Data Transmission in Processor Networks

We investigate the communication capacity and optimal data transmission schedules for processor networks connected by communication links, for example Transputer clusters. Each link allows the two processors at its endpoints to exchange data with a given xed transmission rate d. The communication itself is done in a blocking mode, that means the two processors have to synchronize before starting to exchange data and at any time each processor cannot communicate with more than one other processor. Our eeciency analysis will be more realistic by also taking into account the setup time for a communication, which will be assumed to be a xed constant s > 0. Thus, a large amount of data can be sent from one processor to a neighbour faster by a single long communication step than by a bunch of small data exchange steps: sending p data units in one step takes time s + p d. However, there is a tradeoo since the receiver has to wait until it has received the complete set of data before it can forward pieces to other processors. The following prototype task called scattering will be considered: At the beginning one processor called the source possesses a set of unit size data packets, one for each processor in the network. The goal is to distribute the packets in minimal time to all recipients. Our results concerning the complexity of this problem in arbitrary processor networks are as follows: For the general case, we give lower bounds on the minimal schedule length and show that to determine the length precisely is N P {complete. Special classes of simple strategies are investigated in more detail. For certain networks they turn out to yield optimal schedules. Finally, we investigate optimal schedules that can be computed eeciently and good approximation algorithms for speciic regular networks like hypercubes and multidimensional grids.