We investigate a very basic problem in dynamic speed scaling where a sequence of jobs, each specified by an arrival time, a deadline and a processing volume, has to be processed so as to minimize energy consumption. We study multi-processor environments with m parallel variable-speed processors assuming that job migration is allowed, i.e. whenever a job is preempted it may be moved to a different processor. We first study the offline problem and show that optimal schedules can be computed efficiently in polynomial time, given any convex non-decreasing power function. In contrast to a previously known strategy, our algorithm does not resort to linear programming. For the online problem, we extend two algorithms Optimal Available and Average Rate proposed by Yao et al. [15] for the single processor setting. Here we concentrate on power functions P (s) = s α , where s is the processor speed and α > 1 is a constant.

This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

This paper investigates the problem of scheduling jobs on multiple speedscaled processors without migration, i.e., we have constant α > 1 such that running a processor at speed s results in energy consumption s α per time unit. We consider the general case where each job has a monotonously increasing cost function that penalizes delay. This includes the so far considered cases of deadlines and flow time. For any type of delay cost functions, we obtain the following results: Any β-approximation algorithm for a single processor yields a randomized βB α -approximation algorithm for multiple processors without migration, where B α is the αth Bell number, that is, the number of partitions of a set of size α. Analogously, we show that any β-competitive online algorithm for a single processor yields a βB α -competitive online algorithm for multiple processors without migration. Finally, we show that any β-approximation algorithm for multiple processors with migration yields a deterministic βB α -approximation algorithm for multiple processors without migration. These facts improve several approximation ratios and lead to new results. For instance, we obtain the first constant factor online and offline approximation algorithm for multiple processors without migration for arbitrary release times, deadlines, and job sizes.

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Since its introduction in 2004, the MapReduce framework has become one of the standard approaches in massive distributed and parallel computation. In contrast to its intensive use in practise, theoretical footing is still limited and only little work has been done yet to put MapReduce on a par with the major computational models. Following pioneer work that relates the MapReduce framework with PRAM and BSP in their macroscopic structure, we focus on the functionality provided by the framework itself, considered in the parallel external memory model (PEM). In this, we present upper and lower bounds on the parallel I/O-complexity that are matching up to constant factors for the shuffle step. The shuffle step is the single communication phase where all information of one MapReduce invocation gets transferred from map workers to reduce workers. Hence, we move the focus towards the internal communication step in contrast to previous work. The results we obtain further carry over to the BSP * model. On the one hand, this shows how much complexity can be "hidden" for an algorithm expressed in MapReduce compared to PEM. On the other hand, our results bound the worst-case performance loss of the MapReduce approach in terms of I/O-efficiency.