Semi on-line algorithms for the partition problem

Kellerer, Hans; Котов, В. М.; Speranza, Maria; Tuza, Źsolt

doi:10.1016/s0167-6377(98)00005-4

Cited by 166 publications

(103 citation statements)

References 10 publications

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“…The paper [12] is probably the first paper which studied and compared several notions of semi-online algorithms, including known sum of processing times. Some combination of the previous restrictions were studied in [14] for non-preemptive scheduling on identical machines.…”

Section: Semi-online Restrictions and Previous Workmentioning

confidence: 99%

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

Ebenlendr

Sgall

2010

Theory Comput Syst

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Abstract. We present a unified optimal semi-online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize the makespan. This algorithm works for all types of semi-online restrictions, including the ones studied before, like sorted (decreasing) jobs, known sum of processing times, known maximal processing time, their combinations, and so on. Based on the analysis of this algorithm, we derive some global relations between various semi-online restrictions and tight bounds on the approximation ratios for a small number of machines.

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Section: Semi-online Restrictions and Previous Workmentioning

confidence: 99%

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

Ebenlendr

Sgall

2010

Theory Comput Syst

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“…Kellerer et al [22] and Zhang [25] considered the case of two identical machines, and showed that a buffer of size 1 allows to achieve an approximation ratio of 4 3 , which is best possible. For m identical machines, Englert et al [16] showed that a buffer of size O(m) is sufficient.…”

Section: Introductionmentioning

confidence: 99%

Max-min Online Allocations with a Reordering Buffer

Epstein

Levin

Stee³

2010

Automata, Languages and Programming

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Abstract. We consider online scheduling so as to maximize the minimum load, using a reordering buffer which can store some of the jobs before they are assigned irrevocably to machines. For m identical machines, we show an upper bound of Hm−1 + 1 for a buffer of size m − 1. A competitive ratio below Hm is not possible with any finite buffer size, and it requires a buffer of sizeΩ(m) to get a ratio of O(log m). For uniformly related machines, we show that a buffer of size m + 1 is sufficient to get an approximation ratio of m, which is best possible for any finite sized buffer. Finally, for the restricted assignment model, we show lower bounds identical to those of uniformly related machines, but using different constructions. In addition, we design an algorithm of approximation ratio O(m) which uses a finite sized buffer. We give tight bounds for two machines in all the three models. These results sharply contrast to the (previously known) results which can be achieved without the usage of a reordering buffer, where it is not possible to get a ratio below an approximation ratio of m already for identical machines, and it is impossible to obtain an algorithm of finite approximation ratio in the other two models, even for m = 2. Our results strengthen the previous conclusion that a reordering buffer is a powerful tool and it allows a significant decrease in the competitive ratio of online algorithms for scheduling problems. Another interesting aspect of our results is that our algorithm for identical machines imitates the behavior of the greedy algorithm on (a specific set of) related machines, whereas our algorithm for related machines completely ignores the speeds until the end, and then only uses the relative order of the speeds.

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“…Zhang and Ye [20] studied a model, in which it is known in advance that the largest job is last, and the scheduler is notified when this last job arrives (see also [9]). Some models assume a priori knowledge on the input already in the beginning of the input sequence, such as the total size of all jobs [15,2,3], bounds on possible job sizes [14], the cost of an optimal schedule [4,6], or other properties of the sizes, e.g., an arrival in a sorted order [13,17,7].…”

Section: Introductionmentioning

confidence: 99%

“…This model was first studied for the case of two identical speed machines by Kellerer et al [15], and by Zhang [19]. In both papers, an algorithm of competitive ratio 4 3 , which uses K = 1 was designed.…”

Section: Introductionmentioning

confidence: 99%

“…The second question is whether there exists a constant K, so that a larger buffer is no longer beneficial to an algorithm, that is, increasing the size of the buffer above this threshold would not change the best competitive ratio further. Previous work [15,19,5] shows that in the case s = 1, already K = 1 allows to design a 4 3 -competitive algorithm, which is best possible for any K ≥ 1, whereas the best possible ratio for K = 0 is 3 2 . Similar results have been show for multiple identical machines [5].…”

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confidence: 99%

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Online scheduling with a buffer on related machines

Dósa

Epstein

2008

J Comb Optim

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Online scheduling with a buffer is a semi-online problem which is strongly related to the basic online scheduling problem. Jobs arrive one by one and are to be assigned to parallel machines. A buffer of a fixed capacity K is available for storing at most K input jobs. An arriving job must be either assigned to a machine immediately upon arrival, or it can be stored in the buffer for unlimited time. A stored job which is removed from the buffer (possibly, in order to allocate a space in the buffer for a new job) must be assigned immediately as well. We study the case of two uniformly related machines of speed ratio s ≥ 1, with the goal of makespan minimization.Two natural questions can be asked. The first question is whether this model is different from standard online scheduling, that is, is any size of buffer K > 0 already helpful to the algorithm, compared to the case K = 0. The second question is whether there exists a constant K, so that a larger buffer is no longer beneficial to an algorithm, that is, increasing the size of the buffer above this threshold would not change the best competitive ratio further. Previous work [15,19,5] shows that in the case s = 1, already K = 1 allows to design a 4 3 -competitive algorithm, which is best possible for any K ≥ 1, whereas the best possible ratio for K = 0 is 3 2 . Similar results have been show for multiple identical machines [5]. We answer both questions affirmatively, and show that a buffer of size K = 2 is sufficient to achieve the a competitive ratio which matches the lower bound for K → ∞ for any s > 1. In fact, we show that a buffer of size K = 1 can evidently be exploited by the algorithm for any s > 1, but for a range of values of s, it is still weaker than a buffer of size 2. On the other hand, in the case s ≥ 2, a buffer of size K = 1 already allows to achieve optimal bounds.

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Semi on-line algorithms for the partition problem

Cited by 166 publications

References 10 publications

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

Semi-Online Preemptive Scheduling: One Algorithm for All Variants

Max-min Online Allocations with a Reordering Buffer

Online scheduling with a buffer on related machines

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