The Average Hop Count Measure For Virtual Path Layouts

Abstract

This paper studies the average hop count measure for virtual path layouts of ATM and optical networks. Routing in the ATM and optical network models is based on covering the network with simple virtual paths, under some constraints on the allowed load (i.e., the number of paths that can share an edge). The hop count is the number of edges along the virtual path.Two basic results are established concerning the average hop count parameter. The first concerns comparing the maximum and average hop count measures assuming uniform all-to-all communication requirements. We develop a rather general connection between the two measures for virtual path layouts with bounded maximum load. This connection allows us to extend known lower bounds on the maximum hop count into ones on the average hop count for network families satisfying certain conditions, termed non-condensingly contractable (NCC) graph families. Using this characterization, we establish tight lower bounds on the average hop count of virtual path layouts with bounded maximum load for paths, cycles, and trees.Our second result is an algorithm for designing a virtual path layout of minimum average hop count for a given tree network with general (weighted) one-to-all requirements.