Self -Routing Superconcentrators

Abstract

Superconcentrators are switching systems that solve the generic problem of interconnecting clients and servers during sessions, in situations where either the clients or the servers are interchangeable (so that it does not matter which client is connected to which server). Previous constructions of superconcentrators have required an external agent to find the interconnections appropriate in each instance. We remedy this shortcoming by constructing superconcentrators that are “self -routing,” in the sense that they compute for themselves the required interconnections. Specifically, we show how to construct, for eachn, a systemSnwith the following properties: (1) The systemSnhasninputs,noutputs, andO(n) components, each of which is one of a fixed finite number of finite automata and is connected to a fixed finite number of other components through cables, each of which carries signals from a fixed finite alphabet. (2) When some of the inputs, and an equal number of outputs, are “marked” (by the presentation of a certain signal), then afterO(logn) steps (a time proportional to the “diameter” of the network) the system will establish a set of disjoint paths from the marked inputs to the marked outputs. The sizeO(n) is of course optimal for superconcentrators, as is the diameterO(logn) for superconcentrators of bounded degree.