Malleable Scheduling Beyond Identical Machines

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. Jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S for j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than e/(e-1), unless P = NP. On the positive side, we present polynomial-time algorithms with approximation ratios 2e/(e-1) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding and result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of 1+phi for unrelated speeds (phi is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms (i) for minimizing the sum of weighted completion times; and (ii) a variant where we determine the effective speed of a set of allocated machines based on the L_p norm of their speeds

Scheduling Malleable Tasks with Precedence Constraints

In this paper we propose an approximation algorithm for scheduling malleable tasks with precedence constraints. Based on an interesting model for malleable tasks with continuous processor allotments by Prasanna and Musicus \cite{PrMu91,PrMu94,PrMu96}, we define two natural assumptions for malleable tasks: the processing time of any malleable task is non-increasing in the number of processors allotted, and the speedup is concave in the number of processors. We show that under these assumptions the work function of any malleable task is non-decreasing in the number of processors and is convex in the processing time. Furthermore, we propose a two-phase approximation algorithm for the scheduling problem. In the first phase we solve a linear program to obtain a fractional allotment for all tasks. By rounding the fractional solution, each malleable task is assigned a number of processors. In the second phase a variant of the list scheduling algorithm is employed. %In the phases we use two parameters $\mu\in\{1,\dots\lfloor (m+1)/2\rfloor\}$ and $\rho\in [0,1]$ for the allotment and the rounding, respectively, where $m$ is the number of processors. By choosing appropriate values of the parameters, we show (via a nonlinear program) that the approximation ratio of our algorithm is at most $100/63+100(\sqrt{6469}+13)/5481\approx 3.291919$. We also show that our result is asymptotically tight

Scheduling with Communication Delays

International audiencehanbook on ordonnancemen

Performance optimization and energy efficiency of big-data computing workflows

Next-generation e-science is producing colossal amounts of data, now frequently termed as Big Data, on the order of terabyte at present and petabyte or even exabyte in the predictable future. These scientific applications typically feature data-intensive workflows comprised of moldable parallel computing jobs, such as MapReduce, with intricate inter-job dependencies. The granularity of task partitioning in each moldable job of such big data workflows has a significant impact on workflow completion time, energy consumption, and financial cost if executed in clouds, which remains largely unexplored. This dissertation conducts an in-depth investigation into the properties of moldable jobs and provides an experiment-based validation of the performance model where the total workload of a moldable job increases along with the degree of parallelism. Furthermore, this dissertation conducts rigorous research on workflow execution dynamics in resource sharing environments and explores the interactions between workflow mapping and task scheduling on various computing platforms. A workflow optimization architecture is developed to seamlessly integrate three interrelated technical components, i.e., resource allocation, job mapping, and task scheduling. Cloud computing provides a cost-effective computing platform for big data workflows where moldable parallel computing models are widely applied to meet stringent performance requirements. Based on the moldable parallel computing performance model, a big-data workflow mapping model is constructed and a workflow mapping problem is formulated to minimize workflow makespan under a budget constraint in public clouds. This dissertation shows this problem to be strongly NP-complete and designs i) a fully polynomial-time approximation scheme for a special case with a pipeline-structured workflow executed on virtual machines of a single class, and ii) a heuristic for a generalized problem with an arbitrary directed acyclic graph-structured workflow executed on virtual machines of multiple classes. The performance superiority of the proposed solution is illustrated by extensive simulation-based results in Hadoop/YARN in comparison with existing workflow mapping models and algorithms. Considering that large-scale workflows for big data analytics have become a main consumer of energy in data centers, this dissertation also delves into the problem of static workflow mapping to minimize the dynamic energy consumption of a workflow request under a deadline constraint in Hadoop clusters, which is shown to be strongly NP-hard. A fully polynomial-time approximation scheme is designed for a special case with a pipeline-structured workflow on a homogeneous cluster and a heuristic is designed for the generalized problem with an arbitrary directed acyclic graph-structured workflow on a heterogeneous cluster. This problem is further extended to a dynamic version with deadline-constrained MapReduce workflows to minimize dynamic energy consumption in Hadoop clusters. This dissertation proposes a semi-dynamic online scheduling algorithm based on adaptive task partitioning to reduce dynamic energy consumption while meeting performance requirements from a global perspective, and also develops corresponding system modules for algorithm implementation in the Hadoop ecosystem. The performance superiority of the proposed solutions in terms of dynamic energy saving and deadline missing rate is illustrated by extensive simulation results in comparison with existing algorithms, and further validated through real-life workflow implementation and experiments using the Oozie workflow engine in Hadoop/YARN systems

Scheduling malleable task trees

Solving sparse linear systems can lead to processing tree workflows on a platform of processors. In this study, we use the model of malleable tasks motivated in [Prasanna96,Beaumont07] in order to study tree workflow schedules under two contradictory objectives: makespan minimization and memory minization. First, we give a simpler proof of the result of [Prasanna96] which allows to compute a makespan-optimal schedule for tree workflows. Then, we study a more realistic speed-up function and show that the previous schedules are not optimal in this context. Finally, we give complexity results concerning the objective of minimizing both makespan and memory

Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Theory and Engineering of Scheduling Parallel Jobs

Scheduling is very important for an efficient utilization of modern parallel computing systems. In this thesis, four main research areas for scheduling are investigated: the interplay and distribution of decision makers, the efficient schedule computation, efficient scheduling for the memory hierarchy and energy-efficiency. The main result is a provably fast and efficient scheduling algorithm for malleable jobs. Experiments show the importance and possibilities of scheduling considering the memory hierarchy