1 / 37

A Model for Minimizing Active Processor Time

A Model for Minimizing Active Processor Time . Jessica Chang. Joint work with Hal Gabow and Samir Khuller. Simple Batch Scheduling Problem . n jobs release times and deadlines length batch machine time is slotted in each slot, “active” or “inactive”

audra
Download Presentation

A Model for Minimizing Active Processor Time

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. A Model for Minimizing Active Processor Time Jessica Chang Joint work with Hal Gabow and Samir Khuller

  2. Simple Batch Scheduling Problem • n jobs • release times and deadlines • length • batch machine • time is slotted • in each slot, “active” or “inactive” • “active” at t  can schedule ≤ B jobs • minimize number of “active” slots n=9 B=3

  3. Simple Batch Scheduling Problem • n jobs • release times and deadlines • length • batch machine • time is slotted • in each slot, “active” or “inactive” • “active” at t  can schedule ≤ B jobs • minimize number of “active” slots n=9 B=3 FOUR ACTIVE SLOTS

  4. Of Ovens and Trucks ? B=2 B=2

  5. “Potential benefits include increased data center capacity and reduced capital expenditures as well as reduced power and cooling costs with power-aware job scheduling … However, current batch job scheduling algorithms and configurations are tuned only to optimize performance; energy efficiency has been ignored.” - Intel TR, 2010

  6. Simple Batch Scheduling Problem • n jobs • release times and deadlines • length • batch machine • time is slotted • in each slot, “active” or “inactive” • “active” at t  can schedule ≤ B jobs • minimize number of “active” slots n=9 B=3

  7. View Through Flows J T job 2 1 2 1 2 1 2 1 2 1 timeslot can be scheduled in B=2

  8. View Through Flows J T 2 1 2 1 2 1 2 1 2 1 B=2 Goal: pick the smallest subset of T that can support n units of flow

  9. Batching Algorithms • Wolsey’s greedy algorithm [‘82] • -approximation • Exact alg via Dynamic Programming [Even et. al., ‘08] • Time complexity: • Faster exact algorithm?

  10. Lazy Activation

  11. Lazy Activation • Step I. Scan slots right to left, and decrement deadlines in overloaded slots • favor decrementing deadlines of jobs with earlier B=3

  12. Lazy Activation • Step I. Scan slots right to left, and decrement deadlines in overloaded slots • favor decrementing deadlines of jobs with earlier • Step II. • Order jobs s.t. • Consider deadlines LTR: • Schedule at any outstanding jobs with deadline • Fill the remaining capacity with feasible jobs of later deadline, favoring those with earlier deadline

  13. Example of Step II n=9 B=3

  14. Example of Step II n=9 B=3

  15. Example of Step II n=9 B=3

  16. Example of Step II ACTIVE TIME: 3 n=9 B=3

  17. Optimality of Lazy Activation • Modifying deadlines in Step I does not change optimal active time • After Step I, OPT w.l.o.g. opens only deadlines • At the first deadline, w.l.o.g. OPT satisfies the same set of jobs that Lazy Activation does

  18. Lazy Activation Maximizes Throughput • On infeasible instances, Step I preserves the maximum number of jobs B=3

  19. A more general model • Each job may have a non-unit (integral) processing requirement, but preemption at integral points is allowed • Each job can be done in a subset of time slots (not necessarily one interval)

  20. Main Results Lazy Activation SINGLE WINDOW • An O(n log n) algorithm for unit jobs with one window (any B) • Polynomial DP solution for unit jobs with one window and (time is not slotted, any B) • For B ≥3, NP-hard (even for unit jobs), trivial O(log n)-approx. [Wolsey, ‘82] • Polynomial solution for B=2 and unit length jobs • Extension to non-unit length jobs and to budgeted problem • For (unrestricted) preemption, a fast combinatorial algorithm • For B=2, integrally-preemptive OPT is within 4/3 of the unrestricted-preemptive OPT MULTIPLE WINDOWS

  21. Handling Jobs with Multiple Windows J T B 1 1 B B 1 B 1 B 1

  22. Polynomial Alg. for B=2 J T d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2

  23. Polynomial Alg. for B=2 J T d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 inactive slot d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 • MAX DEGREE SUBGRAPH: • Given a graph G=(V,E) and upper bound on degree constraints, find a max cardinality subgraph satisfying degree constraints. • Reducible to finding max matchings

  24. Need to be a bit careful! J T unscheduled d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 • MAX DEGREE SUBGRAPH: • Given a graph G=(V,E) and upper bound on degree constraints, find a max cardinality subgraph satisfying degree constraints. • Reducible to finding max matchings

  25. Need to be a bit careful! J T

  26. Need to be a bit careful! J T d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 • FIX: • Find a standard max matching M on (J,T) sans loops

  27. Need to be a bit careful! J T d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 • FIX: • Find a standard max matching M on (J,T) sans loops • Augment M to a max DCS including loops • Matched nodes remain matched

  28. Polynomial Alg. for B=2 J T d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 d(v) ≤1 d(v) ≤ 2 |max DCS| = |J|+ # self-loops

  29. Unrestricted Preemption (B=2) • Time slots may be partially active J T

  30. Unrestricted Preemption (B=2) • Time slots may be partially active J T 2-matching: s.t.

  31. Unrestricted Preemption (B=2) • Time slots may be partially active J T 1/2 1/2 1/2 1 1 1/2 1/2 1/2 1 2-matching: s.t.

  32. Unrestricted Preemption (B=2) • Time slots may be partially active J T 1/2 1/2 1/2 1 1 1/2 1/2 1/2 1 Max triangle-free 2-matching: [Babenko, Gusakov, Razenshteyn, ‘10]

  33. Unrestricted Preemption (B=2) • Time slots may be partially active • Can reduce 2 slots to 1.5 active slots B=2

  34. Unrestricted Preemption (B=2) • Time slots may be partially active • Can reduce 2 slots to 1.5 active slots • Theorem: int-pmtv OPT ≤ 4/3 unres-pmtv OPT J T

  35. Related Work • DP improvement [Khuller, Koehler, ‘12] • Batch scheduling to minimize makespan; max tardiness [Ikura, Gimple, ‘86] • improvement [Condotta, Knust, Shakhlevich, ‘10] • does not minimize number of batches • Minimizing busy time [Khandekar, Schieber, Shachnai, Tamir, ‘10] • infinite batch resources • jobs of arbitrary length • 5-approximation • Minimizing gaps [Baptiste, ‘06; Baptiste, Chrobak, Dürr, ‘07]

  36. Future Work • Polynomial time algorithms for jobs of non-unit length and arbitrary B • Online versions • Improved algorithms for minimizing busy time

  37. Thanks. Questions?

More Related