Efficient application scheduling is critical for achieving high performance in heterogeneous computing environments. The application scheduling problem has been shown to be NP-complete in general cases as well as in several restricted cases. Because of its key importance, this problem has been extensively studied and various algorithms have been proposed in the literature which are mainly for systems with homogeneous processors. Although there are a few algorithms in the literature for heterogeneous processors, they usually require significantly high scheduling costs and they may not deliver good quality schedules with lower costs. In this paper, we present two novel scheduling algorithms for a bounded number of heterogeneous processors with an objective to simultaneously meet high performance and fast scheduling time, which are called the Heterogeneous Earliest-Finish-Time (HEFT) algorithm and the Critical-Path-on-a-Processor (CPOP) algorithm. The HEFT algorithm selects the task with the highest upward rank value at each step and assigns the selected task to the processor, which minimizes its earliest finish time with an insertion-based approach. On the other hand, the CPOP algorithm uses the summation of upward and downward rank values for prioritizing tasks. Another difference is in the processor selection phase, which schedules the critical tasks onto the processor that minimizes the total execution time of the critical tasks. In order to provide a robust and unbiased comparison with the related work, a parametric graph generator was designed to generate weighted directed acyclic graphs with various characteristics. The comparison study, based on both randomly generated graphs and the graphs of some real applications, shows that our scheduling algorithms significantly surpass previous approaches in terms of both quality and cost of schedules, which are mainly presented with schedule length ratio, speedup, frequency of best results, and average scheduling time metrics.

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Performance-effective and low-complexity task scheduling for heterogeneous computing

Static scheduling of a program represented by a directed task graph on a multiprocessor system to minimize the program completion time is a well-known problem in parallel processing. Since finding an optimal schedule is an NP-complete problem in general, researchers have resorted to devising efficient heuristics. A plethora of heuristics have been proposed based on a wide spectrum of techniques, including branch-and-bound, integer-programming, searching, graph-theory, randomization, genetic algorithms, and evolutionary methods. The objective of this survey is to describe various scheduling algorithms and their functionalities in a contrasting fashion as well as examine their relative merits in terms of performance and time-complexity. Since these algorithms are based on diverse assumptions, they differ in their functionalities, and hence are difficult to describe in a unified context. We propose a taxonomy that classifies these algorithms into different categories. We consider 27 scheduling algorithms, with each algorithm explained through an easy-to-understand description followed by an illustrative example to demonstrate its operation. We also outline some of the novel and promising optimization approaches and current research trends in the area. Finally, we give an overview of the software tools that provide scheduling/mapping functionalities.

http://charm.cs.uiuc.edu/users/arya/docs/6.pdf

Static scheduling algorithms for allocating directed task graphs to multiprocessors

In this paper, we propose a static scheduling algorithm for allocating task graphs to fully connected multiprocessors. We discuss six recently reported scheduling algorithms and show that they possess one drawback or the other which can lead to poor performance. The proposed algorithm, which is called the Dynamic Critical-Path (DCP) scheduling algorithm, is different from the previously proposed algorithms in a number of ways. First, it determines the critical path of the task graph and selects the next node to be scheduled in a dynamic fashion. Second, it rearranges the schedule on each processor dynamically in the sense that the positions of the nodes in the partial schedules are not fixed until all nodes have been considered. Third, it selects a suitable processor for a node by looking ahead the potential start times of the remaining nodes on that processor, and schedules relatively less important nodes to the processors already in use. A global as well as a pair-wise comparison is carried out for all seven algorithms under various scheduling conditions. The DCP algorithm outperforms the previous algorithms by a considerable margin. Despite having a number of new features, the DCP algorithm has admissible time complexity, is economical in terms of the number of processors used and is suitable for a wide range of graph structures.

