Duality in Chain ATM Virtual Path Layouts

We present certain dualities occuring in the study of routing problems within 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 Given bounds on the load L and number of hops H in a layout and when the messages have to be sent along shortest paths a family of layouts Tshort L H was presented in GWZ and analysis was given for the worst case and for the average case optimal constructions For the same case but without the restriction of shortest paths a family of layouts T L H was presented in DFZ In both cases the results obtained turned out to be symmetric in H and L In this paper we use transformations of these two families of tree layouts into binary and ternary trees correspondingly in order to combinatorially explain the duality between these two measures of hop count and load These correspondences shed more light into the structure of these two families of trees In particular they give simple explanations for the symmetric role played by the hop count and the load in all of the above cases In addition it is shown that an optimal worst case solution for the shortest path and general path cases is also an optimal average case solution in both cases allowing a simpler characterization of these optimal layouts