The Cube-Connected Cycles: A Versatile Network for Parallel Computation,” Comm ACM, vol. 24, no. 5, pp. 300-309, 1981.
[9] J.C. Bermond and C. Delorme,“Strategies for interconnection networks: Some methods from graphtheory,” J. Parallel and Distributed Computing, vol. 3, pp. 433-449, 1986.
[10] N. Biggs,Algebraic Graph Theory.London: Cambridge Univ. Press, 1974.
[11] M. Ben-Ayed,“Dynamic routing for regular direct computer networks,” PhD thesis, Electrical Engineering Dept., College of Engineering andApplied Science, Univ. of Rochester, New York, 1990.
[12] K.W. Tang and B.W. Arden,“Vertex-transitivity and routing for Cayley graphs in GCRrepresentations,” Proc. 1992 Symp. Applied Computing,Kansas City, Mo, pp. 1,180, 1,187, Mar. 1992.
[13] S.B. Akers and B. Krishnamurthy, “A Group-Theoretic Model for Symmetric Interconnection Networks,” IEEE Trans. Computers, vol. 38, no. 4, pp. 555-566, Apr. 1989.
[14] G.E. Carlsson,J.E. Cruthirds,, and H.B. Sexton,“Interconnection networks based on a generalization of cube-connectedcycles,” IEEE Trans. Computers, vol. 334, no. 8, pp. 769-772, Aug. 1985.
[15] D.A. Reed and R.M. Fujimoto,Multicomputer Networks.Cambridge, Mass.: MIT Press, 1987.

Index Terms:
Generalized chordal ring, interconnection network, parallel computer.
Citation:
Bruce W. Arden, K. Wendy Tang, "Class-Congruence Property and Two-Phase Routing of Borel Cayley Graphs," IEEE Transactions on Computers, vol. 44, no. 12, pp. 1462-1468, Dec. 1995, doi:10.1109/12.477252
Usage of this product signifies your acceptance of the Terms of Use.