On constructing DAG-schedules with large AREAs

Abstract

The Area of a schedule ∑ for a dag measures the rate at which ∑ renders 's nodes eligible for execution. Specifically, AREA(∑) is the average number of nodes that are eligible for execution as ∑ executes 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 ®" 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. © 2014 Springer International Publishing Switzerland.