An Efficient Algorithm for Computing Lower Bounds on Time and Processors for Scheduling Precedence Graphs on Multicomputer Systems

Abstract

In this paper, we propose new lower bounds on minimum number of processors and minimum time to execute a given program on a multicomputer system, where the program is represented by a directed acyclic task graph having arbitrary execution time and arbitrary communication delays. Additionally, we propose an O(n2+mlog n) time algorithm to compute these bounds for a task graph with n nodes and m arcs. The key ideas of our approach include: (i) identification of certain points called event points and provingthat the intervals havingev ent points as both ends are enough to compute the desired bounds; and (ii) the use of a sweeping technique. Our bounds are shown to be as sharp as the current best known bounds due to Jain and Rajaraman [7]. However, their approach requires O(n2+mlog n+nWerl2) time, where Werl is the earliest execution time of the task graph when arbitrary number of processors are available. Thus, in general, our algorithm performs as good as their algorithm, and exhibits better time complexity for task graphs having Werl > O(?n).