Give a polynomial time algorithm that finds a schedule with as small a completio

Question

Give a polynomial time algorithm that finds a schedule with as small a completion time as possible The wildly popular Spanish-language search engine El Goog needs to do a serious amount of computation every time it recompiles its index. Fortunately, the company has at its disposal a single large supercomputer, together with an essentially unlimited supply of high-end PCs. They've broken the overall computation into n distinct jobs, labeled J1, J2.Jn, which can be performed completely independently of one another. Each job consists of two stages: first it needs to be preprocessed on the supercomputer, and then it needs to be finished on one of the PCs. Let's say that job Ji needs pi seconds of time on the supercomputer, followed by fi seconds of time on a PC. Since there are at least n PCs available on the premises, the finishing of the jobs can be performed fully in parallel-all the jobs can be processed at the same time. However, the supercomputer can only work on a single job at a time, so the system managers need to work out an order in which to feed the jobs to the supercomputer. As soon as the first job in order is done on the supercomputer, it can be handed off to a PC for finishing; at that point in time a second job can be fed to the supercomputer; when the second job is done on the supercomputer, it can proceed to a PC regardless of whether or not the first job is done (since the PCs work in parallel); PCs Lets say that jobJi noods pi seconds of time on tho supercompute, folowed by i seconds and so on. Let's say that a schedule is an ordering of the jobs for the supercomputer, and the completion time of the schedule is the earliest time at which all jobs will have finished processing on the PCs. This is an important quantity to minimize, since it determines how rapidly El Goog can generate a new index. ive a polynomial-time algorithm that finds a schedule with as small a completion time as possible.

Explanation / Answer

Polynomial Time Algorithms for Minimum Energy Scheduling

Conference paper

DOI: 10.1007/978-3-540-75520-3_14

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4698)

Cite this paper as:

Baptiste P., Chrobak M., Dürr C. (2007) Polynomial Time Algorithms for Minimum Energy Scheduling. In: Arge L., Hoffmann M., Welzl E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg

Abstract

The aim of power management policies is to reduce the amount of energy consumed by computer systems while maintaining satisfactory level of performance. One common method for saving energy is to simply suspend the system during the idle times. No energy is consumed in the suspend mode. However, the process of waking up the system itself requires a certain fixed amount of energy, and thus suspending the system is beneficial only if the idle time is long enough to compensate for this additional energy expenditure. In the specific problem studied in the paper, we have a set of jobs with release times and deadlines that need to be executed on a single processor. Preemptions are allowed. The processor requires energy L to be woken up and, when it is on, it uses the energy at a rate of R units per unit of time. It has been an open problem whether a schedule minimizing the overall energy consumption can be computed in polynomial time. We solve this problem in positive, by providing an O(n5)-time algorithm. In addition we provide an O(n4)-time algorithm for computing the minimum energy schedule when all jobs have unit length.

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