Lower Bound on the Number of Processors and Time for Scheduling Precedence Graphs with Communication Costs

This paper proposes a new lower bound on the number of processors and finish time for the problem of scheduling precedence graphs with communication costs. An algorithm (ETF) has been proposed by Hwang [1] for scheduling precedence graphs in systems with interprocessor communication times. In this paper the notion of the earliest starting time of a task is formulated for the context of lower bounds. A lower bound on the completion time is proposed. A task delay which does not increase the earliest completion time of a schedule is denned. Each task can then be scheduled within a time interval without affecting the lower bound performance on the finish time. This leads to definition of a new lower bound on the number of processors required to process the task graph. A derivation of the minimum time increase over the earliest completion time is also proposed for the case of smaller number of processors. Finally the paper proposes a lower bound on the minimum number of interprocessor communication links required to achieve optimum performance. Evaluation has been carried out by using a set of 360 small graphs. The bound on the finish time deviates at most by 5% from the optimum solution in 96% of the cases and performs well with respect to the minimum number of processors and communication links.