Kirk Pruhs

Speed is as powerful as clairvoyance

We introduce resource augmentation as a method for analyzing online scheduling problems. In resource augmentation analysis the on-line scheduler is given more resources, say faster processors or more processors, than the adversary. We apply this analysis to two well-known on-line scheduling problems, the classic uniprocessor CPU scheduling problem 1 |<italic>r<subscrpt>i</subscrpt>, pmtn|&Sgr; F<subscrpt>i</subscrpt>,</italic> and the best-effort firm real-time scheduling problem 1|<italic>r<subscrpt>i</subscrpt>, pmtn</italic>| &Sgr; <italic>w<subscrpt>i</subscrpt></italic>( 1- <italic>U<subscrpt>i</subscrpt></italic>). It is known that there are no constant competitive nonclairvoyant on-line algorithms for these problems. We show that there are simple on-line scheduling algorithms for these problems that are constant competitive if the online scheduler is equipped with a slightly faster processor than the adversary. Thus, a moderate increase in processor speed effectively gives the on-line scheduler the power of clairvoyance. Furthermore, the on-line scheduler can be constant competitive on all inputs that are not closely correlated with processor speed. We also show that the performance of an on-line scheduler is best-effort real time scheduling can be significantly improved if the system is designed in such a way that the laxity of every job is proportional to its length.

Speed scaling to manage energy and temperature

Speed scaling is a power management technique that involves dynamically changing the speed of a processor. We study policies for setting the speed of the processor for both of the goals of minimizing the energy used and the maximum temperature attained. The theoretical study of speed scaling policies to manage energy was initiated in a seminal paper by Yao et al. [1995], and we adopt their setting. We assume that the power required to run at speed s is P(s) = sα for some constant α > 1. We assume a collection of tasks, each with a release time, a deadline, and an arbitrary amount of work that must be done between the release time and the deadline. Yao et al. [1995] gave an offline greedy algorithm YDS to compute the minimum energy schedule. They further proposed two online algorithms Average Rate (AVR) and Optimal Available (OA), and showed that AVR is 2α − 1 αα-competitive with respect to energy. We provide a tight αα bound on the competitive ratio of OA with respect to energy. We initiate the study of speed scaling to manage temperature. We assume that the environment has a fixed ambient temperature and that the device cools according to Newtons law of cooling. We observe that the maximum temperature can be approximated within a factor of two by the maximum energy used over any interval of length 1/b, where b is the cooling parameter of the device. We define a speed scaling policy to be cooling-oblivious if it is simultaneously constant-competitive with respect to temperature for all cooling parameters. We then observe that cooling-oblivious algorithms are also constant-competitive with respect to energy, maximum speed and maximum power. We show that YDS is a cooling-oblivious algorithm. In contrast, we show that the online algorithms OA and AVR are not cooling-oblivious. We then propose a new online algorithm that we call BKP. We show that BKP is cooling-oblivious. We further show that BKP is e-competitive with respect to the maximum speed, and that no deterministic online algorithm can have a better competitive ratio. BKP also has a lower competitive ratio for energy than OA for α ≥5. Finally, we show that the optimal temperature schedule can be computed offline in polynomial-time using the Ellipsoid algorithm.

Algorithmic problems in power management

We survey recent research that has appeared in the theoretical computer science literature on algorithmic problems related to power management. We will try to highlight some open problem that we feel are interesting. This survey places more concentration on lines of research of the authors: managing power using the techniques of speed scaling and power-down which are also currently the dominant techniques in practice.

Dynamic speed scaling to manage energy and temperature

We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed s is P(s) = s/sup /spl alpha//. We provide a tight /spl alpha//sup /spl alpha// bound on the competitive ratio of the previously proposed optimal available algorithm. This improves the best known competitive ratio by a factor of 2/sup /spl alpha//. We then introduce an online algorithm, and show that this algorithms competitive ratio is at most 2(/spl alpha//(/spl alpha/ - 1))/sup /spl alpha//e/sup /spl alpha//. This competitive ratio is significantly better and is approximately 2e/sup /spl alpha/+1/ for large /spl alpha/. Our result is essentially tight for large /spl alpha/. In particular, as /spl alpha/ approaches infinity, we show that any algorithm must have competitive ratio e/sup /spl alpha// (up to lower order terms). We then turn to the problem of dynamic speed scaling to minimize the maximum temperature that the device ever reaches, again subject to the constraint that all jobs finish by their deadlines. We assume that the device cools according to Fouriers law. We show how to solve this problem in polynomial time, within any error bound, using the ellipsoid algorithm.

