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27
Algorithmic problems in power management
 SIGACT News
, 2005
"... We survey recent research that has appeared in the theoretical computer science literature on algorithmic ..."
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Cited by 72 (4 self)
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We survey recent research that has appeared in the theoretical computer science literature on algorithmic
Getting the Best Response for Your Erg
"... We consider the speed scaling problem of minimizing the average response time of a collection of dynamically released jobs subject to a constraint A on energy used. We propose an algorithmic approach in which an energy optimal schedule is computed for a huge A, and then the energy optimal schedule ..."
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Cited by 66 (11 self)
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We consider the speed scaling problem of minimizing the average response time of a collection of dynamically released jobs subject to a constraint A on energy used. We propose an algorithmic approach in which an energy optimal schedule is computed for a huge A, and then the energy optimal schedule is maintained as A decreases. We show that this approach yields an efficient algorithm for equiwork jobs. We note that the energy optimal schedule has the surprising feature that the job speeds are not monotone functions of the available energy. We then explain why this algorithmic approach is problematic for arbitrary work jobs. Finally, we explain how to use the algorithm for equiwork jobs to obtain an algorithm for arbitrary work jobs that is O(1)approximate with respect to average response time, given an additional factor of (1 + ffl)energy.
Speed Scaling of Tasks with Precedence Constraints
, 2005
"... We consider the problem of speeding scaling to conserve energy in a distributedsetting where there are precedence constraints between tasks, and where the performance measure is the makespan. That is, we consider an energy bounded versionof the classic problem P  prec  Cmax. We show that, without ..."
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Cited by 46 (2 self)
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We consider the problem of speeding scaling to conserve energy in a distributedsetting where there are precedence constraints between tasks, and where the performance measure is the makespan. That is, we consider an energy bounded versionof the classic problem P  prec  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 Q  prec  Cmax to obtain a polylog(m)approximation algorithm.
Scheduling for speed bounded processors
 In Proc. ICALP
, 2008
"... Abstract. We consider online scheduling algorithms in the dynamic speed scaling model, where a processor can scale its speed between 0 and some maximum speed T. The processor uses energy at rate s α when run at speed s, where α> 1 is a constant. Most modern processors use dynamic speed scaling to ..."
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Cited by 39 (12 self)
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Abstract. We consider online scheduling algorithms in the dynamic speed scaling model, where a processor can scale its speed between 0 and some maximum speed T. The processor uses energy at rate s α when run at speed s, where α> 1 is a constant. Most modern processors use dynamic speed scaling to manage their energy usage. This leads to the problem of designing execution strategies that are both energy efficient, and yet have almost optimum performance. We consider two problems in this model and give essentially optimum possible algorithms for them. In the first problem, jobs with arbitrary sizes and deadlines arrive online and the goal is to maximize the throughput, i.e. the total size of jobs completed successfully. We give an algorithm that is 4competitive for throughput and O(1)competitive for the energy used. This improves upon the 14 throughput competitive algorithm of Chan et al. [10]. Our throughput guarantee is optimal as any online algorithm must be at least 4competitive even if the energy concern is ignored [7]. In the second problem, we consider optimizing the tradeoff between the total flow time incurred and the energy consumed by the jobs. We give a 4competitive algorithm to minimize total flow time plus energy for unweighted unit size jobs, and a (2 + o(1))α / ln αcompetitive algorithm to minimize fractional weighted flow time plus energy. Prior to our work, these guarantees were known only when the processor speed was unbounded (T = ∞) [4]. 1
Energy efficient online deadline scheduling
 IN PROC. SODA
, 2007
"... This paper extends the study of online algorithms for energyefficient deadline scheduling to the overloaded setting. Specifically, we consider a processor that can vary its speed between 0 and a maximum speed T to minimize its energy usage (of which the rate is roughly a cubic function of the speed ..."
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Cited by 30 (11 self)
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This paper extends the study of online algorithms for energyefficient deadline scheduling to the overloaded setting. Specifically, we consider a processor that can vary its speed between 0 and a maximum speed T to minimize its energy usage (of which the rate is roughly a cubic function of the speed). As the speed is upper bounded, the system may be overloaded with jobs and no scheduling algorithms can meet the deadlines of all jobs. An optimal schedule is expected to maximize the throughput, and furthermore, its energy usage should be the smallest among all schedules that achieve the maximum throughput. In designing a scheduling algorithm, one has to face the dilemma of selecting more jobs and being conservative in energy usage. Even if we ignore energy usage, the best possible online algorithm is 4competitive on throughput [12]. On the other hand, existing work on energyefficient scheduling focuses on minimizing the energy to complete all jobs on a processor with unbounded speed, giving several O(1)competitive algorithms with respect to the energy usage [2,20]. This paper presents the first online algorithm for the more realistic setting where processor speed is bounded and the system may be overloaded; the algorithm is O(1)competitive on both throughput and energy usage. If the maximum speed of the online scheduler is relaxed slightly to (1+ǫ)T for some ǫ> 0, we can improve the competitive ratio on throughput to arbitrarily close to one, while maintaining O(1)competitive on energy usage.
An efficient algorithm for computing optimal discrete voltage schedules
 SIAM J. on Computing
"... Abstract. We consider the problem of job scheduling on a variable voltage processor with d discrete voltage/speed levels. We give an algorithm which constructs a minimum energy schedule for n jobs in O(dn log n) time. Previous approaches solve this problem by first computing the optimal continuous ..."
