# On the Use of Duality and Geometry in Layouts for ATM Networks

## Abstract

We show how duality properties and geometric considerations are used in studies related to virtual path layouts of ATM networks. We concentrate on the one-to-many problem for a chain network, in which one constructs a set of paths, that enable connecting one vertex with all others in the network. We consider the parameters of load (the maximum number of paths that go through any single edge) and hop count (the maximum number of paths traversed by any single message). Optimal results are known for the cases where the routes are shortest paths and for the general case of unrestricted paths. These solutions are symmetric with respect to the two parameters of load and hop count, and thus suggest duality between these two. We discuss these dualities from various points of view. The trivial ones follow from corresponding recurrence relations and lattice paths. We then study the duality properties using trees; in the case of shortest paths layouts we use binary trees, and in the general case we use ternary trees. In this latter case we also use embedding into high dimensional spheres.

The duality nature of the solutions, together with the geometric approach, prove to be extremely useful tools in understanding and analyzing layout designs. They simplify proofs of known results (like the best average case designs for the shortest paths case), enable derivation of new results (like the best average case designs for the general paths case), and improve existing results (like for the all-to-all problem).

## Keywords

Short Path Recurrence Relation Binary Tree Optimal Layout Lattice Path## Preview

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## References

- 1.W. Aiello, S. Bhatt, F. Chung, A. Rosenberg, and R. Sitaraman, Augmented
*rings networks, 11th Intl. Conf. on Math. and Computer Modelling and Scientific Computing (ICMCM & SC)*1997; also: Proceedings of the*6th International Colloquium on Structural Information and Communication Complexity (SIROCCO)*, Lacanau-Océan, France, 1999, pp. 1–16.Google Scholar - 2.L. Becchetti, P. Bertolazzi, C. Gaibisso and G. Gambosi,
*On the design of efficient ATM routing schemes*, submitted, 1997.Google Scholar - 3.L. Beccheti and C. Gaibisso,
*Lower bounds for the virtual path layout problem in ATM networks*, Proceedings of the 24th Seminar on Theory and Practice of Informatics (SOFSEM), Milovny, The Czech Republic, November 1997, pp. 375–382.Google Scholar - 4.J-C. Bermond, N. Marlin, D. Peleg and S. Pérennes,
*Directed virtual path layout in ATM networks*, Proceedings of the 12th International Symposium on Distributed Computing (DISC), Andros, Greece, September 1998, pp. 75–88.Google Scholar - 5.I. Cidon, O. Gerstel and S. Zaks,
*A scalable approach to routing in ATM networks. 8th International Workshop on Distributed Algorithms (WDAG)*, Lecture Notes in Computer Science 857, Springer Verlag, Berlin, 1994, pp. 209–222.Google Scholar - 6.Y. Dinitz, M. Feighelstein and S. Zaks,
*On optimal graph embedded into path and rings, with analysis using*l_{1}-spheres,*23th International Workshop on Graph-Theoretic Concepts in Computer Sciences (WG)*, Berlin, Germany, June 1997.Google Scholar - 7.T. Eilam, M. Flammini and S. Zaks,
*A Complete Characterization of the Path Layout Construction Problem for ATM Networks with Given Hop Count and Load*, Proceedings of the*24th International Colloquium on Automata, Languages and Programming (ICALP)*, Bologna, Italy, pp. 527–537, July 1997.Google Scholar - 8.M. Feighelstein,
*Virtual path layouts for ATM networks with unbounded stretch factor*, M.Sc. Dissertation, Department of Computer Science, Technion, Haifa, Israel, May 1998.Google Scholar - 9.M. Flammini, E. Nardelli and G. Proietti,
*ATM layouts with bounded hop count and congestion*, Proceedings of the 11th International Workshop on Distributed Algorithms (WDAG), Saarbrüecken, Germany, September 1997, pp. 24–26.Google Scholar - 10.M. Feighelstein and S. Zaks,
*Duality in chain ATM virtual path layouts*, Proceedings of the*4th International Colloquium on Structural Information and Communication Complexity (SIROCCO)*, Monte Verita, Ascona, Switzerland, July 24–26, 1997, pp. 228–239.Google Scholar - 11.O. Gerstel,
*Virtual Path Design in ATM Networks*, Ph.D. thesis, Department of Computer Science, Technion, Haifa, Israel, December 1995.Google Scholar - 12.S. W. Golomb and L. R. Welch,
*Perfect Codes in the Lee Metric and the Packing of Polyominoes*. SIAM Journal on Applied Math., vol. 18, no. 2, January, 1970, pp. 302–317.zbMATHCrossRefMathSciNetGoogle Scholar - 13.O. Gerstel, A. Wool and S. Zaks,
*Optimal layouts on a chain ATM network*, Discrete Applied Mathematics, special issue on Network Communications, 83, 1998, pp. 157–178.Google Scholar - 14.O. Gerstel, A. Wool and S. Zaks,
*Optimal Average-Case Layouts on Chain Networks*, Proceedings of the*Workshop on Algorithmic Aspects of Communication*, Bologna, Italy, July 11–12, 1997.Google Scholar - 15.O. Gerstel and S. Zaks,
*The Virtual Path Layout Problem in Fast Networks*, Proceedings of the*13th ACM Symposium on Principles of Distributed Computing (PODC)*, Los Angeles, CA, U.S.A., August 1994, pp. 235–243.Google Scholar - 16.E. Kranakis, D. Krizanc and A. Pelc,
*Hop-congestion tradeoffs for ATM networks*,*7th IEEE Symp. on Parallel and Distributed Processing*, pp. 662–668.Google Scholar - 17.L. Stacho and I. Vrt’o,
*Virtual Path Layouts for Some Bounded Degree Networks*. 3rd International Colloquium on Structural Information and Communication Complexity (SIROCCO), Siena, Italy, June 1996.Google Scholar - 18.S. Zaks,
*Path Layout in ATM Networks*, Proceedings of the*24th Annual Conference on Current Trends in Theory and Practice of Informatics (SOFSEM)*, Lecture Notes in Computer Science 1338, Springer Verlag, Milovy, The Czech Republic, November 22–29, 1997, pp. 144–160.Google Scholar