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(n ₂+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 proving that the intervals having event 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(n ₂+m log n+nW ₂ erl) time, where W_erl 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 $W_{erl} > O\left( {\sqrt n } \right)$.

Part of this work was done while the first author was visiting the Department of Computer Science and Engineering, the University of Texas at Arlington. This work was supported by NASA Ames Research Center under Cooperative Agreement Number NCC 2-5395.