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Abstract. We consider the problem of job scheduling on a variable voltage processor with d discrete voltage/speed levels. We give an algorithm which constructs a minimum energy schedule for n jobs in O(dn log n) time. Previous approaches solve this problem by first computing the optimal continuous solution in O(n3) time and then adjusting the speed to discrete levels. In our approach, the optimal discrete solution is characterized and computed directly from the inputs. We also show that O(n log n) time is required, hence the algorithm is optimal for fixed d. 1
Improved bounds for speed scaling in devices obeying the cuberoot rule
, 2012
"... scaling is a power management technology that involves dynamically changing the speed of a processor. This technology gives rise to dualobjective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedul ..."
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Cited by 22 (6 self)
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scaling is a power management technology that involves dynamically changing the speed of a processor. This technology gives rise to dualobjective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedule. In the most investigated speed scaling problem in the literature, the QoS constraint is deadline feasibility, and the objective is to minimize the energy used. The standard assumption is that the processor power is of the form sα where s is the processor speed, and α> 1 is some constant; α ≈ 3 for CMOS based processors. In this paper we introduce and analyze a natural class of speed scaling algorithms that we call qOA. The algorithm qOA sets the speed of the processor to be q times the speed that the optimal offline algorithm would run the jobs in the current state. When α = 3, we show that qOA is 6.7competitive, improving upon the previous best guarantee of 27 achieved by the algorithm Optimal Available (OA). We also give almost matching upper and lower bounds for qOA for general α. Finally, we give the first nontrivial lower bound, namely eα−1 /α, on the competitive ratio of a general deterministic online algorithm for this problem. ACM Classification: F.2.2
Competitive Nonmigratory Scheduling for Flow Time and Energy
 SPAA'08
, 2008
"... Energy usage has been an important concern in recent research on online scheduling. In this paper we extend the study of the tradeoff between flow time and energy from the singleprocessor setting [8, 6] to the multiprocessor setting. Our main result is an analysis of a simple nonmigratory online ..."
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Cited by 19 (7 self)
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Energy usage has been an important concern in recent research on online scheduling. In this paper we extend the study of the tradeoff between flow time and energy from the singleprocessor setting [8, 6] to the multiprocessor setting. Our main result is an analysis of a simple nonmigratory online algorithm called CRR (classified round robin) on m ≥ 2 processors, showing that its flow time plus energy is within O(1) times of the optimal nonmigratory offline algorithm, when the maximum allowable speed is slightly relaxed. This result still holds even if the comparison is made against the optimal migratory offline algorithm (the competitive ratio increases by a factor of 2.5). As a special case, our work also contributes to the traditional online flowtime scheduling. Specifically, for minimizing flow time only, CRR can yield a competitive ratio one or even arbitrarily smaller than one, when using sufficiently faster processors. Prior to our work, similar result is only known for online algorithms that needs migration [21, 23], while the best nonmigratory result can achieve an O(1) competitive ratio [14]. The above result stems from an interesting observation that there always exists some optimal migratory schedule S that can be converted (in an offline sense) to a nonmigratory schedule S ′ with a moderate increase in flow time plus energy. More importantly, this nonmigratory schedule always dispatches jobs in the same way as CRR.
Discrete and Continuous MinEnergy Schedules for Variable Voltage Processors ∗
"... Current dynamic voltage scaling techniques allow the speed of processors to be set dynamically in order to save energy consumption, which is a major concern in microprocessor design. A theoretical model for minenergy job scheduling was first proposed a decade ago, and it was shown that for any conv ..."
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Cited by 17 (0 self)
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Current dynamic voltage scaling techniques allow the speed of processors to be set dynamically in order to save energy consumption, which is a major concern in microprocessor design. A theoretical model for minenergy job scheduling was first proposed a decade ago, and it was shown that for any convex energy function, the minenergy schedule for a set of n jobs has a unique characterization and is computable in O(n 3) time. This algorithm has remained as the most efficient known despite many investigations of this model. In this paper we give a new algorithm with running time O(n 2 log n) for finding the minenergy schedule. In contrast to the previous algorithm which outputs optimal speed levels from high to low iteratively, the new algorithm is based on finding successive approximations to the optimal schedule. At the core of the approximation is an efficient partitioning of the job set into high and low speed subsets by any speed threshold, without computing the exact speed function.
Average rate speed scaling
 In Latin American Theoretical Informatics Symposium, 2008. Nikhil Bansal, HoLeung Chan, Kirk Pruhs, and Dmitriy RogozhnikovKatz. Improved
, 2007
"... Speed scaling is a power management technique that involves dynamically changing the speed of a processor. This gives rise to dualobjective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedule. Yao ..."
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Cited by 15 (6 self)
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Speed scaling is a power management technique that involves dynamically changing the speed of a processor. This gives rise to dualobjective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedule. Yao, Demers, and Shenker [4] considered the problem where the QoS constraint is deadline feasibility and the objective is to minimize the energy used. They proposed an online speed scaling algorithm Average Rate (AVR) that runs each job at a constant speed between its release and its deadline. They showed that the competitive ratio of AVR is at most (2α) α /2 if a processor running at speed s uses power s α. We show the competitive ratio of AVR is at least ((2 − δ)α) α /2, where δ is a function of α that approaches zero as α approaches infinity. This shows that the competitive analysis of AVR by Yao, Demers, and Shenker is essentially tight, at least for large α. We also give an alternative proof that the competitive ratio of AVR is at most (2α) α /2 using a potential function argument. We believe that this analysis is significantly simpler and more elementary than the original analysis of AVR in [4]. 1