This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
DCC Linear Congruential Graphs: A New Class of Interconnection Networks
February 1996 (vol. 45 no. 2)
pp. 156-164

Abstract—Let n be an integer and F = {fi : 1 ≤it for some integer t} be a finite set of linear functions. We define a linear congruential graph G(F, n) as a graph on the vertex set V = {0, 1, ..., n - 1}, in which any xV is adjacent to fi(x) mod n, 1 ≤it. For a linear function $\sl g$, and a subset V1 of V we define a linear congruential graph $G(F,\,\,n,\,\,{\sl g},\,\,V_1)$ as a graph on vertex set V, in which any xV is adjacent to fi(x) mod n, 1 ≤it , and any xV1 is also adjacent to ${\sl g}(x)$ mod n.

These graphs generalize several well known families of graphs, e.g., the de Bruijn graphs. We give a family of linear functions, called DCC linear functions, that generate regular, highly connected graphs which are of substantially larger order than de Bruijn graphs of the same degree and diameter. Some theoretical and empirical properties of these graphs are given and their structural properties are studied.

[1] 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.
[2] B.W. Arden and H. Lee, "Analysis of chordal ring network," IEEE Trans. Computers, vol. 30, no. 4, pp. 291-295, Apr. 1981.
[3] J.C. Bermond and B. Bollobas, "The diameter of graphs-a survey," Proc. Congressus Numerantium, vol. 32, pp. 3-27, 1981.
[4] J.-C. Bermond, F. Comellas, and D.F. Hsu, “Distributed Loop Computers: A Survey,” J. Parallel and Distributed Computing, vol. 24, pp. 2-10, 1995.
[5] J.C. Bermond, C. Delorme, and G. Farhi, "Large graphs with given degree and diameter III," Proc. Coll. Cambridge (1981). Ann. Discrete Math. 13, North-Holland, pp. 23-32, 1982.
[6] 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.
[7] J.C. Bermond, N. Homobono, and C. Peyrat, "Large fault-tolerant interconnection networks," First Japan Conf. Graph Theory and Applications,Hakone, Japan, June 1986.
[8] J.C. Bermond and C. Peyrat, "deBruijn, and Kautz networks: A competitor for the hypercube?" Proc. Conf. Hypercube and Distributed Computers, pp. 279-294, 1989.
[9] B. Bollobas and W.F. de la Vega, "The diameter of random graphs," Combinatorica 2, pp. 125-134, 1982.
[10] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications. NorthHolland, 1976.
[11] N.G. deBruijn, "A combinatorial problem," Koninklije Nedderlandse Academie van Wetenshappen Proc., vol. serA49, pp. 758-764, 1946.
[12] F.R.K. Chung, "Diameters of graphs: Old problems and new results," Proc. 18th South-Eastern Conf. Combinatorics, Graph Theory, and Computing, Congressus Numerantium, pp. 295-317, 1987.
[13] C. Delorme, "A table of large graphs of small degrees and diameters," personnal communication, 1990.
[14] D.Z. Du and F.K. Hwang, "Generalized de Bruijn Digraphs," Networks, vol. 18, pp. 27-38, 1988.
[15] D.Z. Du, D.F. Hsu, and G.W. Peck, "Connectivity of consecutive-d digraphs," Discrete Applied Math., to appear.
[16] B. Elpas, "Topological constraints on interconnection limited logic," Switching Circuits Theory and Logical Design 5, pp. 133-147, 1964.
[17] D.M. Gordon, "Parallel sorting on Cayley graphs, abstract," The Capital City Conf. Combinatorics and Theoretical Computer Science, May 1989.
[18] W.D. Hillis, The Connection Machine.Cambridge, Mass.: The MIT Press, 1985.
[19] M. Imase and M. Itoh, "Design to minimize diameter on building block network," IEEE Trans. Computers, vol. 30, pp. 439-442, 1981.
[20] M. Jerrum and S. Skyum, "Families of fixed degree graphs for processor interconnection," IEEE Trans. Computers, vol. 33, pp. 190-194, 1984.
[21] W.H. Kautz, "Bounds on directed (d, k) graphs," Theory of Cellular Logic Networks and Machines, SRI Project 7258, pp. 20-28, 1968.
[22] M. Klawe, "Limitations on explicit constructions of expanding graphs," Siam J. Computing, vol. 13, no. 1, pp. 156-166, 1984.
[23] D. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1998.
[24] W. Leland and M. Solomon, "Dense trivalent graphs for processor interconnection," IEEE Trans. Computers, vol. 31, no. 3, pp. 219-222, Mar. 1982.
[25] J. Opatrny and D. Sotteau, "Linear congruential graphs, graph theory, combinatorics, algorithms, and applications," SIAM Proc. series, pp. 404-426, 1991.
[26] D.K. Pradhan, Fault-Tolerant Multiprocessor and VLSI Based Systems Communication Architecture, Fault-Tolerant Computing, Theory and Techniques vol. 2. Prentice-Hall, 1986.
[27] M.R. Samatham and D.K. Pradhan, "The de Bruijn Multiprocessor Network: A Versatile Parallel Processing and Sorting Network for VLSI," IEEE Trans. Computers, vol. 38, no. 4, pp. 567-581, Apr. 1989.
[28] M.R. Samantham and D.K. Pradhan, Correction to The de Bruijn multiprocessor network: A versatile parallel processing and sorting network for VLSI," IEEE Trans. Computers, vol. 40, no. 1, p. 122, Jan. 1991.

Index Terms:
Graph theory, interconnectons networks, network design, parallel processing, computer networks.
Citation:
J. Opatrny, D. Sotteau, N. Srinivasan, K. Thulasiraman, "DCC Linear Congruential Graphs: A New Class of Interconnection Networks," IEEE Transactions on Computers, vol. 45, no. 2, pp. 156-164, Feb. 1996, doi:10.1109/12.485369
Usage of this product signifies your acceptance of the Terms of Use.