Sequencing Asynchronous Pipeline

Asynchronous pipelining is a form of parallelism in which processors execute diierent loop tasks, as opposed to diierent loop iterations. An asynchronous pipeline schedule for a loop is a generalization of a noniterative DAG schedule, and consists of a processor assignment|an assignment of loop tasks to processors; plus a task sequence for each processor|an order on instances of tasks assigned to a processor. This variant of pipeline parallelism is particularly relevant in distributed memory systems (since pipeline control may be distributed across processors), but may also be used in shared memory systems. Results for asynchronous pipelining may also provide insights into other forms of pipelining, such as software pipelining. We show that the problem of nding an optimal asynchronous pipeline schedule is NP-hard under three diierent deenitions of optimality, even when the loop body is self-cyclic, i.e., has no multi-task dependence cycles. A number of pipeline scheduling subproblems are themselves NP-hard, but given a processor assignment, the problem of nding an asymptotically optimal task sequence for a self-cyclic loop can be solved in polynomial time. Self-cyclic loops represent a signiicant subset of loops which may contain cross-iteration dependences, I/O, and simple reductions. Assuming that a processor assignment for a loop is speciied, we design a heuristic task sequencing algorithm which can be customized to trade oo algorithm execution time against the quality of the resulting task sequence. The algorithm is asymptotically optimal for self-cyclic loops, and can be used as a component of a complete pipeline scheduling algorithm which generates schedules for self-cyclic loops that are guaranteed to be within 18% of optimal. For loops which contain multi-task cycles, the algorithm generates task sequences which average within 3.5{6.8% of optimal for a set of 10,800 random loops. The primary determinant of the execution time of the algorithm is the time required to determine the maximum cycle ratio of a loop, which in practice takes O(ve) time for a loop with v tasks and e dependence edges.