On Constructing DAG-Schedules with Large AREAs.

The Area of a schedule Sigma for a DAG G measures the rate at which Sigma renders G's nodes eligible for execution. Specifically, AREA(Sigma) is the average number of nodes that are eligible for execution as S executes G node by node. Extensive simulations suggest that, for many distributions of processor availability and power, schedules having larger Areas execute DAGs faster on platforms that are dynamically heterogeneous: their processors change power and availability status in unpredictable ways and at unpredictable times. While Area-maximal schedules exist for every DAG, efficient generators of such schedules are known only for well-structured DAGs. We prove that the general problem of crafting Area-maximal schedules is NP-complete, hence likely computationally intractable. This situation motivates the development of heuristics for producing DAG-schedules that have large Areas. We build on the Sidney decomposition of a DAG to develop a polynomial-time heuristic, SIDNEY, whose schedules have quite large Areas. (1) Simulations on DAGs having random structure indicate that SIDNEY's schedules have Areas: (a) at least 85% of maximal; (b) at least 1.25 times larger than those produced by previous heuristics. (2) Simulations on DAGs having the structure of random "LEGO (R)" DAGs indicate that SIDNEY's schedules have Areas that are at least 1.5 times larger than those produced by previous heuristics. The "85%" result emerges from an LP-based formulation of the Area-maximization problem. (3) Our results on random DAGs are roughly matched by a second heuristic that emerges directly from the LP formulation.