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Ways of decreasing the number of operations needed to compute the lower bounds of optimal schedules, by reducing the number of time intervals that must be considered, are presented. The bounds apply to a system of identical processors executing a partially ordered set of tasks, with known execution times, using a non-preemptive scheduling strategy. In one approach we find that the required number of intervals depends on the graph. In our other approach, which subsumes the first, the number of intervals is decreased to at most min [D2/2, n2], where D is the deadline to complete the tasks and n is the number of tasks. The actual number of intervals for a particular graph can be considerably smaller than this worst case.
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