Online optimization of max stretch on clusters Erik Saule , Doruk Bozdag, Umit Catalyurek Department of Biomedical Informatics, The Ohio State University { esaule,bozdagd,umit } @bmi.osu.edu Scheduling in Aussois 2010 Supported by the U.S. DOE, the U.S. National Science Foundation and the Ohio Supercomputing Center Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule :: 1 / 25 HPC Lab http://bmi.osu.edu/hpc
Outline Problem Definition 1 The max stretch objective What’s known Approximation Results 2 Counter Examples First-Come First-Serve DASEDF Summing up Resource Augmentation 3 Faster Machines More Machines Experimental Validation 4 Conclusion 5 Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule :: 2 / 25 HPC Lab http://bmi.osu.edu/hpc
The P m | r i , p i , online | max S i problem Cluster scheduling A cluster accepts jobs submitted over time. The jobs are independent and uses a single machine. No preemption is allowed. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Problem Definition::The max stretch objective 3 / 25 HPC Lab http://bmi.osu.edu/hpc
The P m | r i , p i , online | max S i problem Cluster scheduling A cluster accepts jobs submitted over time. The jobs are independent and uses a single machine. No preemption is allowed. The flow time index The flow-time F i = C i − r i is the classical choice in cluster scheduling. But it is unfair for small jobs since a 10-hours job waiting for an hour as the same weight as a 10-minutes job waiting for an hour. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Problem Definition::The max stretch objective 3 / 25 HPC Lab http://bmi.osu.edu/hpc
The P m | r i , p i , online | max S i problem Cluster scheduling A cluster accepts jobs submitted over time. The jobs are independent and uses a single machine. No preemption is allowed. The flow time index The flow-time F i = C i − r i is the classical choice in cluster scheduling. But it is unfair for small jobs since a 10-hours job waiting for an hour as the same weight as a 10-minutes job waiting for an hour. The stretch performance index The stretch S i = C i − r i normalizes flow-time and corrects the unfairness of p i flow-time. But makes the scheduling way more difficult. ∆ denotes the ratio max p i min p i . Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Problem Definition::The max stretch objective 3 / 25 HPC Lab http://bmi.osu.edu/hpc
Previous results Theorem 1 | r i , p i | max S i is NP-Complete in the strong sense [BCM98]. Theorem There is no Ω( n 1 − ǫ ) approximation algorithm for 1 | r i , p i | max S i for constant ǫ unless P = NP [BCM98]. Theorem 1 | r i , p i , online , pmpt | max S i can not be approximated within √ 2 − 1 ∆ [LSV08]. 2 Theorem First-Come First-Serve is a ∆ -approximation of 1 | r i , p i , online , pmpt | max S i [LSV08]. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Problem Definition::What’s known 4 / 25 HPC Lab http://bmi.osu.edu/hpc
Outline of the Talk Problem Definition 1 The max stretch objective What’s known Approximation Results 2 Counter Examples First-Come First-Serve DASEDF Summing up Resource Augmentation 3 Faster Machines More Machines Experimental Validation 4 Conclusion 5 Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results:: 5 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). A large task enters in the system. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). If it is scheduled immediately, a small task is sent. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). It suffers a large delay (and an unbounded stretch). Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). If the large task is scheduled later, a small task is sent accordingly. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). It suffers a large delay (and an unbounded stretch). Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online one machine case Theorem 1 | r i , p i , online | max S i can not be approximated within 1+∆ 2 . Proof. (the adversary technique). It suffers a large delay (and an unbounded stretch). The optimal stretch is always less than 2 and the schedule stretch is always 1 + ∆. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 6 / 25 HPC Lab http://bmi.osu.edu/hpc
The online m machines case Theorem ∆ 1+ P m | r i , p i , online | max S i can not be approximated within m +1 . 2 Proof. ∆ m +1 Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::Counter Examples 7 / 25 HPC Lab http://bmi.osu.edu/hpc
First-Come First-Serve on one machine Theorem ([LSV08]) FCFS is a ∆ -approximation algorithm for 1 | r i , p i , online | max S i . Proof. r l C i i FCFS j i Optimal If optimal has a better stretch for a task (red) then one of the task scheduled between r l and C i (the blue tasks) will complete after C i (green). C ∗ j − r j S ∗ ≥ C i − r i = C i − r i p i p i j = p j = S i p j p j p i p j j ≤ p j S i Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters p i ≤ ∆ Erik Saule Approximation Results::First-Come First-Serve 8 / 25 HPC Lab http://bmi.osu.edu/hpc S ∗
First-Come First-Serve on m machines Theorem FCFS is a ∆ + (1 − 1 m )(∆ + 1) -approximation algorithm for P m | r i , p i , online | maxS i . Proof. ∆ (1 − 1 m )∆ (1 − 1 m ) Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::First-Come First-Serve 9 / 25 HPC Lab http://bmi.osu.edu/hpc
Dual-approximation Algorithm for Stretch using EDF DASEDF ( S ) It targets a maximum stretch S . Task i must complete before the deadline D i = r i + p i S . Solves the deadline problem. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::DASEDF 10 / 25 HPC Lab http://bmi.osu.edu/hpc
Dual-approximation Algorithm for Stretch using EDF DASEDF ( S ) It targets a maximum stretch S . Task i must complete before the deadline D i = r i + p i S . Solves the deadline problem. Earliest Deadline First (EDF) Considers the tasks in order of non-decreasing deadline. Schedules the tasks as soon as possible. If a task starts after its deadline, declares the schedule infeasible. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::DASEDF 10 / 25 HPC Lab http://bmi.osu.edu/hpc
Dual-approximation Algorithm for Stretch using EDF DASEDF ( S ) It targets a maximum stretch S . Task i must complete before the deadline D i = r i + p i S . Solves the deadline problem. Earliest Deadline First (EDF) Considers the tasks in order of non-decreasing deadline. Schedules the tasks as soon as possible. If a task starts after its deadline, declares the schedule infeasible. DASEDF Find the smallest maximum stretch S ∗ such that the deadline problem is feasible using a binary search. Use that schedule until an other tasks is released. Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::DASEDF 10 / 25 HPC Lab http://bmi.osu.edu/hpc
Dual-approximation Algorithm for Stretch using EDF Lemma If a schedule that completes each task i before D i exists, then EDF creates a schedule where each task i completes before D i + (1 − 1 m ) p i . Ohio State University, Biomedical Informatics Online optimization of max stretch on clusters Erik Saule Approximation Results::DASEDF 11 / 25 HPC Lab http://bmi.osu.edu/hpc
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