On Constructing DAG-Schedules with Large AREAs

Abstract

The Area of a schedule Σ for a DAG G is a quality metric that measures the rate at which Σ renders G's nodes eligible for execution. Specifically, AREA(Σ) is the average number of nodes of G that are eligible for execution as Σ executes G 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 every DAG, 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. 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; (b) are at least 1.25 times greater than previously known heuristics. (2) Simulations on DAGs having the structure of random "LEGO R " 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 Areamaximization 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.

An AREA-Oriented Heuristic for Scheduling DAGs on Volatile Computing Platforms

“…The problem of finding truly AREA-maximizing schedules has recently been shown to be computationally intractable, specifically NP-complete [34]. AO is the first efficient scheduling mechanism that provably enhances the rate of producing allocationeligible chores for every computation-DAG.…”

“…Yet, the impetus for the definition is the intuition that increasing the rate of rendering chores eligible for execution will have a benign impact on the performance of any volatile platform-and the experiments in [6] support this intuition. The current paper is motivated by the fact that, while all computations admit A-M schedules, the problem of finding such schedules for general computations is likely computationally intractable, being NP-complete [34]. We respond to the need for efficiently generated schedules by building on the results in [9] to craft an AREA-Oriented (A-O) scheduling heuristic that we call AO.…”

“…Algorithms reported in [6] efficiently find A-M schedules for several significant families of DAGs. However, it has recently been reported in [34] that the problem of finding A-M schedules for general DAGs is NPcomplete, hence likely computationally intractable. The importance of being able to schedule DAGs efficiently led us in [9] to parlay the main result in that paper-an efficient algorithm that crafts an A-M schedule for any series-parallel DAG-into an efficient heuristic, called AO, that crafts AREAOriented schedules for general DAGs.…”