Efficient approximation algorithms for scheduling malleable tasks

A parallelizable (or malleable) task is one which can be run on an arbitrary number of processors, with a task execution time that depends on the number of processors allotted to it. Consider a system of M independent parallelizable tasks which are to be scheduled without preemption on a parallel computer consisting of P identical processors. For each task, the execution time is a known function of the number of processors allotted to it. The goal is to find (1) for each task i, an allotment of processors β, and (2) overall, a non-preemptive schedule assigning the tasks to the processors which minimizes the average response time of the tasks. Equivalently, we can minimize the flow time which is the sum of the completion times of each of the tasks. In this paper we tackle the problem of finding a schedule with minimum average response time in the special case where each task in the system has sublinear speedup. This natural restriction on the task execution time means simply that the efficiency of a task decrease or remains constant as the number of processors allotted to it increases. The scheduling problem with sublinear speedups has been shown to be NP-complete in the strong sense. We therefore focus on finding a polynomial time algorithm whose solution comes within a fixed multiplicative constant of optimal. In particular, we given algorithm which finds a schedule having a response time that is within 2 times that of the optimal schedule and which runs in O(M(M2 + P)) time.

Solving a Multiprocessor Problem by Column Generation and Branch-and-price

We consider the problem of scheduling tasks on multiprocessor architectures in the presence of communication delays. Given a set of dependent tasks, the scheduling problem is to allocate the tasks to processors such that the pre-specified precedence constraints among the tasks are obeyed and certain cost-measures (such as the computation time) are minimized. Several cases of the scheduling problem have

A parallel circuit simulator, iPRIDE, which uses a direct solution method and runs on a shared-memory multiprocessor is described. The simulator is based on a multilevel node tearing approach which produces a nested bordered-block-diagonal (BBD) form of the circuit equation matrix. The parallel solution of the nested BBD matrix is described. Its efficiency is shown to depend on how the circuit is partitioned as well as the sequence in which the concurrent tasks (scheduling) are solved. A partitioning heuristic is proposed, assuming that an arbitrary number of processors is available. Scheduling methods are studied when the number of processors is fixed. A method for determining the optimal level of partitioning, which depends on the number of processors as well as the scheduling methods, is described. The program is implemented on an ALLIANT FX/8 multiprocessor with shared memory. The performance of the program is reported. >

Scientific workloads are often described by directed acyclic task graphs. This is in particular the case for multifrontal factorization of sparse matrices—the focus of this paper—whose task graph is structured as a tree of parallel tasks. Prasanna and Musicus [19, 20] advocated using the concept of malleable tasks to model parallel tasks involved in matrix computations. In this powerful model each task is processed on a time-varying number of processors. Following Prasanna and Musicus, we consider malleable tasks whose speedup is $$p^\alpha $$ , where p is the fractional share of processors on which a task executes, and $$\alpha $$ ( $$0 < \alpha \le 1$$ ) is a task-independent parameter. Firstly, we use actual experiments on multicore platforms to motivate the relevance of this model for our application. Then, we study the optimal time-minimizing allocation proposed by Prasanna and Musicus using optimal control theory. We greatly simplify their proofs by resorting only to pure scheduling arguments. Building on the insight gained thanks to these new proofs, we extend the study to distributed (homogeneous or heterogeneous) multicore platforms. We prove the NP-completeness of the corresponding scheduling problem, and we then propose some approximation algorithms.

Minimum execution time scheduling of unit execution time (UET) task systems with resources has been the subject of several papers over the past few years. Because such scheduling problems are, in general, NP-hard, a variety of heuristic methods for producing schedules have been studied, among them, critical path scheduling. The strongest results to date have been for systems where there is no processor constraint. These results may be utilized for systems with a processor constraint by treating the processors as an additional resource. Unfortunately, in those cases where the number of processors is close to the number of resources, this results in an upper bound which is somewhat misleading. In this paper we investigate the performance of critical path scheduling for UET task systems with resources and a fixed number of processors. An upper bound for the worst case performance of critical path scheduling is given. This bound depends both on the number of processors and on the number of different resources. Moreover, we show that this is the best possible (asymptotic) upper bound.

