This Article 
 Bibliographic References 
 Add to: 
Augmented Ring Networks
June 2001 (vol. 12 no. 6)
pp. 598-609

Abstract—We study four augmentations of ring networks which are intended to enhance a ring's efficiency as a communication medium significantly, while increasing its structural complexity only modestly. Chordal rings add “shortcut” edges, which can be viewed as chords, to the ring. Express rings are chordal rings whose chords are routed outside the ring. Multirings append subsidiary rings to edges of a ring and, recursively, to edges of appended subrings. Hierarchical ring networks (HRN's) append subsidiary rings to nodes of a ring and, recursively, to nodes of appended subrings. We show that these four modes of augmentation are very closely related: 1) Planar chordal rings, planar express rings, and multirings are topologically equivalent families of networks with the “cutwidth” of an express ring translating into the “tree depth” of its isomorphic multiring and vice versa. 2) Every depth-d HRN is a spanning subgraph of a ${\rm depth} \hbox {-} (2d-1)$ multiring. 3) Every depth-d multiring ${\cal M}$ can be embedded into a d-dimensional mesh with dilation 3 in such a way that some node of ${\cal M}$ resides at a corner of the mesh. 4) Every depth-d HRN ${\cal H}$ can be embedded into a d-dimensional mesh with dilation 2 in such a way that some node of ${\cal H}$ resides at a corner of the mesh. In addition to demonstrating that these four augmented ring networks are grid graphs, our embedding results afford us close bounds on how much decrease in diameter is achievable for a given increase in structural complexity for the networks. Specifically, we derive upper and lower bounds on the optimal diameters of N-node depth-d multirings and HRNs that are asymptotically tight for large N and d.

