Link-Connectivities of Extended Double Loop Networks

  • Frank K. Hwang
  • Wen-Ching Winnie Li
Part of the Applied Optimization book series (APOP, volume 1)

Abstract

The notion of extended double loop networks (EDLN) was introduced in [5]. Such a network, denoted by G(n; a, e; b, f), is a 2-regular digraph (each node has 2 inlinks and 2 outlinks) with n nodes labelled by the residues 0,1, ... , n-1, of integers modulo n, and 2n links i ai + e, i → bi + f, for i = 0,1, ... , n - 1. Many 2-regular digraphs popular as topologies for interconnecting networks are special EDLNs. For example, G(n; 2, 0; 2, 1) is the generalized de Bruijn network [6],[9], G(n; -2, -1; -2, -2) is the Imase-Itoh network [7], G(n;1, e; 1, f) is the usual double loop network [3], and G(n;1,1;1, f) is the FLBH (forward loop backward hop) network [9],[11]. EDLNs are interesting not only because they are a natural generalization of the networks well studied before, but also because their graph structures, despite simple linking patterns, are reasonably complicated due to the “noncommutative” nature of the two types of links (that is, the two paths of length two starting from a node using both types of links do not usually terminate at the same node) and the fact that they are not necessarily Cayley graphs [5]. Our ultimate goal is to investigate various properties of EDLNs to see if certain EDLNs will serve as good interconnection networks. In this paper we study the connectivities of such networks.

Keywords

Cayley Graph Line Graph Small Positive Integer Residue Modulo Link Connectivity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Y. Cheng, F. K. Hwang, I. F. Akyildiz and D. F. Hsu, “Routing Algorithms for Double Loop Networks,” Inter. J. Found. Comput. Sci. Google Scholar
  2. [2]
    D. Z. Du and F. K. Hwang, “Generalized de Bruijn Digraphs,” Networks 18, 27–38, 1988.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. A. Fiol, M. Valero, J. L. A. Yebra, I. Alegre and T. Lang, “Optimization of Double-Loop Structures for Local Networks,” Proc. XIX Int. Symp. MIMI ‘82, Paris, 1982, pp. 37–41.Google Scholar
  4. [4]
    F. K. Hwang, “The Hamiltonian Property of Linear Functions,” Oper. Res. Letters 6, 125–127, July 1987.MATHCrossRefGoogle Scholar
  5. [5]
    F. K. Hwang and W.-C. W. Li, “Hamiltonian Circuits for 2-Regular Interconnection Networks,” in Networks Optimization, Ed: D. Z. Du and P. Pardalos, World Scientific, River Edge, NJ.Google Scholar
  6. [6]
    M. Imase and M. Itoh, “Design to Minimize Diameter on Building-Block Network,” IEEE Trans. Comput. C-30, 439–442, June 1981.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Imase and M. Itoh, “A Design for Directed Graphs with Minimum Diameters,” IEEE Trans. Comput. C-32, 782–784, August 1983.CrossRefGoogle Scholar
  8. [8]
    D. E. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, MA 1972.Google Scholar
  9. [9]
    C. S. Raghavendra, M. Gerla and A. Avienis, “Reliable Loop Topologies for Large Local Computer Networks,” IEEE Trans. Comput. C-34, 46–54, January 1985.CrossRefGoogle Scholar
  10. [10]
    S. M. Reddy, D. K. Pradhan and J. G. Kuhl, “Direct Graphs with Minimum Diameter and Maximal Connectivity,” School of Eng., Oakland Univ. Tech. Rep., July 1980.Google Scholar
  11. [11]
    C. K. Wong and D. Coppersmith, “A Combinatorial Problem Related to Multinodule Memory Organizations,” J. Assoc. Comput. Mach. 21, 392–402, July, 1974.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    E. A. van Doorn, “Connectivity of Circulant Digraphs,” J. Graph Theory10, 9–14, 1986.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Frank K. Hwang
    • 1
  • Wen-Ching Winnie Li
    • 2
  1. 1.AT&T Bell LaboratoriesMurray HillUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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