Scheduling UET-UCT DAGs of Depth Two on Two Processors

Given unit execution time (UET) tasks whose precedence constraints form a directed acyclic graph (DAG), the arcs are associated with unit communication time (UCT) delays. The problem is to schedule the tasks on two processors in order to minimize the makespan. Several polynomial algorithms in the literature are proposed for special classes of digraphs, but the complexity of solving this problem in general case stills a challenging open question. We propose in this paper a linear time algorithm to compute an optimal schedule for the class of DAGs of depth two.