How To Share a Divisible Load in a Hypercube

abstract

We study the problem of scheduling divisible loads in a multiprocessor system with hypercube interconnection topology and linear communication delays. A divisible load consists of large number of fine grain data elements, each requiring the same type of processing [1]. In the absence of precedence constraints such a load may be partitioned among processors in the system in every suitable way so that each part may be processed separately of and simultaneously with all other parts of the load. Initially, the whole load resides in the local memory of a selected node, called originator. The goal is to partition the load into fractions and communicate them to other processors in such way so that the entire load is transmitted and processed in the shortest possible time. Earlier solutions for this optimization problem were obtained under the assumption that each processor in the system can use all its communication links at the same time [2,3,4]. While this is not a problem in small hypercubes, it can present some difficulties in very large machines (e.g. machines with thousands of processors). We present a scheduling algorithm that restricts each processor to use only one communication link at a time. We model the process of data dissemination and computation with a set of recursive equations. By solving it we obtain a closed-form expression for the schedule length as a function of the load size and key parameters of the system such as hypercube dimension, processing speed of one node and communication bandwidth of one link.