The virtual path layout problem in fast networks (extended abstract)
PODC '94 Proceedings of the thirteenth annual ACM symposium on Principles of distributed computing
Optimal layouts on a chain ATM network
Discrete Applied Mathematics - Special issue: network communications broadcasting and gossiping
A Scalable Approach to Routing in ATM Networks
WDAG '94 Proceedings of the 8th International Workshop on Distributed Algorithms
ATM Layouts with Bounded Hop Count and Congestion
WDAG '97 Proceedings of the 11th International Workshop on Distributed Algorithms
Directed Virtual Path Layouts in ATM Networks
DISC '98 Proceedings of the 12th International Symposium on Distributed Computing
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
SOFSEM '97 Proceedings of the 24th Seminar on Current Trends in Theory and Practice of Informatics: Theory and Practice of Informatics
Lower Bounds for the Virtual Path Layout Problem in ATM Networks
SOFSEM '97 Proceedings of the 24th Seminar on Current Trends in Theory and Practice of Informatics: Theory and Practice of Informatics
On Optimal Graphs Embedded into Path and Rings, with Analysis Using l1-Spheres
WG '97 Proceedings of the 23rd International Workshop on Graph-Theoretic Concepts in Computer Science
Hop-congestion trade-offs for high-speed networks
SPDP '95 Proceedings of the 7th IEEE Symposium on Parallel and Distributeed Processing
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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).