Due to the ubiquitous batch data processing in cloud computing, the fundamental model of scheduling malleable batch tasks and its extensions have received significant attention recently. In this model, a set of n tasks is to be scheduled on C identical machines and each task is specified by a value, a workload, a deadline and a parallelism bound. Within the parallelism bound, the number of the machines allocated to a task can vary over time and its workload will not change accordingly. In this paper, the two core results of this paper are to quantitatively characterize a sufficient and necessary condition such that a set of malleable batch tasks with deadlines can be feasibly scheduled on C machines, and to propose a polynomial time algorithm to produce such a feasible schedule. The core results provide a conceptual tool and an optimal scheduling algorithm to enable proposing new analysis and design of algorithms or improving existing algorithms for extensive scheduling objectives.
In @cite_15 , consider an extension of our task model, i.e., DAG-structured malleable tasks, and, based on randomized rounding of linear programming , they propose an algorithm with an expected approximation ratio of @math for every @math , where @math . The online version of our task model is considered in @cite_2 @cite_9 ; again based on the dual-fitting technique , two weighted greedy algorithms are proposed respectively for non-committed and committed scheduling and achieve the competitive ratios of @math where @math @cite_3 and @math where @math and @math .
@cite_9 @cite_15 @cite_3 @cite_2
2047204419 2076885483 165387623 2054626117
We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below @math and show several lower bounds on the competitive ratios. First, we give a barely random @math -competitive algorithm that uses only one random bit. We also show a lower bound of @math on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to @math . Second, we give a deterministic @math -competitive algorithm in the model that allows restarts, and we show that in this model the ratio @math is optimal. For randomized algorithms with restarts we show a lower bound of @math . This paper presents a novel algorithm for scheduling big data jobs on large compute clusters. In our model, each job is represented by a DAG consisting of several stages linked by precedence constraints. The resource allocation per stage is malleable, in the sense that the processing time of a stage depends on the resources allocated to it (the dependency can be arbitrary in general).The goal of the scheduler is to maximize the total value of completed jobs, where the value for each job depends on its completion time. We design an algorithm for the problem which guarantees an expected constant approximation factor when the cluster capacity is sufficiently high. To the best of our knowledge, this is the first constant-factor approximation algorithm for the problem. The algorithm is based on formulating the problem as a linear program and then rounding an optimal (fractional) solution into a feasible (integral) schedule using randomized rounding. We propose a general dual-fitting technique for analyzing online scheduling algorithms in the unrelated machines setting where the objective function involves weighted flow-time, and we allow the machines of the on-line algorithm to have (1 + e)-extra speed than the offline optimum (the so-called speed augmentation model). Typically, such algorithms are analyzed using non-trivial potential functions which yield little insight into the proof technique. We propose that one can often analyze such algorithms by looking at the dual (or Lagrangian dual) of the linear (or convex) program for the corresponding scheduling problem, and finding a feasible dual solution as the on-line algorithm proceeds. As representative cases, we get the following results: • For the problem of minimizing weighted flow-time, we give an O (1 e)-competitive greedy algorithm. This is an improvement by a factor of 1 e on the competitive ratio of the greedy algorithm of Chadha-Garg-Kumar-Muralidhara. • For the problem of minimizing weighted lk norm of flow-time, we show that a greedy algorithm gives an O (1 e)-competitive ratio. This marginally improves the result of Im and Moseley. • For the problem of minimizing weighted flow-time plus energy, and when the energy function f(s) is equal to sγ, γ > 1, we show that a natural greedy algorithm is O(γ2)-competitive. Prior to our work, such a result was known for the related machines setting only (Gupta-Krishnaswamy-Pruhs). In this paper, we study online algorithms that schedule malleable jobs, i.e., jobs that can be parallelized on any subset of the available @math m identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job @math j has processing requirement @math pj and is assigned to run on @math kj machines, then @math j's execution time becomes @math pj kj+(kj-1)c, where @math c is a constant parameter to the problem. For @math m=2, we present an online algorithm OCS that has a strong competitive ratio of 3 2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically @math ((4-∈) (3-∈))-competitive when @math m=2, where @math 0<∈≤2 is a parameter to the algorithm, improving upon an existing asymptotically (3 2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly @math 2-competitive when @math m=3, improving upon the previous best upper bound of @math 9 4.