The problem of optimal scheduling n tasks in a parallel processor system is studied. The tasks are malleable, i.e., a task may be executed by several processors simultaneously and the processing speed of a task is a nonlinear function of the number of processors allocated to it. The total number of processors is m and it is an upper bound on the number of processors that can be used by all the tasks simultaneously. It is assumed that the number of processors is sufficient to process all the tasks simultaneously, i.e. n≤m. The objective is to find a task schedule and a processor allocation such that the overall task completion time, i.e. the makespan, is minimized. The problem is motivated by real-life applications of parallel computer systems in scientific computing of highly parallelizable tasks. An O(n) algorithm is presented to solve this problem when all the processing speed functions are convex. If these functions are all concave and the number of tasks is a constant, the problem can be solved in polynomial time. A relaxed problem, in which the number of processors allocated to each task is not required to be integer, can be solved in O(nmax {m,nlog 2 m}) time. It is proved that the minimum makespan values for the original and relaxed problems coincide. For n=2 or n=3, an optimal solution for the relaxed problem can be converted into an optimal solution for the original problem in a constant time.

The aim of the paper is to introduce general techniques in order to optimize the parallel execution time of sorting on a distributed architectures with processors of various speeds. Such an application requires a partitioning step. For uniformly related processors (processors speeds are related by a constant factor), we develop a constant time technique for mastering processor load and execution time in an heterogeneous environment and also a technique to deal with unknown cost functions. For non uniformly related processors, we use a technique based on dynamic programming. Most of the time, the solutions are in \({\mathcal O}\)(p) (p is the number of processors), independent of the problem size n. Consequently, there is a small overhead regarding the problem we deal with but it is inherently limited by the knowing of time complexity of the portion of code following the partitioning.

A model is considered for representing recurrent precedence-constrained tasks that are to execute on multiprocessor platforms. A recurrent task is specified as a directed acyclic graph (DAG), a period, and a relative deadline. Each vertex of the DAG represents a sequential job, while the edges of the DAG represent precedence constraints between these jobs. All the jobs of the DAG are released simultaneously and need to complete execution within the specified relative deadline of their release. Each task may release jobs in this manner an unbounded number of times, with successive releases occurring at least the specified period apart. Conditional control structures are also allowed. The scheduling problem is to determine whether a set of such recurrent tasks can be scheduled to always meet all deadlines upon a specified number of identical processors. This problem is shown to be computationally intractable, but amenable to efficient approximate solutions. Earliest Deadline First (EDF) and Deadline Monotonic (DM) are shown to be good approximate global scheduling algorithms. Polynomial and pseudo-polynomial time schedulability tests, of differing effectiveness, are presented for determining whether a given task set can be scheduled by EDF or DM to always meet deadlines on a specified number of processors.

Performing direct simulation Monte Carlo (DSMC) calculations on parallel processors may result in a strongly unbalanced distribution of work. This situation affects the performances leading to very low speedups. We present a method which makes use of adaptive domain decomposition to dynamically distribute the computational work load. The major advantage of this method is the possibility to redistribute the fraction of work which depends on the domain decomposition only. The method has been implemented on a transputer system with up to 8 processors (20 MHz Inmos T800), using the FORTRAN version of Parasoft Express Library for parallel compilers. The problems studied were hypersonic flows of a chemically reacting gas in a two-dimensional domain. The adaptive decomposition method worked well in uniformly distributing the computational work over an arbitrary number of processors and reduced the average execution time by over the 38% in certain cases. >

Models for energy-efficient static scheduling of parallelizable tasks with deadlines on frequency-scalable parallel machines comprise moldable vs. malleable tasks and continuous vs. discrete frequency levels, plus preemptive vs. non-preemptive task execution with or without task migration. We investigate the tradeoff between scheduling time and energy efficiency when going from continuous to discrete core allocation and frequency levels on a multicore processor, and from preemptive to non-preemptive task execution. To this end, we present a tool to convert a schedule computed for malleable tasks on machines with continuous frequency scaling [Sanders and Speck, Euro-Par (2012)] into one for moldable tasks on a machine with discrete frequency levels. We compare the energy efficiency of the converted schedule to the energy consumed by a schedule produced by the integrated crown scheduler [Melot et al., ACM TACO (2015)] for moldable tasks and a machine with discrete frequency levels. Our experiments with synthetic and application-based task sets indicate that the converted Sanders Speck schedules, while computed faster, consume more energy on average than crown schedules. Surprisingly, it is not the step from malleable to moldable tasks that is responsible but the step from continuous to discrete frequency levels. One-time frequency scaling during a task’s execution can compensate for most of the energy overhead caused by frequency discretization.

