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Lower Bound on the Number of Processors and Time for Scheduling Precedence Graphs with Communication Costs
December 1990 (vol. 16 no. 12)
pp. 1390-1401

A lower bound on the number of processors and finish time for the problem of scheduling precedence graphs with communication costs is presented. 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 defined. 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 a smaller number of processors. A lower bound on the minimum number of interprocessor communication links required to achieve optimum performance is proposed. Evaluation had 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.

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Index Terms:
scheduling; precedence graphs; communication costs; earliest starting time; lower bounds; completion time; task delay; time interval; finish time; task graph; minimum time increase; interprocessor communication links; optimum performance; small graphs; graph theory; scheduling
M.A. Al-Mouhamed, "Lower Bound on the Number of Processors and Time for Scheduling Precedence Graphs with Communication Costs," IEEE Transactions on Software Engineering, vol. 16, no. 12, pp. 1390-1401, Dec. 1990, doi:10.1109/32.62447
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