ATM layouts with bounded hop count and congestion

In this paper we consider two new cost measures related to the communication overhead and the space requirements associated to virtual path layouts in ATM networks, that is the edge congestion and the node congestion. Informally, the edge congestion of a given edge e at an incident node u is defined as the number of VPs terminating or starting from e at u, while the node congestion of a node v is defined as the number of VPs having v as an endpoint. We investigate the problem of constructing virtual path layouts allowing to connect a specified root node to all the others in at most h hops and with maximum edge or node congestion c, for two given integers h and c. We first give tight results concerning the time complexity of the construction of such layouts for both the two congestion measures, that is we exactly determine all the tractable and intractable cases. Then, we provide some combinatorial bounds for arbitrary networks, together with optimal layouts for specific topologies such as chains, rings, grids and tori. Extensions to d-dimensional grids and tori with d > 2 are also discussed.