Task Graph Scheduling in Single-Bus Parallel Systems

Consider a scheduling problem of parallel computations in multiprocessor systems. Let a parallel program be modeled by a task graph, where vertices represent tasks and arcs the communications between tasks. An interprocessor communication time incurs when two tasks assigned to two diierent processors have to communicate. Such a scheduling problem has recently been studied in the literature, mostly for the case where interprocessor communication times are fully determined. In this paper, we consider the scheduling problem with resource constraints. More speciically, we consider the case where all interprocessor communications take place on a single bus. We consider two variants of the problem: communications with independent-data semantics or common-data semantics. We show that even for very speciic subproblems, viz. scheduling of general graphs on two processors and scheduling of binary trees on innnite number of processors, the minimization of the makespan of parallel programs in such a single-bus multiprocessor system is strongly NP-hard. These results are to be contrasted with the corresponding scheduling problems without bus constraint where the former has unknown time complexity and the latter is polynomial. As a consequence, the general scheduling problem of parallel programs with communication resource constraints is strongly NP-hard. Our results are then extended to the case of broadcasting communications, and can be applied to multiprocessor systems with shared memory.