Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Energy-efficient scheduling Guillaume Aupy 1 , Anne Benoit 1 , 2 , Paul Renaud-Goud 1 and Yves Robert 1 , 2 , 3 1 . Ecole Normale Sup´ erieure de Lyon, France 2 . Institut Universitaire de France 3 . University of Tennessee Knoxville, USA Anne.Benoit@ens-lyon.fr http://graal.ens-lyon.fr/~abenoit/ Dagstuhl Seminar 13381, September 2013 Algorithms and Scheduling Techniques for Exascale Systems Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 1/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Energy: a crucial issue Data centers 330 , 000 , 000 , 000 Watts hour in 2007: more than France 533 , 000 , 000 tons of CO 2 : in the top ten countries Exascale computers (10 18 floating operations per second) Need effort for feasibility 1% of power saved � 1 million dollar per year Lambda user 1 billion personal computers 500 , 000 , 000 , 000 , 000 Watts hour per year � crucial for both environmental and economical reasons Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 2/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Energy: a crucial issue Data centers 330 , 000 , 000 , 000 Watts hour in 2007: more than France 533 , 000 , 000 tons of CO 2 : in the top ten countries Exascale computers (10 18 floating operations per second) Need effort for feasibility 1% of power saved � 1 million dollar per year Lambda user 1 billion personal computers 500 , 000 , 000 , 000 , 000 Watts hour per year � crucial for both environmental and economical reasons Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 2/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Power dissipation of a processor P = P leak + P dyn • P leak : constant • P dyn = B × V 2 × f supply constant frequency voltage Standard approximation: P = P leak + f α (2 ≤ α ≤ 3) Energy E = P × time D ynamic V oltage and F requency S caling Real life: discrete speeds Continuous speeds can be emulated Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 3/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Outline Revisiting the greedy algorithm for independent jobs 1 Reclaiming the slack of a schedule 2 Tri-criteria problem: execution time, reliability, energy 3 Checkpointing and energy consumption 4 Conclusion 5 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 4/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Framework Scheduling independent jobs Greedy algorithm: assign next job to least-loaded processor Two variants: OnLine-Greedy : assign jobs on the fly OffLine-Greedy : sort jobs before execution Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 5/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Classical problem n independent jobs { J i } 1 ≤ i ≤ n , a i = size of J i p processors {P q } 1 ≤ q ≤ p allocation function alloc : { J i } → {P q } load of P q = load ( q ) = � { i | alloc ( J i )= P q } a i a 1 a 2 load (1) a 3 a 1 a 10 a 3 a 13 P 1 a 4 a 7 a 6 P 2 a 5 P 3 a 9 a 12 a 8 a 6 a 7 P 4 a 4 a 8 P 5 a 2 a 11 a 5 a 9 a 10 Execution time: a 11 max 1 ≤ q ≤ p load ( q ) a 12 a 13 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 6/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion OnLine-Greedy Theorem OnLine-Greedy is a 2 − 1 p approximation (tight bound) Optimal solution OnLine-Greedy Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 7/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion OffLine-Greedy Theorem OffLine-Greedy is a 4 3 − 1 3 p approximation (tight bound) — OffLine-Greedy Optimal solution Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 8/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Bi-criteria problem Minimizing (dynamic) power consumption: P dyn = f α = f 3 ⇒ use slowest possible speed Bi-criteria problem: Given bound M = 1 on execution time, minimize power consumption while meeting the bound Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 9/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Bi-criteria problem statement n independent jobs { J i } 1 ≤ i ≤ n , a i = size of J i p processors {P q } 1 ≤ q ≤ p allocation function alloc : { J i } → {P q } load of P q = load ( q ) = � { i | alloc ( J i )= P q } a i ( load ( q )) 3 power dissipated by P q � p q =1 ( load ( q )) 3 max 1 ≤ q ≤ p load ( q ) Power Execution time Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 10/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Same Greedy algorithm . . . Strategy: assign next job to least-loaded processor Natural for execution-time smallest increment of maximum load minimize objective value for currently processed jobs Natural for power too smallest increment of total power (convexity) minimize objective value for currently processed jobs Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 11/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion . . . but different optimal solution! Makespan 10, power 2531.441 Makespan 10.1, power 2488.301 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 12/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Greedy and L r norms 1 r p � ( load ( q )) r N r = q =1 Execution time N ∞ = lim r →∞ N r = max 1 ≤ q ≤ p load ( q ) Power ( N 3 ) 3 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 13/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Known results N 2 , OffLine-Greedy Chandra and Wong 1975: upper and lower bounds Leung and Wei 1995: tight approximation factor N 3 , OffLine-Greedy Chandra and Wong 1975: upper and lower bounds N r Alon et al. 1997: PTAS for offline problem Avidor et al. 1998: upper bound 2 − Θ( ln r r ) for OnLine-Greedy Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 14/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Contribution N 3 Tight approximation factor for OnLine-Greedy Tight approximation factor for OffLine-Greedy Greedy for power fully solved! Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 15/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Approximation for OnLine-Greedy � (1 + ( p − 1) β ) 3 + ( p − 1) (1 − β ) 3 � 1 P online p 3 ≤ β 3 + (1 − β ) 3 P opt ( p − 1) 2 � �� � f ( on ) ( β ) p Theorem f ( on ) has a single maximum in β ( on ) ∈ [ 1 p , 1] p p OnLine-Greedy is a f ( on ) ( β ( on ) ) approximation p p This approximation factor is tight Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 16/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Approximation for OffLine-Greedy �� � 3 � � 3 � 1 + ( p − 1) β 1 − β 1 + ( p − 1) p 3 3 3 P offline ≤ β 3 + (1 − β ) 3 P opt ( p − 1) 2 � �� � f ( off ) ( β ) p Theorem f ( off ) has a single maximum in β ( off ) ∈ [ 1 p , 1] p p OffLine-Greedy is a f ( off ) ( β ( off ) ) approximation p p This approximation factor is tight Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 17/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Numerical values of approximation ratios OnLine-Greedy OffLine-Greedy p 2 1 . 866 1 . 086 3 2 . 008 1 . 081 4 2 . 021 1 . 070 5 2 . 001 1 . 061 6 1 . 973 1 . 054 7 1 . 943 1 . 048 8 1 . 915 1 . 043 64 1 . 461 1 . 006 512 1 . 217 1 . 00083 2048 1 . 104 1 . 00010 2 24 1 . 006 1 . 000000025 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 18/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Large values of p Asymptotic approximation factor 4 1 OnLine-Greedy 3 2 1 OffLine-Greedy ↑ optimal Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 19/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Outline Revisiting the greedy algorithm for independent jobs 1 Reclaiming the slack of a schedule 2 Tri-criteria problem: execution time, reliability, energy 3 Checkpointing and energy consumption 4 Conclusion 5 Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 20/ 52
Introduction Greedy Slack-reclaiming Tri-criteria Checkpointing Conclusion Motivation Mapping of tasks is given (ordered list for each processor and dependencies between tasks) If deadline not tight, why not take our time? Slack: unused time slots Goal: efficiently use speed scaling (DVFS) D D Anne.Benoit@ens-lyon.fr Dagstuhl 2013 Energy-efficient scheduling 21/ 52
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