Speed scaling with an arbitrary power function

All of the theoretical speed scaling research to date has assumed that the power function, which expresses the power consumption P as a function of the processor speed s, is of the form P = sα, where α > 1 is some constant. Motivated in part by technological advances, we initiate a study of speed scaling with arbitrary power functions. We consider the problem of minimizing the total flow plus energy. Our main result is a (3+e)-competitive algorithm for this problem, that holds for essentially any power function. We also give a (2+e)-competitive algorithm for the objective of fractional weighted flow plus energy. Even for power functions of the form sα, it was not previously known how to obtain competitiveness independent of α for these problems. We also introduce a model of allowable speeds that generalizes all known models in the literature.

Speed scaling for weighted flow time

In addition to the traditional goal of efficiently managing time and space, many computers now need to efficiently manage power usage. For example, Intels SpeedStep and AMDs PowerNOW technologies allow the Windows XP operating system to dynamically change the speed of the processor to prolong battery life. In this setting, the operating system must not only have a job selection policy to determine which job to run, but also a speed scaling policy to determine the speed at which the job will be run. These policies must be online since the operating system does not in general have knowledge of the future. In current CMOS based processors, the speed satisfies the well known cube-root-rule, that the speed is approximately the cube root of the power [Mud01, BBS+00]. Thus, in this work, we make the standard generalization that the power is equal to speed to some power α ≥ 1, where one should think of α as being approximately 3 [YDS95, BKP04]. Energy is power integrated over time. The operating system is faced with a dual objective optimization problem as it both wants to conserve energy, and optimize some Quality of Service (QoS) measure of the resulting schedule.

Online weighted matching

Abstract We introduce and study online versions of weighted matching problems in metric spaces. We present a simple 2 k − 1 competitive algorithm for online minimum weighted bipartite matching where 2 k is the number of nodes. We show that this competitiveness is optimal. For online maximum matching, we prove that the greedy algorithm achieves an optimal competitive factor of 3. In contrast, we prove that the greedy algorithm performs exponentially poorly for online minimum matching.

On-Line Load Balancing of Temporary Tasks

This paper considers the nonpreemptive on-line load balancing problem where tasks havelimited durationin time. Upon arrival, each task has to be immediately assigned to one of the machines, increasing the load on this machine for the duration of the task by an amount that depends on both the machine and the task. The goal is to minimize the maximum load. Azar, Broder, and Karlin studied theunknown durationcase where the duration of a task is not known upon its arrival (On-line load balancingin“Proc. 33rd IEEE Annual Symposium on Foundations of Computer Science, 1992,” pp. 218Â?225). They focused on the special case in which for each task there is a subset of machines capable of executing it, and the increase in load due to assigning the task to one of these machines depends only on the task and not on the machine. For this case, they showed anO(n2/3)- competitive algorithm, and anÂ?(n)lower bound on the competitive ratio, wherenis the number of the machines. This paper closes the gap by giving anO(n)-competitive algorithm. In addition, trying to overcome theÂ?(n)lower bound for the case of unknown task duration, this paper initiates a study of the load balancing problem for tasks withknown duration(i.e., the duration of a task becomes known upon its arrival). For this case we show anO(lognT)-competitive algorithm, whereTis the ratio of the maximum possible duration of a task to the minimum possible duration of a task. The paper explores an alternative way to overcome theÂ?(n)bound; it considers therelated machinescase with unknown task duration. In the related machines case, a task can be executed by any machine and the increase in load depends on the speed of the machine and the weight of the task. For this case the paper gives a 20-competitive algorithm and shows a lower bound of 3Â?o(1) on the competitive ratio.

An optimal deterministic algorithm for online b -matching

Abstract We consider the natural online version of the well-known unweighted b -matching problem. We present a deterministic algorithm B ALANCE whose competitive ratio is 1−1/(1+1/b) a , where a is the number of online servers per site, and b is the number of adversarial servers per site. We show that the competitive ratio of every deterministic online algorithm is at least 1−1/(1+1/b) a . Hence, B ALANCE is optimally competitive, including low-order terms. In the case a=b , the competitive ratio of B ALANCE approaches 1−1/e≈0.63 as b grows.

Speed Scaling of Tasks with Precedence Constraints

Abstract We consider the problem of speed scaling to conserve energy in a multiprocessor setting where there are precedence constraints between tasks, and where the performance measure is the makespan. That is, we consider an energy bounded version of the classic problem Pm|prec|Cmax . We extend the standard 3-field notation and denote this problem as Sm|prec, energy|Cmax . We show that, without loss of generality, one need only consider constant power schedules. We then show how to reduce this problem to the problem Qm|prec|Cmax  to obtain a poly-log(m)-approximation algorithm.