[1] W. Aiello, S.N. Bhatt, F.R.K. Chung, A.L. Rosenberg, and R.K. Sitaraman, “Augmented Ring Networks,” Invited Talk, Int'l Conf. Math. Computer Modeling and Scientific Computing, (no proceedings) 1997.
[2] F.S. Annexstein, M. Baumslag, and A.L. Rosenberg, “Group Action Graphs and Parallel Architectures,” SIAM J. Computing, vol. 19 pp. 544-569, 1990.
[3] B.W. Arden and H. Lee, “Analysis of Chordal Ring Networks,” IEEE Trans. Computers, vol. 30 pp. 291-295, 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, J.M. Fourneau, and A. Jean-Marie, “A Graph Theoretical Approach to the Equivalence of Multistage Interconnection Networks,” Discrete Applied Math., vol. 22, pp. 201-214, 1988/89.
[6] S.N. Bhatt, F.R.K. Chung, J.-W. Hong, F.T. Leighton, B. Obrenic, A.L. Rosenberg, and E.J. Schwabe, “Optimal Emulations by Butterfly-Like Networks,” J. ACM, vol. 43, pp. 293-330, 1996.
[7] M. Blum and D. Kozen, “On the Power of the Compass,” Proc. 19th IEEE Symp. Foundations of Computer Science, pp. 132-142, 1978.
[8] B. Bollobás and F.R.K. Chung, “The Diameter of a Cycle Plus a Random Matching,” SIAM J. Discrete Math., vol. 1, pp. 328-333, 1988.
[9] H. Burkhardt et al., “Overview of the KSR1 Computer System,” Technical Report KSR-TR 9202001, Kendall Square Research, 1992.
[10] S. Cosares, I. Saniee, and O. Wasem, “Network Planning with the SONET Toolkit,” Bellcore Exchange, pp. 8-15, Sept./Oct. 1992.
[11] W.J. Dally, “Express Cubes: Improving the Performance ofk-Aryn-Cube Interconnection Networks,” IEEE Trans. Computers, vol. 40, pp. 1016-1023, 1991.
[12] Y. Dinitz, M. Feighelstein, and S. Zaks, “On Optimal Graphs Embedded into Paths and Rings Using$l_1 \hbox {-} {\rm Spheres}$,” Proc. 23rd Int'l Workshop Graph-Theoretic Concepts in Computer Science, 1997.
[13] R. Feldmann and W. Unger, “The Cube-Connected Cycles Network Is a Subgraph of the Butterfly Network,” Parallel Processing Letters, vol. 2, pp. 13-19, 1992.
[14] A. Frank, T. Nishizeki, N. Saito, H. Suzuki, and É. Tardos, “Algorithms for Routing around a Rectangle,” Discrete Applied Math., vol. 40, pp. 363-378, 1992.
[15] O. Gerstel, I. Cidon, and S. Zaks, “The Layout of Virtual Paths in ATM Networks,” ACM/IEEE Trans. Networking, vol. 4, pp. 873-884, 1996.
[16] O. Gerstel and S. Zaks, “The Virtual Path Layout Problem in Fast Networks,” Proc. 13th ACM Symp. Principles of Distributed Computing, pp. 235-243, 1994.
[17] O. Gerstel and S. Zaks, “The Virtual Path Layout Problem in ATM Networks,” Proc. First Int'l Colloquium Structure, Information and Comm. Complexity, pp. 151-166, 1994.
[18] O. Gerstel, A. Wool, and S. Zaks, “Optimal Layouts on a Chain ATM Network,” Discrete Applied Math., vol. 83, no. 1-3, pp. 157-178, 1998.
[19] V.C. Hamacher and H. Jiang, “Comparison of Mesh and Hierarchical Networks for Multiprocessors,” Proc. Int'l Conf. Parallel Processing, vol. I pp. 67-71, 1994.
[20] F. Harary, Graph Theory. Reading, Mass.: Addison-Wesley, 1969.
[21] C.-T. Ho and S.L. Johnsson, “Embedding Meshes in Boolean Cubes by Graph Decomposition,” J. Parallel and Distributed Computing, vol. 8, pp. 325-339, 1990.
[22] R. Koch, F.T. Leighton, B.M. Maggs, S.B. Rao, A.L. Rosenberg, and E.J. Schwabe, “Work-Preserving Emulations of Fixed-Connection Networks,” J. ACM, vol. 44, pp. 104-147, 1997.
[23] E. Kranakis, D. Krizanc, and A. Pelc, “Hop-Congestion Tradeoffs for High-Speed Networks,” Int'l J. Foundations of Computer Science, vol. 8, pp. 117-126, 1997.
[24] C.P. Kruskal and M. Snir, “A Unified Theory of Interconnection Network Structure,” Theoretical Computer Science, vol. 48, pp. 75-94, 1986.
[25] D.-M. Kwai and B. Parhami, “Periodically Regular Chordal Rings: Generality, Scalability, and VLSI Layout,” Proc. Eighth IEEE Symp. Parallel and Distributed Processing, pp. 148-151, 1996.
[26] C. Lam, H. Jiang, and V.C. Hamacher, “Design and Analysis of Hierarchical Ring Networks for Shared-Memory Multiprocessors,” Proc. Int'l Conf. Parallel Processing, pp. 46-50, 1995.
[27] A. Litman and A.L. Rosenberg, “Balancing Communication in Ring-Structured Networks,” Technical Report 93-80, Univ. of Massachusetts, 1993.
[28] A. Michail, “Optimal Broadcast and Summation on Hierarchical Ring Architectures,” Parallel Processing Letters, vol. 8, 1998.
[29] E.J. Schwabe, “On the Computational Equivalence of Hypercube-Derived Networks,” Proc. Second ACM Symp. Parallel Algorithms and Architectures, pp. 388-397, 1990.
[30] Z.G. Vranesic, M. Stumm, D.M. Lewis, and R. White, “Hector: A Hierarchically Structured Shared-Memory Multiprocessor,” Computer, vol. 24, no. 1, pp. 72-79, Jan. 1991.
[31] S. Zaks, “Path Layout in ATM Networks—A Survey,” Proc. DIMACS Workshop Networks in Distributed Computing, 1997.
[32] C. Zamfirescu and T. Zamfirescu, “Hamiltonian Properties of Grid Graphs,” SIAM J. Discrete Math., vol. 5, pp. 564-570, 1992.

Index Terms:
Ring networks, chordal rings, express rings, multirings, hierarchical ring networks, grid graphs, graph embedding, diameter trade-offs.
William Aiello, Sandeep N. Bhatt, Fan R.K. Chung, Arnold L. Rosenberg, Ramesh K. Sitaraman, "Augmented Ring Networks," IEEE Transactions on Parallel and Distributed Systems, vol. 12, no. 6, pp. 598-609, June 2001, doi:10.1109/71.932713
Usage of this product signifies your acceptance of the Terms of Use.