Duality in ATM Layout Problems

Abstract

We present certain dualities occuring in problems of design of virtual path layouts in ATM networks. We concentrate on the one- to-many problem on 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 one follows from corresponding recurrence relations. We then present var- ious one-to-one correspondences. In the case of shortest paths layouts we use binary trees and lattice paths (that use horizontal and vertical steps). In the general case we use ternary trees, lattice paths (that use horizontal, vertical and diagonal steps), and high dimensional spheres. These correspondences shed light on the structure of the optimal solu- tions, and simplify some of the proofs, especially for the optimal average case designs.