On constructing DAG‐schedules with large areas

Abstract

SummaryThe Area of a schedule Σ for a directed acyclic graph (DAG) is a quality metric that measures the rate at which Σ renders 's nodes eligible for execution. Specifically, AREA(Σ) is the average number of nodes of that are eligible for execution as Σ executes node by node. Extensive simulations suggest that, for many distributions of processor availability and power, DAG‐schedules having larger Areas execute DAGs faster on platforms that are dynamically heterogeneous: the platform's processors change power and availability status in unpredictable ways and at unpredictable times. (Clouds and desktop grids exemplify such platforms.) While Area‐maximal schedules can provably be found for everyDAG, efficient generators of such schedules are known only for families of well‐structured DAGs. Our first result shows that the problem of crafting Area‐maximal schedules for general DAGs is NP‐complete, hence likely computationally intractable. We also provide an efficient algorithm that approximates optimal Area to within a factor of , where n is the number of tasks in the DAG—a factor that is likely interesting only for small DAGs. The lack of efficient Area‐maximizing schedulers for general DAGs has instigated the development of several heuristics for producing DAG‐schedules that have large Areas. We propose a novel polynomial‐time heuristic that produces schedules having quite large Areas; the heuristic is based on the Sidney decomposition of a DAG. (1) Simulations on DAGs having random structure yield the following results. The SIDNEY heuristic produces schedules whose Areas: (a) are at least 85% of maximal; and (b) are at least 1.25 times greater than previously known heuristics. (2) Simulations on DAGs having the structure of random LEGO®;DAGs (as formulated in earlier studies) indicate that the schedules produced by the SIDNEY heuristic have Areas that are at least 1.5 times greater than previously known heuristics. The ‘85%’ result is obtained from formulating the Area‐maximization problem as a linear program (LP); the Areas of DAG‐schedules produced by the SIDNEY heuristic are at least 85% of the Area value produced by the (unrounded) LP. (3) The reported results on random DAGs are essentially matched by a second heuristic, which produces DAG‐schedules by rounding the results of the LP formulation. Copyright © 2015 John Wiley & Sons, Ltd.