Preemptive Online Scheduling with Reordering

Dósa, György; Epstein, Leah

doi:10.1137/090766139

“…Without a reordering buffer the competitive ratio of the problem is m m /(m m − (m − 1) m ). However, similar to the result for non-preemptive scheduling with reordering, Dósa and Epstein [12] show that -no online algorithm can achieve a competitive ratio of less than 4/3 for even m and to 4m 2 /(3m 2 + 1) for odd m for preemptive scheduling on m identical machines and a reordering buffer whose size does not depend on the input sequence, -there is an efficient algorithm matching this lower bound with a reordering buffer of size m/2 − 1 , and -a buffer of size Ω(m) is necessary to achieve this competitive ratio.…”

Section: Preemptive Schedulingsupporting

confidence: 70%

“…These results are mainly concerned with only two machines. Recently, the case of non-preemptive [13] and shortly afterwards the preemptive case [12] was finally well-understood for an arbitrary number of machines.…”

Section: Minimum Makespan Schedulingmentioning

confidence: 99%

An overview of some results for reordering buffers

Englert

¹

2011

Comput Sci Res Dev

2

0

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Lookahead is a classic concept in the theory of online scheduling. An online algorithm without lookahead has to process tasks as soon as they arrive and without any knowledge about future tasks. With lookahead, this strict assumption is relaxed. There are different variations on the exact type of information provided to the algorithm under lookahead but arguably the most common one is to assume that, at every point in time, the algorithm has knowledge of the attributes of the next k tasks to arrive. This assumption is justified by the fact that, in practice, tasks may not always strictly arrive one-by-one and therefore, a certain number of tasks are always waiting in a queue to be processed.In recent years, so-called reordering buffers have been studied as a sensible generalization of lookahead. The basic idea is that, in problem settings where the order in which the tasks are processed is not important, we can permit a scheduling algorithm to choose to process any task waiting in the queue. This stands in contrast to lookahead, where the algorithm has knowledge of all the tasks in the queue but still has to process them in the order they arrived. We discuss some of the results for reordering buffers for different scheduling problems.

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“…Englert et al [2014] characterize the best ratio achievable with a buffer of size Θ(m), where the ratio is between 4/3 and 1.4659 depending on the number of machines m. When both preemption and migration are allowed, Chen et al [1995] give a 1.58-competitive algorithm without buffer, matching the previous lower bound by Chen et al [1994b]. Dósa and Epstein [2011] achieve a ratio of 4/3 with a buffer of size Θ(m).…”

Section: Introductionmentioning

confidence: 90%

Online Makespan Minimization: The Power of Restart

Huang,

Kang,

Tang

et al. 2018

Preprint

0

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We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of (5 − √ 5)/2 ≈ 1.382, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios.We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.

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“…even a larger buffer cannot result in a smaller competitive ratio) and, in the case s ≥ 2, a buffer of size 2 already allows to achieve an optimal ratio. Dósa and Epstein [11] further study preemptive scheduling, as opposed to nonpreemptive scheduling, on m identical machines with a reordering buffer. They present a tight bound on the competitive ratio for any m. This bound is 4/3 for even values of m and slightly lower for odd values of m. They show that a buffer of size Θ(m) is sufficient to achieve this bound, but a buffer of size o(m) does not reduce the best overall competitive ratio of e/(e − 1) that is known for the case without reordering.…”

Section: Related Workmentioning

confidence: 99%

Online Makespan Scheduling with Job Migration on Uniform Machines

Englert

¹

,

Mezlaf

²

,

Westermann

³

2021

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In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\approx 1.4659$$ ≈ 1.4659 . They show that $$k = O(m)$$ k = O ( m ) is sufficient to achieve this bound and no $$k = o(n)$$ k = o ( n ) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\delta = \varTheta (1)$$ δ = Θ ( 1 ) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \delta $$ 1.4659 + δ with $$k = o(n)$$ k = o ( n ) . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\approx 1.7992$$ ≈ 1.7992 with $$k = O(m)$$ k = O ( m ) . We also show that $$k = \varOmega (m)$$ k = Ω ( m ) is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.

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Preemptive Online Scheduling with Reordering

Cited by 11 publications

References 17 publications

An overview of some results for reordering buffers

An overview of some results for reordering buffers

Online Makespan Minimization: The Power of Restart

Online Makespan Scheduling with Job Migration on Uniform Machines

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