Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines

Xiao, Min; Liu, Jing; Wang, Yuqing

doi:10.1016/j.ipl.2011.01.002

Cited by 14 publications

(7 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…• 最后重排 -机器限制 (reassignFE) [67] : 所有工件到达后可重排每台机器 (或一台机器 [68] ) 上最晚到达的 δ 个工件. 对 P m|reassignSP|C max , Sanders 等 [65] 17 16 的在线算法.…”

Section: 特殊加工机制下的在线排序问题unclassified

Online scheduling on parallel machines: A survey

Ling¹,

Tan²

2020

Sci. Sin.-Math.

View full text Add to dashboard Cite

关键词排序在线竞争比 MSC (2010) 主题分类 90B35, 90C27 1 引言在线问题 (online problem) 是 20 世纪 80 年代起逐渐形成的一个新的研究方向, 可用于求解在线问题的算法称为在线算法 (online algorithm). 在线的两个基本特征是实例信息随算法执行过程逐渐披露、以及算法决策不可因后续出现的信息而更改. 相应地, 实例信息在算法执行前全部已知的问题称为离线问题 (offline problem). 在线问题及算法可抽象为在算法设计者和对手 (adversary) 之间进行的一个需求-应答博弈 (request-answer game). 对手每给出一个需求, 算法根据此前双方的应对和当前需求决定如何应答, 而对手再根据至目前为止双方的应对给出下一个需求. 算法的目标是使其性能尽可能好, 而对手的目标则恰好相反. 为对一个在线算法求解在线问题的性能作出定量的评估, 最坏情况比被移植到在线问题的研究中, 并结合其特点称其为竞争比 (competitive ratio). 称 ρ = inf{r 1 | C A (I) rC * (I), ∀ I} (ρ = inf{r 1 | C * (I) rC A (I), ∀ I})

show abstract

Section: 特殊加工机制下的在线排序问题unclassified

Online scheduling on parallel machines: A survey

Ling¹,

Tan²

2020

Sci. Sin.-Math.

View full text Add to dashboard Cite

show abstract

“…The authors design strategies that perform well with respect to the best possible solution that can be constructed with the budget. Online makespan minimization on m = 2 machines was considered in [17,22]. The best competitiveness is 4/3.…”

Section: Previous Workmentioning

confidence: 99%

On the Value of Job Migration in Online Makespan Minimization

Albers

Hellwig

2016

Algorithmica

View full text Add to dashboard Cite

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of n jobs has to be scheduled non-preemptively on m machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range [1. 88, 1.9201]. Currently no randomized online algorithm with a smaller competitiveness is known, for general m.In this paper we explore the power of job migration, i.e. an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is α m -competitive, for any m ≥ 2, where α m is the solution of a certain equation. For m = 2, α 2 = 4/3 and lim m→∞ α m = W −1 (−1/e 2 )/(1 + W −1 (−1/e 2 )) ≈ 1.4659. Here W −1 is the lower branch of the Lambert W function. For m ≥ 11, the algorithm uses at most 7m migration operations. For smaller m, 8m to 10m operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses o(n) job migrations can achieve a competitive ratio smaller than α m . We finally trade performance for migrations. We give a family of algorithms that is c-competitive, for any 5/3 ≤ c ≤ 2. For c = 5/3, the strategy uses at most 4m job migrations. For c = 1.75, at most 2.5m migrations are used. Algorithm ALG(α m ):Job arrival phase. Each J t , 1 ≤ t ≤ n, is scheduled as follows.• J t is small: Assign J t to an M j with ℓ s (j, t) ≤ β(j)L * t . • J t is large: Assign J t to a least loaded machine. Lemma 11In the job removal step ALG(1.75) removes at most four jobs from each machine M j ∈ A.Proof. We show that, for any M j ∈ A, it suffices to remove at most four jobs from M j such that the resulting load is upper bounded by 0.75L.First assume that ℓ s (j, n+1) ≤ 0.75L. Then it suffices to remove all jobs that are large at time n+1. Each such job has a processing time greater than 0.5L and was large at the time it was assigned to M j . Consider the last time when such a job was assigned to M j . At that time M j had a load of at most 1.25L and hence could contain no more than two jobs of processing time greater than 0.5L. Thus at time n + 1 machine M j contains at most three of these large jobs.Next assume ℓ s (j, n + 1) > 0.75L. If ℓ s (j, n) ≤ 0.75L n , then J n is assigned to M j because L = L n . Hence it suffices to remove J n and, as shown in the last paragraph, three additional jobs of processing time greater than 0.5L n = 0.5L.We concentrate on the case that ℓ s (j, n + 1) > 0.75L and ℓ s (j, n) > 0.75L n . Let t * be the earliest time such that ℓ s (j, t) > 0.75L t holds for all times t ≥ t * . We partition the jobs that reside on M j at time n + 1 into three sets. Set T 1 (set T 2 ) contains those jobs that were assigned to M j at or before time t * −...