Dynamic critical-path scheduling: an effective technique for allocating task graphs to multiprocessors

We present a low-complexity heuristic, named the dominant sequence clustering algorithm (DSC), for scheduling parallel tasks on an unbounded number of completely connected processors. The performance of DSC is on average, comparable to, or even better than, other higher-complexity algorithms. We assume no task duplication and nonzero communication overhead between processors. Finding the optimum solution for arbitrary directed acyclic task graphs (DAG's) is NP-complete. DSC finds optimal schedules for special classes of DAG's, such as fork, join, coarse-grain trees, and some fine-grain trees. It guarantees a performance within a factor of 2 of the optimum for general coarse-grain DAG's. We compare DSC with three higher-complexity general scheduling algorithms: the ETF by J.J. Hwang, Y.C. Chow, F.D. Anger, and C.Y. Lee (1989); V. Sarkar's (1989) clustering algorithm; and the MD by M.Y. Wu and D. Gajski (1990). We also give a sample of important practical applications where DSC has been found useful. >

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DSC: scheduling parallel tasks on an unbounded number of processors

Much previous work in artificial intelligence has neglected representing time in all its complexity. In particular, it has neglected continuous change and the indeterminacy of the future. To rectify this, I have developed a first-order temporal logic, in which it is possible to name and prove things about facts, events, plans, and world histories. In particular, the logic provides analyses of causality, continuous change in quantities, the persistence of facts (the frame problem), and the relationship between tasks and actions. It may be possible to implement a temporal-inference machine based on this logic, which keeps track of several “maps” of a time line, one per possible history.

A Temporal Logic for Reasoning about Processes and Plans.

The problem of nonpreemptively scheduling a set of m partially ordered tasks on n identical processors subject to interprocessor communication delays is studied in an effort to minimize the makespan. A new heuristic, called Earliest Task First (ETF), is designed and analyzed. It is shown that the makespan $\omega _{{\text{ETF}}} $ generated by ETF always satisfies $\omega _{{\text{ETF}}} \leqq ({{2 - 1} / n})\omega _{{\text{opt}}}^{(i)} + C$, where $\omega _{{\text{opt}}}^{(i)} $ is the optimal makespan without considering communication delays and C is the communication requirements over some immediate predecessor-immediate successor pairs along one chain. An algorithm is also provided to calculate C. The time complexity of Algorithm ETF is $O(nm^2 )$.

https://ir.nctu.edu.tw:443/bitstream/11536/4388/1/A1989T808500004.pdf

Scheduling precedence graphs in systems with interprocessor communication times

Multiprocessor scheduling with interprocessor communication delays

Scheduling with sufficient loosely coupled processors

Leslie Lamport presented a set of axioms in 1979 that capture the essential properties of the temporal relationships between complex and perhaps unspecified activities within any system, and proceeded to use this axiom system to prove the correctness of sophisticated algorithms for reliable communication and mutual exclusion in systems without shared memory. As a step toward a more complete metatheory of Lamport's axiom system, this paper determines the extent to which that system differs from systems based on “atomic,” or indivisible, actions. Theorem 1 shows that only very weak conditions need be satisfied in addition to the given axioms to guarantee the existence of an atomic “model,” while Proposition 1 gives sufficient conditions under which any such model must be a “faithful” representation. Finally, Theorem 2 restates a result of Lamport showing exactly when a system can be thought of as made up of a set of atomic events that can be totally ordered temporally. A new constructive proof is offered for this result.

On Lamport's interprocessor communication model

In 1983, Allen presented an ingenious method for the representation and maintenance of temporal information in the presence of imprecise, uncertain, and relative knowledge about time of occurrence. He introduced 13 relations between his primitive “temporal intervals,” providing for the expression of “any relationship which can hold between two intervals.” the model, however, did not address the problem of temporally incomparable events, such as events occurring in a distributed system without a common clock. Lamport's interprocessor communication model furnishes an axiomatic system for describing such events and their possible relationships. This article demonstrates that Allen's temporal model can be subsumed in a more general model based on Lamport's axiomatics. It is further suggested that this extended model can provide the underpinnings of a temporal knowledge base containing time-dependent information measured by unsynchronized clocks or in relativistic space-time. In this model, the number of relations between intervals increases dramatically from Allen's 13 or Lamport's 2 or 3 to over 80. Within this context, a modification of Allen's algorithm for the maintenance of a temporal reasoning system is presented, thus permitting the advantages of such a system to extend to reasoning about a wider range of phenomena.

Frank D. Anger

Papers

Scheduling precedence graphs in systems with interprocessor communication times

Multiprocessor scheduling with interprocessor communication delays

Scheduling with sufficient loosely coupled processors

On Lamport's interprocessor communication model

Temporal reasoning: A relativistic model