Geometric approach for optimal routing on mesh with buses

Recently, the architecture of 'mesh of buses' is becoming quite popular in parallel computing. Its main advantage is the limited broadcast capability that is used to overcome the main disadvantage of the mesh, namely the relatively big diameter. We show that in such networks busses indeed accelerate the time for the fundamental problem of routing. Furthermore, unlike in the 'store and forward' model, the time becomes proportional to the network load. Namely, small number of packets is faster to route. We consider 1 - 1 routing of m packets in a d-dimensional mesh with n^d processors and d·n^d-1 buses (one per each row, column). The two standard models of accessing the buses are considered and compared. CREW in which only one processor may transmit at any given time, and the CRCW model in which several processors may attempt to transmit at the same time (getting a noise signal as a result). We design a routing algorithm that routes m packets in the CREW model in O(max(m^1÷d, n^1÷d+1)) steps. A matching lower bound is also proved. In the CRCW case we show an algorithm of O(m^1÷d log n) and a lower bound of Ω(m^1÷d). It is shown that the difference between the models is essentially due to the improved capability of estimating threshold functions in the CRCW case.