show abstract

“…They demonstrate algorithms to minimize maximum machine load with optimal competitive ratios. Min et al (2011) [15] proposed a fourth variant where the last task of only one machine can be reassigned, and showed that it has the same competitive ratio as the second variant. The fifth variant is like the fourth but the total length of all tasks is known beforehand, and they presented an optimal algorithm with competitive ratio 5 4 .…”

Section: Delayed Reschedulingmentioning

confidence: 99%

Dynamic Reallocation Problems in Scheduling

Lim¹,

Gilbert²,

Zhong³

2015

Preprint

View full text Add to dashboard Cite

In this paper we look at the problem of scheduling tasks on a single-processor system, where each task requires unit time and must be scheduled within a certain time window, and each task can be added to or removed from the system at any time. On each operation, the system is allowed to reschedule any tasks, but the goal is to minimize the number of rescheduled tasks. Our main result is an allocator that maintains a valid schedule for all tasks in the system if their time windows have constant size and reschedules O 1 ε log( 1 ε ) tasks on each insertion as ε → 0, where ε is a certain measure of the schedule flexibility of the system. We also show that it is optimal for any allocator that works on arbitrary instances. We also briefly mention a few variants of the problem, such as if the tasks have time windows of difference sizes, for which we have an allocator that we conjecture reschedules only 1 task on each insertion if the schedule flexibility remains above a certain threshold.Let span( S) = max X∈S X.end− min X∈S X.start for any finite set/sequence of intervals S.Let ( P ? X : Y ) evaluate to X iff P = true and Y otherwise for any boolean P and expressions X, Y . Reallocation problem terminologyTake any instance I = (n, T, W ) that comprises a set of n unit tasks T [1..n] and their windows W [1..n]. For convenience we shall often not mention the tasks but associate an allocated slot directly with the task's window.Call S a valid allocation for I iff it is an allocation of tasks in I such that each taskis unallocated, and no two tasks in I are allocated to overlapping slots. For convenience we shall use "slot" to refer to a unit interval unless otherwise specified.Call I ordered iff W [1..n] is ordered. If so, call a valid allocation S for I ordered iff S[1..n] is ordered. (Unallocated tasks are ignored.) Call S a solution for I iff S is a valid allocation for I that allocates all tasks in I.Call S a partial solution for (I, k) iff S is a valid allocation for I that allocates all tasks inCall S a γ-solution for I iff S is a solution for I ′ where I ′ is I with all task lengths multiplied by γ.Call S a γ-partial solution for (I, k) iff S is a partial solution for (I ′ , k) where I ′ is I with all task lengths multiplied by γ.Call I feasible iff there is a solution for I, and call I γ-underallocated iff there is a γ-solution for I.Call I ε-slack iff I is (1 + ε)-underallocated, and let the underallocation of I be the maximum such ε, which clearly exists.Call (I, S, k) an insert state iff S is a partial solution for (I, k), and call it ordered iff I is ordered, feasible iff I is feasible, and (1 + ε)-underallocated or equivalently ε-slack iff there is a (1 + ε)-partial solution for (I, k).Call an insertion of a new task feasible iff the resulting instance is feasible, or equivalently iff it creates a feasible insert state.Call an algorithm A an allocator iff it maintains a solution on any feasible task insertion.

show abstract

Optimal semi-online algorithms for scheduling problems with reassignment on two identical machines

Cited by 14 publications

References 9 publications

Online scheduling on parallel machines: A survey

Online scheduling on parallel machines: A survey

On the Value of Job Migration in Online Makespan Minimization

Dynamic Reallocation Problems in Scheduling

Contact Info

Product

Resources

About