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<p>The lower and upper bounds on the minimum time needed to process a given directedacyclic task graph for a given number of processors are derived. It is proved that theproposed lower bound on time is not only sharper than the previously known values butalso easier to calculate. The upper bound on time, which is useful in determining theworst case behavior of a given task graph, is presented. The lower and upper bounds onthe minimum number of processors required to process a given task graph in the minimum possible time are also derived. It is seen with a number of randomly generated dense task graphs that the lower and upper bounds we derive are equal, thus giving the optimal time for scheduling directed acyclic task graphs on a given set of processors.</p>
Index Termsparallel algorithms; scheduling; directed graphs; minimisation; lower time bound; upper time bound; multiprocessor optimal schedules; minimum processing time; directed acyclic task graph; worst case behavior; minimum processor number; randomly generated dense task graphs; parallel processing; performance evaluation

V. Rajaraman and K. Jain, "Lower and Upper Bounds on Time for Multiprocessor Optimal Schedules," in IEEE Transactions on Parallel & Distributed Systems, vol. 5, no. , pp. 879-886, 1994.
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