Tree task structures occur frequently in many applications where parallelization may be desirable. We present a formal treatment of non-preemptively scheduling task trees on distributed memory multiprocessors and show that the fundamental problems of scheduling (i) a task tree in absence of any inter-task communication on a fixed number of processors and (ii) a task tree with inter-task communication on an unbounded number of processors are NP-complete. For task trees that satisfy certain constraints, we present an optimal scheduling algorithm. The algorithm is shown optimal over a wider set of task trees than previous works.

Scheduling directed a-cyclic task graphs (DAGs) onto multiprocessors is known to be an intractable problem. Although there have been several heuristic algorithms for scheduling DAGs onto multiprocessors, few address the mapping onto a given number of completely connected processors with an objective of minimizing the finish time. We present an efficient algorithm called ClusterMerge to statically schedule directed a-cyclic task graphs onto a homogeneous completely connected MIMD system with a given number of processors. The algorithm clusters tasks in a DAG using a longest path heuristic and then iteratively merges these clusters to give a number of clusters identical to the number of available processors. Each of these clusters is then scheduled on a separate processor. Using simulations, we demonstrate that ClusterMerge schedules task graphs yielding the same or lower execution times than those of other researchers, but using fewer processors. We also discuss pitfalls in the various approaches to defining the longest path in a directed a-cyclic task graph.