We consider the problem of on-line scheduling of a set of hard real-time tasks in an environment where future task parameters are not known a-priori. First, we derive necessary and sufficient conditions to determine whether a given set of tasks with known parameters is feasible using a set of m processors. Next, we consider two sets of tasks T/sub 0/ and T/sub t/ where T/sub 0/ is the set of tasks that need to be scheduled at time 0 such that any future task set T/sub t/, t>0 can be scheduled with or without disturbing the schedule for T/sub 0/. In this context, we derive necessary and sufficient conditions applicable to the schedule obtained for T/sub 0/ at time 0 such that, any future task set T/sub t/ can be scheduled provided the task set T/sub 0//spl cup/T/sub t/ is feasible using m processors.

Dynamic load balancing techniques prevents computer nodes from overloading unevenly while leaving other idle. It is considered as one of the most challenging topics in parallel computing. Moreover, it is essential for increasing the efficiency of highly parallel systems especially in solving multitask problems with unpredictable load estimates. Particularly, over each processor in the parallel systems and interconnection networks. This paper focuses on developing an efficient algorithm for load balancing on Hyper Hexa Cell (HHC) interconnection network, namely, HHCLB algorithm. Basically, the Dimension Exchange Method (DEM) approach is used in this paper to construct a new load balancing approach on the network of HHC interconnections. Thus, an algorithm was introduced and simulated using java threads, where the performance of the algorithm is evaluated both analytically and experimentally. The evaluation was in terms of various performance metrics, including, execution time, load balancing accuracy, communication cost. By implementing the proposed load balancing algorithm to the HHC network, a high degree of accuracy and minimal execution time was achieved. It is important to highlight that the algorithm recorded small gap between the execution time for small number of processors and large number of processors. For instance, the algorithm achieved 0.14 seconds for balancing the load of 6 processors while 0.59 seconds for balancing the load of 3072 processors. This proves how effective the algorithm is in balancing the load for different network sizes from small to large number of processors, with a slight difference in execution time.

This paper presents a brief formal semantics of constructors for ordered sequences (cons) and for unordered multisets (frons) followed by a detailed operational semantics for both. A multiset is a generalization of a list structure which lacks order a priori; its order is determined by the a posteriori migration of computationally convergent elements to the front. The introductory material includes an example which demonstrates that a multiset of yet-unconverged values and a timing primitive may be used to implement the scheduler for an operating system in an applicative style. The operational semantics, given in PASCAL-like code, is described in two detailed steps: first a uniprocessor implementation of the cons/frons constructors and the first/rest probes, followed by an extension to a multiprocessor implementation. The center of either implementation is the EUREKA structure transformation, which brings convergent elements to the fore while preserving order of shared structures. The multiprocessor version is designed to run on an arbitrary number of processors with only one semaphore but makes heavy use of the sting memory store primitive. Stinging is a conditional store operation which is carried out independently of its dispatching processor so that shared nodes may be somewhat altered without interfering with other processors. An appendix presents the extension of this code to a ‚fair“ implementation of multisets.

PurposeA computational fluid dynamics code for the calculation of laminar hypersonic multi‐species gas flows in chemical non‐equilibrium in axisymmetric or two‐dimensional configuration on shared and distributed memory parallel computers is presented and validated. The code is designed to work efficiently in combination with an automatic domain decompositioning method developed to facilitate efficient parallel computations of various flow problems.Design/methodology/approachThe baseline implicit numerical method developed is the lower‐upper symmetric Gauss‐Seidel scheme, which is combined with a sub‐iteration scheme to achieve time‐accuracy up to third‐order. The spatial discretisation is based on Roe's flux‐difference splitting and various non‐linear flux limiters maintaining total‐variation diminishing properties and up to third‐order spatial accuracy in continuous regions of flow. The domain subdivision procedure is designed to work for single‐ and multi‐block domains without being constrained by the block boundaries, and an arbitrary number of processors used for the computation.FindingsThe code developed reproduces accurately various types of flows, e.g. flow over a flat plate, diffusive mixing and oscillating shock induced combustion around a projectile fired into premixed gas, and demonstrates close to linear scalability within limits of load imbalance.Research limitations/implicationsThe cases considered are axisymmetric or two‐dimensional, and assume laminar flow. An extension to three‐dimensional turbulent flows is left for future work.Originality/valueResults of a parallel computation, utilising a newly developed automatic domain subdivision procedure, for oscillating shock‐induced combustion around a projectile and various other cases are presented. The influence of entropy correction in Roe's flux‐difference splitting algorithm on diffusive mixing of multi‐species flows was examined.